Differential Equations with Impulse Effects: Multivalued Right-hand Sides with Discontinuities

De Gruyter Studies in Mathematics 40 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesz...

Author: Nikolai A. Perestyuk

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De Gruyter Studies in Mathematics 40 Editors Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Nikolai A. Perestyuk Viktor A. Plotnikov Anatolii M. Samoilenko Natalia V. Skripnik

Differential Equations with Impulse Effects Multivalued Right-hand Sides with Discontinuities

De Gruyter

Mathematics Subject Classification 2010: 34A37, 34A60, 34C29, 34A30, 34A12.

ISBN 978-3-11-021816-9 e-ISBN 978-3-11-021817-6 ISSN 0179-0986 Library of Congress Cataloging-in-Publication Data Differential equations with impulse effects : multivalued right-hand sides with discontinuities / by Nikolai A. Perestyuk … [et al.]. p. cm. ⫺ (De Gruyter studies in mathematics ; 40) Includes bibliographical references and index. ISBN 978-3-11-021816-9 (alk. paper) 1. Impulsive differential equations. I. Perestyuk, N. A. (Nikolai Alekseevich) QA377.D557 2011 5151.353⫺dc22 2011007994

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

To the Memory of Viktor Aleksandrovich Plotnikov

Introduction

Signiﬁcant interest in the investigation of systems with discontinuous trajectories is explained by the development of equipment in which signiﬁcant role is played by impulsive control systems and impulsive computing systems. Impulsive systems are also encountered in numerous problems of natural sciences described by mathematical models with conditions reﬂecting the impulsive action of external forces with pulses whose duration can be neglected. It was discovered that the presence of a pulse action may signiﬁcantly complicate the behavior of trajectories of these systems even in the case of quite simple differential equations. Individual impulsive systems were studied by numerous researchers. Various examples of problems of this sort can be found in the works by N. N. Bogolyubov and N. M. Krylov [72], N. N. Bautin [16], B. S. Kalitin [60–62], A. E. Kobrinskii and A. A. Kobrinskii [68], N. A. Perestyuk and A. M. Samoilenko [142], and D. D. Bainov and A. B. Dishliev [11, 39]. In the works by N. N. Bogolyubov and N. M. Krylov [72], S. T. Zavalishchin and A. N. Sesekin [151, 152], and A. Halanay and D. Wexler [56], systems with pulse action were described by differential equations with generalized functions on the righthand side. In these works, the differential equations describe pulses occurring at ﬁxed moments of time, and the case where the times of pulse action depend on the phase vector is not investigated. Another approach to the investigation of impulsive differential equations is based on the application of the classical methods of the theory of ordinary differential equations. As the ﬁrst works in this direction, we can mention the works by A. D. Myshkis and A. M. Samoilenko [86, 87, 94], in which the general concepts of the theory of systems with pulse action are formulated from a new point of view and their basic speciﬁc features are investigated. Later, numerous works of many mathematicians were devoted to the analysis of the problems of stability of solutions of differential equations with pulse action, development of the theory of periodic and almost periodic solutions of impulsive systems, determination of invariant sets, construction of asymptotic expansions by the Krylov–Bogolyubov–Mitropol’skii method of small parameter, application of the method of comparison, solution of problems of the theory of optimal control, and investigation of impulsive systems with random perturbations [27,56,72,74,86–88,94,137,142,143,151,152]. The monographs [27,74,88,142,143] contain an extensive list of references in this ﬁeld. It is worth noting that the analysis of the dynamics of any real processes with the help of differential equations with univalent right-hand sides corresponds to the ideal model that does not take into account the action of random noises, errors of

viii

Introduction

measurement in specifying the coefﬁcients, and errors of specifying the functions on the right-hand sides of differential equations. If the probabilistic characteristics of the model are known, then the inﬂuence of random factors is taken into account by using stochastic differential equations. The theory of these equations is now rapidly developed and is extensively used in practice [1, 55, 64]. As a natural generalization of differential equations, we can mention differential inclusions capable of description of the dynamics of nondeterministic processes without using the probabilistic characteristics of the model. In numerous cases, this enables one to avoid the necessity of application of various a priori assumptions about these characteristics. The results of investigation of the model performed by the method of differential inclusions enable one to establish direct upper bounds for all results obtained by using probabilistic models, which is sometimes sufﬁcient for applications. The ﬁrst investigations of differential equations with set-valued right-hand sides were carried out by S. Zaremba [158, 159] and A. Marchaud [80–83]. In these works, the authors made an attempt to extend the available results in the theory of differential equations to a more general case. Thus, S. Zaremba introduced the notion of differential equations in paratingents, and A. Marchaud proposed the notion of differential equations in contingents. For the next 25 years, no works were published in this direction (we can mention only the works by A. D. Myshkis [92, 93]). This was explained by the absence of applications. At the beginning of the 1960s, new fundamental results on the existence and properties of solutions of differential equations with set-valued right-hand sides (differential inclusions) were obtained in the cycles of works by T. Wazewski [157] and A. F. Filippov [49]. As one of the most important results obtained in the cited works, we can mention the established relationship between differential inclusions and problems of optimal control, which led to the extensive development of the theory of differential inclusions. The interest in the problems of control after the Second World War was connected with urgent needs of new technologies developed in the aviation, spacecraft engineering, and power-generating industry. This period was characterized by the appearance of new general methods for the solution of optimization problems of control, including the Pontryagin maximum principle, the Bellman method of dynamic programming, etc. The principal results of the theory of differential equations with set-valued righthand sides are presented in the works by A. F. Filippov [23, 48, 49, 51], T. Wazewski [157], V. I. Blagodatskikh [21–23], T. Donchev [41], M. Z. Zgurovskii, V. S. Mel’nik [153], A. I. Panasyuk and V. I. Panasyuk [101, 103], V. A. Plotnikov, A. V. Plotnikov, and A. N. Vityuk [115], A. A. Tolstonogov [145], O. P. Khapaev and M. M. Filatov [47], J.-P. Aubin and H. Frankovska [9], K. Deimling [36], and M. Kisielewicz [67]. The authors studied the problems of existence of solutions of differential inclusions

Introduction

ix

and boundary-value problems, the problems of existence of monotone, bounded, and periodic solutions, stability of solutions, properties of solutions and integral funnels (compactness, connectedness, dependence on initial conditions and conditions on the right-hand side of the inclusion, and the relationship between the sets of solutions of the inclusions xP 2 F .t; x/ and xP 2 co F .t; x/), the problems of determination of the boundary of the set of attainability, conditions for the convexity of the set of solutions, the problems of averaging of differential inclusions, etc. The investigation of properties of the integral funnels of differential inclusions is of high signiﬁcance for the qualitative theory. In this connection, numerous researchers studied the properties of the set of attainability [6, 91, 101, 145] and various approximate methods for its construction, including the method of ellipsoids for linear systems [30, 73, 98], asymptotic methods [41, 115], and numerical methods [91, 96, 149]. In [145], it was shown that the integral funnel is a subset of the solution of the corresponding equation with Hukuhara derivative. The ﬁrst results in the theory of differential equations with Hukuhara derivative were obtained by F. S. de Blasi and F. Iervolino [25] and covered the problems related to the existence of solutions, their uniqueness, and continuous dependence on initial conditions and parameters. The possibility of application of the method of averaging to this class of problems was considered by M. Kisielewicz [66] and A. V. Plotnikov [109]. At present, the methods of the theory of differential equations with set-valued righthand sides and differential equations with Hukuhara derivative are extensively used in the investigation of the dynamics of systems under the conditions of uncertainty, ambiguity, and incompleteness of information (so-called fuzzy systems) [75, 76]. The investigations of differential equations with discontinuous right-hand sides in the case of “sliding modes” carried out by A. F. Filippov [51], M. A. Aizerman [2], L. T. Ashchepkov [5], and V. I. Utkin [147] were also based on the theory of differential inclusions. Note that numerous important engineering problems related, e.g., to the motion of ﬂying vehicles, propagation of seismic oscillations, development of shock and explosive processes, and control over manipulators can also be formulated in terms of discontinuous systems. Discontinuous systems are widely used in economics, chemical technology, theory of automated control, theory of systems with variable structure, and other ﬁelds of science. The theory of impulsive differential equations and theory of differential inclusions were naturally developed in the works devoted to the investigation of differential inclusions with pulse action [7, 17–20, 43, 110–126, 156] dealing with the problems of existence of solutions of Cauchy and boundary-value problems, stability of solutions, existence of periodic solutions, and extendability and continuous dependence of solutions on the initial conditions and the right-hand sides of impulsive differential inclusions. Moreover, the hybrid control systems were also studied by the methods of impulsive differential inclusions.

x

Introduction

Chapters 1 and 5 were written by A. M. Samoilenko and N. A. Perestyuk, Chapter 2 was written by V. A. Plotnikov, Chapters 3, 4, and 7 were written by N. V. Skripnik (née Plotnikova), and Chapter 6 was written by V. A. Plotnikov and N. V. Skripnik.

Notation ¿ ¹xº kxk kM k Br .a/ Sr .a/ mes.A/ co A @A int A A .x; A/ comp.Rn / conv.Rn / h.A; B/ jAj c.A; / C Œa; b M Œa; b

empty set singleton set x 2 Rn Euclidean norm of a vector x 2 Rn spectral norm of a matrix closed ball of radius r centered at a point a 2 Rn sphere of radius r centered at a point a 2 Rn Lebesgue measure of a set A convex hull of a set A boundary of a set A interior of a set A closure of a set A distance from a point x to a set A space of nonempty compact subsets of Rn with Hausdorff metric subspace of comp.Rn / that consists of convex sets Hausdorff distance between sets A and B modulus of a set A support function of a set A space of continuous functions with uniform metric on a segment Œa; b space of bounded functions with uniform metric on a segment Œa; b

Contents

Introduction

vii

Notation

xi

1 Impulsive Differential Equations 1.1 General Characterization of Systems of Impulsive Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 23

2 Impulsive Differential Inclusions 2.1 Differential Inclusions with Fixed Times of Pulse Action . . . . . . . 2.2 Differential Inclusions with Nonﬁxed Times of Pulse Action . . . . . 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 42 48 56

3 Linear Impulsive Differential Inclusions 3.1 Statement of the Problem. Theorem on Existence and Uniqueness . 3.2 Stability of Solutions of Linear Impulsive Differential Inclusions . . 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions . . . 3.4 Linear Differential Equations with Pulse Action at Indeﬁnite Times . 4

5

Linear Systems with Multivalued Trajectories 4.1 Differential Equations with Hukuhara Derivative . . . . . . . . . 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion with the Help of Systems of Differential Equations with Hukuhara Derivative . . . . . . . . . . . . . . . . . . . . . . 4.3 Linear Differential Equations with -Derivative . . . . . . . . . . 4.4 Extension of the Space conv.Rn / for n D 1 . . . . . . . . . . . . 4.5 Approximation of the Integral Funnel of a Linear Differential Inclusion with the Help of Systems of Differential Equations with -Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 . 66 . 72 . 88 . 119

124 . . 124

. . 130 . . 151 . . 155

. . 159

Method of Averaging in Systems with Pulse Action 169 5.1 Oscillating System with One Degree of Freedom . . . . . . . . . . . 169 5.2 Systems with Fixed Times of the Pulse Action . . . . . . . . . . . . . 194 5.3 Systems with Nonﬁxed Times of the Pulse Action . . . . . . . . . . . 204

xiv 6

Contents

Averaging of Differential Inclusions 6.1 Averaging of Inclusions with Pulses at Fixed Times . . . . . . 6.2 Krasnosel’skii–Krein Theorem for Differential Inclusions . . . 6.3 Averaging of Inclusions with Pulses at Nonﬁxed Times . . . . 6.4 Averaging of Impulsive Differential Equations with Hukuhara Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . .

220 . . . . 220 . . . . 229 . . . . 241 . . . . 250

7 Differential Equations with Discontinuous Right-Hand Side 257 7.1 Motions and Quasimotions . . . . . . . . . . . . . . . . . . . . . . . 257 7.2 Impulsive Motions and Quasimotions . . . . . . . . . . . . . . . . . 270 7.3 Euler Quasibroken Lines . . . . . . . . . . . . . . . . . . . . . . . . 273 A Some Elements of Set-Valued Analysis

276

B Differential Inclusions

283

References

295

Index

305

Chapter 1

Impulsive Differential Equations

1.1

General Characterization of Systems of Impulsive Differential Equations

Description of a Mathematical Model. Let M be the phase space of a certain evolution process, i.e., the set of all possible states of this process. By x.t / we denote a point that represents the state of this process at time t. We assume that the process is ﬁnite-dimensional, i.e., the description of its state at a ﬁxed time requires a ﬁnite number, say n, of parameters. Under this assumption, the point x.t / for a ﬁxed t can be interpreted as an n-dimensional vector of the Euclidean space Rn , and M can be regarded as a set from Rn . The topological product M R of the phase space M and the real axis R is called the extended phase space of the evolution process under consideration. Assume that the law of evolution of the process is described by (a) a system of differential equations dx D f .t; x/; dt

x 2 M; t 2 R;

(1.1)

(b) a certain set t given in the extended phase space, and (c) an operator A t given on the set t and mapping it onto the set t0 D A t t of the extended phase space. The process itself runs as follows: a representative point P t D .t; x.t // leaves a point .t0 ; x0 / and moves along the curve ¹t; x.t /º determined by the solution x.t / D x.t; t0 ; x0 / of the system of equations (1.1). The motion along this curve lasts up to a time t D t1 > t0 at which the point .t; x.t // meets the set t (hits a point of the set t /. At time t D t1 , the point P t is “instantaneously” transferred by the operator A t from the location P t1 D .t1 ; x.t1 // to the location P tC D A t1 P t1 D .t1 ; x C .t1 // 2 t01 1 and then moves along the curve ¹t; x.t /º described by the solution x.t / D x.t; t1 ; x C .t1 // of the system of equations (1.1). The motion along the indicated curve lasts up to a time t2 > t1 at which the point P t meets the set t again. At this time, the point P t jumps “instantaneously” from the location P t2 D .t2 ; x.t2 // to the location P tC D A t2 P t2 D .t2 ; x C .t2 // under the action of the operator A t and moves further 2 along the curve ¹t; x.t /º described by the solution x.t / D x.t; t2 ; x C .t2 // of the system of equations (1.1) up to a new contact with the set t , and so on.

2

Chapter 1 Impulsive Differential Equations

In what follows, the collection of relations (a)–(c) characterizing the evolution of a process is called a system of differential equations with pulse action. The trajectory ¹t; x.t /º of a point P t in the extended phase space is called an integral curve, and the function x D x.t / that deﬁnes this curve is called a solution of this system. A system of differential equations with pulse action, i.e., the collection of relations (a)–(c), can be rewritten in a more compact form: dx D f .t; x/; .t; x/ … t ; dt xj.t;x/2 t D A t x x:

(1.2)

Thus, a solution x D '.t / of the system of equations (1.2) is a function that satisﬁes Eq. (1.1) outside the set t and has discontinuities of the ﬁrst kind at the points of t with jumps x D '.t C 0/ '.t 0/ D A t '.t 0/ '.t 0/:

(1.3)

A priori, solutions of Eqs. (1.2) may be of one of the following types: (i) solutions not subjected to instantaneous changes; in this case, the integral curve of the system of equations (1.1) does not intersect the set t or intersects it at ﬁxed points of the operator A t ; (ii) solutions subjected to ﬁnitely many instantaneous changes; in this case, the integral curve intersects the set t at ﬁnitely many points that are not ﬁxed points of the operator A t ; (iii) solutions subjected to countably many instantaneous changes; in this case, the integral curve intersects the set t at countably many points that are not ﬁxed points of the operator A t . Among the solutions whose integral curves pass through countably many points of t , we separate solutions that are absorbed by the set t (they remain in t beginning with a certain time t1 > t0 ) or have an accumulation point. The motion along a trajectory absorbed by the set t consists, beginning with a certain time t1 > t0 , of successive transitions of the representative point P t from the location .t1 ; x1 / to the location .t1 ; A t1 x1 /, then from the latter to .t1 ; A2t1 x1 /, then to .t1 ; A3t1 x1 /, and so on. The motion along a trajectory having an accumulation point in t is a motion that meets and leaves the set t countably many times as time approaches a certain moment t1 > t0 . Therefore, this motion cannot be extended to the time moment t D t1 . The consideration of systems with pulse action meets the same problems as those for ordinary differential equations. However, some speciﬁc problems also arise. The character of these problems depends to a signiﬁcant extent on properties of the operator A t . For example, if A t is not assumed to be one-to-one, then we encounter

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

3

problems related to the study of motions for which the representative point can “instantaneously” split into several points at the times of contact with the set t . If the operator A t is not assumed to be bijective, then we can consider problems related to motions for which independently moving points merge “instantaneously” into a single one at the time of contact with t . Similar speciﬁc problems arise if we assume that the set A t ‡ t is empty for some ‡ t t . This assumption allows one to consider “mortal” systems: a representative point P t that hits ‡ t is transferred by the operator A t to the empty set, i.e., it “dies” according to Vogel [150], and ‡ t serves as the set of “death” of trajectories. For systems of this type, it is natural to pose the problems of the mean lifetime of a moving point, the probability of its “death” in time t0 t T , etc. Unfortunately, the wide variety of systems of differential equations that describe the evolution of a process between two successive times when a representative point hits the set t and the variety of sets t and mappings A t W t ! t0 do not allow one to give a deep classiﬁcation of systems of differential equations with pulse action according to their speciﬁc properties. Depending on the character of pulse action, three essentially different classes (types) of systems of equations under study can be distinguished: (i) systems subjected to pulse action at ﬁxed times; (ii) systems subjected to pulse action at the times when a representative point P t hits given surfaces t D i .x/ of the extended phase space; (iii) discontinuous dynamical systems. Prior to giving a brief description of these classes of systems, we present several examples that illustrate the variety of motions and trajectories in a system with pulse action and their essential dependence on the operator A t and the set t . Example 1. Assume that the phase space of a process is a straight line, the set t is given by the relation t D ¹.t; x/ 2 R2 W x D arctan.tan t /º; the operator A t is deﬁned by the equality A t .t; x/ D .t; x 2 sign x/; and the system of differential Eqs. (1.1) has the form dx D 0: dt In other words, we consider the following system of differential equations with pulse action: dx D 0; .t; x/ … t ; (1.4) dt xj.t;x/2t D x 2 sign x x:

4

Chapter 1 Impulsive Differential Equations

We now study the integral curves and possible motions described by this system. In this system, every motion that starts at t D 0 from a point x0 , jx0 j 2 , corresponds to the state of rest because the integral curve of this motion (the straight line x D x0 ) does not hit the set t for any t 0. The trajectory of each motion of this sort is the point x0 (Figure 1). The motion that starts at t D 0 from a point x0 , 1 < jx0 j < 2 , is subjected to ﬁnitely many pulse actions. The integral curve of this motion hits the set t ﬁnitely many times. For each motion of this sort, one can indicate the time t1 D t1 .x0 / beginning with which the integral curve stays in the set jxj 2 , and, hence, this motion is not subjected to pulse action for t > t1 .x0 /. The trajectory of each motion of this type is a ﬁnite number of points. p For example, the trajectorypof the motion that starts at t D 0 from the point x D 2 consists of two points x D 2 and xpD 2, whereas the trajectory p of the motion p p that starts at t D 0 from the point 8 8 4 x D 2 consists of four points: ¹ 2; 2; 2; 2º. The motion that starts at t D 0 from a point x0 2 .0; 1/ is subjected to countably many pulse actions. The integral curve of this motion intersects the set t countably many times. In this case, one has x.t; x0 / ! 0 as t ! 1. The trajectory of this motion consists of countably many points from the interval .0; 1/. For example, the trajectory of the motion that starts at t D 0 from the point x D 12 is the set of points x D 21n , n D 0; 1; 2; : : : . The integral curves that pass through the points x D 0 and x D ˙1 also intersect the set t countably many times, but the motions corresponding to them are not subjected to pulse action and correspond to the state of rest. This is explained by the fact that the integral curves of these motions intersect the set t at ﬁxed points of the operator A t .

Figure 1. Integral curves (1.4) under different initial conditions.

The motions that start at t D 0 from points of the interval .1; 0/ are subjected to countably many pulse actions on the segment . 3 4 ; /. The sequence of times at which the motion is subjected to pulse action has the limit point t D . Hence, the solution that corresponds to this motion cannot be extended to the interval t .

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

5

The example of these motions illustrates the phenomenon of beating of solutions of impulsive systems against the set t : on a small time interval, the integral curve hits the set t inﬁnitely (countably) many times. In addition to the variety of types of motions and integral curves, this example also shows that, in systems with pulse action, two integral curves can merge into a single one atpa certain time. For example, the integral curves of motions that leave pthe points x D 2 and x D 2 at t D 0 merge into a single curve x D 2 at time t D 2. Example 2. In the theory of optimal control, the following model problems are extensively studied: Find a control u.t / 2 U that minimizes the functional Z I.u/ D

T

x 2 dt

(1.5)

0

on trajectories of the system x .k/ D u;

x.0/ D x 0 ;

x 0 .0/ D x10 ;

::: ;

0 x .k1/ .0/ D xk1 :

(1.6)

For k D 1 and U D Œ1; 1, this example was ﬁrst studied by L. I. Rozenoer in [130] for the illustration of the possibility of appearance of particular controls in the sense of the Pontryagin maximum principle. The optimal solution of the system has the form ´ x 0 t sign x 0 ; 0 t jx 0 j; x .t / D (1.7) 0; jx 0 j t T; ´ sign x 0 ; 0 t jx 0 j; u .t / D 0; jx 0 j t T: If U D ¹1; 1º, then a solution does not exist in the class of absolutely continuous functions for T > jx 0 j, and the so-called sliding mode begins at t > jx 0 j. The equation of motion along the optimal trajectory can be written in the form of an impulsive differential inclusion: xP D u; xjxD0 D 0; x.0/ D x 0 ;

uP D 0;

x ¤ 0;

ujxD0 D sign x 0 ;

(1.8)

u.0/ D sign x 0 :

The solution of the impulsive differential Eq. (1.8) obviously coincides with (1.7); it is subjected to a single pulse action and then remains on the surface x D 0. Problem (1.5), (1.6) corresponds to the motion of an object without regard for its inertia.

6

Chapter 1 Impulsive Differential Equations

For k D 2 and U D Œ1; 1, this example was studied in detail in [53,54,79]. In this case, the surface of control switching has the form x D xP 2 sign x, P and the control has countably many switching points accumulated near the point 0 . For t > 0 , the control satisﬁes the relation u.t / 0 (special mode). Note that if the problem is posed so that x.0/ D x 0 and x.T / D x 1 , then the second point of accumulation of switching points can appear for some 1 2 .0 ; T / (Figure 2) [1].

Figure 2. Integral curve (1.6) in the presence of the sliding mode.

This behavior of systems of optimal control is typical of a certain class of problems that take the inertia of an object into account. The equation of motion along the optimal trajectory can be written in the form of an impulsive differential equation: xR D u; uP D 0;

x.0/ D x 0 ;

x.0/ P D xP 0 ;

u.0/ D sign x 0 ;

x D 0;

xP D 0;

x ¤ xP 2 sign x; P .x; x/ P ¤ 0; ´ P 2u; x D xP 2 sign x; u D u; x D xP D 0:

(1.9)

It is obvious that, for T > 0 , the solution of the impulsive differential Eq. (1.9) has the point of accumulation of switching points, and then the trajectory is located on the switching surface. If the problem is posed so that the ﬁnal point x.T / D x 1 is also given, then there may exist the second point of accumulation of switching points t D 1 < T . Systems Subjected to Pulse Action at Fixed Times. If a real process described by the system of equations (1.1) is subjected to pulse action at ﬁxed times, then the mathematical model of this process is given by the following system of differential equations with pulse action: dx D f .t; x/; dt xj tDi D Ii .x/:

t ¤ i ;

(1.10)

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

7

In this system, the set t is a sequence of hyperplanes t D i of the extended phase space, where ¹i º is a given (ﬁnite or inﬁnite) sequence of times. In this case, it is sufﬁcient to deﬁne the operator A t only for t D i . In other words, it is sufﬁcient to consider only its restriction to the hyperplanes t D i , A ti W M ! M . The most convenient is to consider the sequence of operators Ai W M ! M deﬁned as follows: Ai W x ! Ai x D x C Ii .x/:

(1.11)

Deﬁnition 1. The solution of Eqs. (1.10) is deﬁned as a piecewise-continuous function '.t / with discontinuities of the ﬁrst kind at the points t D i for which the following conditions are satisﬁed: (1) ' 0 .t / D f .t; '.t // for all t ¤ i ; (2) for t D i , the following jump condition is satisﬁed: 'j tDi D '.i C 0/ '.i 0/ D Ii .'.i 0//:

(1.12)

In what follows, the value of the function '.t / at a point t 0 is understood as lim t"t 0 '.t /, i.e., if i is a point of discontinuity of '.t /, then we assume that '.t / is left-continuous and (1.13) '.i / D '.i 0/ D lim '.t /: t "i

Following [94], we present several general theorems on properties of solutions of the systems of Eqs. (1.10). We assume that the function f .t; x/ is deﬁned in the entire space .t; x/ 2 RnC1 (the case where it is deﬁned in a certain domain of this space can be considered by analogy). We also assume that the solutions of the system of equations (1.1) possess the following properties: (i) extendability: every solution x.t / is a continuous function deﬁned on an interval .a; b/, 1 a < b 1, which is individual for every solution; in this case, if a > 1 .b < 1/, then kx.a C 0/k D 1 (kx.b 0/k D 1, respectively); (ii) local character: if a function x.t /, a < t < b, satisﬁes condition (i) and, for any t0 2 .a; b/, there exists " > 0 such that the function x.t / coincides with a certain solution on each of the intervals .t0 "; t0 / and .t0 ; t0 C "/, then x.t / is also a solution; (iii) solvability of the Cauchy problem: for any t0 and x0 , there exists at least one solution x.t /, a < t < b, for which a < t0 < b and x.t0 / D x0 . These conditions are satisﬁed, in particular, for system (1.1) whose right-hand side is continuous or satisﬁes the Carathéodory conditions. Generally speaking, the operators Ai are not assumed to be one-to-one, i.e., for any x 2 Rn , i 2 K, Ai x is a certain (possibly empty) subset of Rn . The deﬁnition of impulsive system and the assumptions concerning solutions of system (1.1) yield the following statement:

8

Chapter 1 Impulsive Differential Equations

Theorem 1 ([142]). If the solutions of the system of equations (1.1) satisfy conditions (i)–(iii), then, for any t0 2 R and x0 2 Rn , there exists at least one solution x.t /, a < t < b, of the impulsive system (1.10) for which a < t0 b and either x.t0 / D x0 .a 1, b 1/ (for t0 < b) or x.t0 0/ D x0 (for t0 D b). In this case, the following assertions are true: (a) if a > 1, then either kx.a C 0/k D 1 or a D i , x.a C 0/ exists (as a ﬁnite limit), and x.a C 0/ … Ai Rn ; (b) if b < 1, then either kx.b 0/k D 1 or b D j , x.b 0/ exists, and Aj x.b 0/ D ¿. A solution x.t / of this type cannot be extended. For any M Rn , we denote by g.t; t0 /M the set of values of x.t / for all solutions of system (1.1) for which x.t0 / 2 M . Then an analogous set for solutions of system (1.10) takes the form G.t; t0 /M , where the mapping G is deﬁned by the following relation for t > t0 : G.t; t0 /M D g.t; i /Ai g.i ; i1 /Ai1 Aj g.j ; t0 /M

(1.14)

.i < t < iC1 ; i D j 1; j; : : : ; j D min¹i W i t0 º/: Furthermore, if there are only ﬁnitely many times i for t > t0 and m D max¹i º, then relation (1.14) with i D m holds for m < t < 1 (in what follows, we do not make a special mention of this fact). For the construction of a solution of system (1.10) in the case where t decreases, i.e., for t < t0 , an analogous formula is valid, in which Ai should be replaced by the naturally introduced mappings A1 i . The introduction of the operator G.t; t0 / of shift along the trajectories of a system with pushes allows one to reformulate, in an obvious manner, the conditions of boundedness, stability, etc., of solutions of this system in terms of properties of this operator. As a remark on Theorem 1, we note that if one additionally assumes that Ai x ¤ ¿, then kx.b 0/k D 1 for b < 1. If one assumes instead that Ai Rn D Rn .i 2 K/, then kx.a C 0/k D 1 for a > 1. The case where Ai x D ¿ corresponds, according to Vogel, to the “death” of a trajectory that hits the point x at time i . Thus, the set ¹xW Aj x D ¿º serves as the “set of death” of trajectories at time j . For example, a solution x D '.t /, '.0/ D 0, of the impulsive equation dx D 1; dt

t ¤ i ;

xj tDi D ln.1 x/;

(1.15)

where i D i , i D 1; 2; : : : , cannot be extended to the interval Œ0; 2, and the time t D 2 is the time of death of this solution. Indeed, for 0 t < 2, this solution is determined by the equality x D '.t / D t (for t D 1 D 1, one has '.1 / D 1, and, therefore, this solution does not have a discontinuity at t D 1 because ln.2 '.1// D 0). For t D 2 D 2, one has '.2/ D 2, and the function ln.2 x/ is not deﬁned at the point x D '.2/. Thus, this solution dies at time t D 2 .

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

9

Theorem 2 ([142]). For the uniqueness of a solution of the Cauchy problem for the impulsive system (1.10) with arbitrary initial data in the case where t increases, it is necessary and sufﬁcient that system (1.1) possess this property for any t0 ¤ i and that, for any t0 D i and x0 2 Ai Rn , each of the sets Ai x contain at most one element. For the uniqueness of a solution of the Cauchy problem for system (1.10) in the case where t decreases, it is necessary and sufﬁcient that system (1.1) possess this property for any t0 and that each of the sets A1 i x contain at most one point. Thus, even if a solution of the Cauchy problem for system (1.1) is unique, solutions of an impulsive system can split or merge in the course of their extension under the action of the operators Ai . For the unbounded extendability of all solutions of system (1.10) forward (back) in time, it is necessary and sufﬁcient that the solutions of system (1.1) possess this property and Ai x ¤ ¿ (respectively, Ai Rn D Rn ) for all i 2 K. One should not think that if a solution of the Cauchy problem for the system of equations (1.1) cannot be extended, say, to the interval Œt0 ; t0 C h, h > 0, then a solution of the corresponding Cauchy problem for the system of equations (1.10) cannot also be extended to this interval. For example, a solution x D '.t /, '.0/ D 0, of the equation dx D 1 C x2 dt cannot be extended to the interval Œ0; =2 (this solution goes to inﬁnity in ﬁnite time: '.t / D tan t ! 1 as t " =2). However, considering a solution x D '.t /, '.0/ D 0, of the impulsive equation dx D 1 C x2; dt

t ¤ i ;

xj t Di D 1;

i D

i ; 4

we conclude that this solution is extendable for all t 0. It is easy to prove that this solution is periodic with period =4 for t 0. For t 2 .0; =4, this function is determined by the equality '.t / D tan t, i.e., i '.t / D tan t for t 2 .i ; iC1 : 4 We assume that the solutions of system (1.1) also possess the following property: (iv) local compactness: for any t0 and x0 , there exists " > 0 such that if jt 0 t0 j < " and kx 0 x0 k ", then any solution x.t / for which x.t 0 / D x 0 exists on the interval Œt0 "; t0 C ", and the set of these solutions for ﬁxed t0 , x0 , and " is compact (in itself) in the metric of C Œt0 "; t0 C ". We also assume that the mappings Ai are upper semicontinuous. Theorem 3 ([142]). Suppose that, under the assumptions introduced above, for given t0 and x.t0 / and a nonempty compact set K Rn all solutions of system (1.1) for

10

Chapter 1 Impulsive Differential Equations

which x.t0 / 2 K exist on a certain interval t0 t T , T < 1. Then, for some " > 0, any solution x.t / satisfying the condition .x.t 0 /; K/ ", jt 0 t0 j ", exists on the entire interval t0 " t T , and the set of these solutions for ﬁxed t0 , K, T , and " is compact in the metric of uniform deviations for discontinuous functions. If t0 D i , then the assertion presented above is valid under the additional condition t 0 t0 . Note that if T D i , then we can take the segment Œt0 "; T C" instead of Œt0 "; T . To extend Theorem 3 to the segment T1 t t0 , T1 < t0 , one should assume that the mappings A1 are upper semicontinuous and take the segment ŒT1 "; t0 C ", i t0 ¤ i , or ŒT1 "; t0 , t0 D i , instead of Œt0 "; T . Corollary 1. Under the additional assumption that a solution of the Cauchy problem for system (1.1) is unique and the mappings Ai are bijective, the solution x.t; t0 ; x0 / of the impulsive system (1.10) depends continuously on t0 ¤ i and x0 on every closed interval of the axis t on which it is deﬁned; for t0 D i , this dependence is left continuous. Corollary 2. Under the conditions of Theorem 3, the set G.t; t0 /K, t0 t T , is compact for every t and depends continuously on t ¤ i ; furthermore, for t D i , this dependence is left continuous and G.i C 0; t0 /K D Ai G.i ; t0 /K: The dependence of G.t; t0 /K on K is upper semicontinuous uniformly in t. If system (1.1) possesses the Knezer property of connectedness of a section of an integral funnel, all sets Ai x are connected, and K is connected, then the set G.t; t0 /K is connected for every t 2 Œt0 ; T . We now present sufﬁcient conditions that must be satisﬁed by the system of Eqs. (1.10) in order that its solutions depend continuously on the initial data and righthand sides. For what follows, we need the lemmas presented below. Lemma 1 ([142]). Suppose that a nonnegative piecewise-continuous function u.t / satisﬁes the following inequality for t t0 : Z t X v.s/u.s/ds C ˇi u.i /; u.t / C C t0

t0 i
where C 0, ˇi 0, v.t / is a positive continuous function, and i are the points of discontinuity of the ﬁrst kind of the function u.t /. Then the following estimate holds for the function u.t /: Z t Y u.t / C .1 C ˇi / exp v.s/ds : t0 i
t0

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

11

Lemma 2 ([142]). Suppose that a nonnegative piecewise-continuous function u.t / satisﬁes the following inequality for t t0 : Z t X u.t / C C u.s/ds C ˇu.i /; t0

t0 i
where C 0, ˇ 0, > 0, and i are the points of discontinuity of the ﬁrst kind of the function u.t /. Then the following estimate holds for the function u.t /: u.t / C.1 C ˇ/i.t0 ;t/ e .tt0 / ; where i.t0 ; t / is the number of points i on the interval Œt0 ; t /. Lemma 3 ([142]). Suppose that a nonnegative piecewise-continuous function u.t / satisﬁes the following inequality for t t0 : Z t X Œˇ C u.s/ds C Œˇ C u.i /; u.t / ˛ C t0

t0 i
where ˛ 0, ˇ 0, > 0, and i are the points of discontinuity of the ﬁrst kind of the function u.t /. Then the following estimate is true: ˇ ˇ .1 C /i.t0 ;t/ e .tt0 / : u.t / ˛ C Using these lemmas, we estimate the variation of solutions of the system of Eqs. (1.10) corresponding to the variation of the initial data and right-hand sides of this system. Assume that the functions f .t; x/ and Ii .x/ are continuous in their variables for x 2 M and t 2 I D Œt0 ; t0 C T and satisfy the Lipschitz condition with respect to x uniformly in t 2 I and i 2 K, i.e., kf .t; x 0 / f .t; x 00 /k Lkx 0 x 00 k;

(1.16)

kIi .x 0 / Ii .x 00 /k Lkx 0 x 00 k: Parallel with Eq. (1.10), we consider the system of equations dy D f .t; y/ C R.t; y/; dt yj tDi D Ii .y/ C Ri .y/;

t ¤ i ;

(1.17)

where the functions R.t; y/ and Ri .y/ are such that solutions of system (1.17) exist. Assume that the inequalities kR.t; y/k

and kRi .y/k

(1.18)

12

Chapter 1 Impulsive Differential Equations

hold for all x 2 M and t 2 I . Consider solutions x.t; x0 / of the system of equations (1.10) and solutions y.t; x0 / of the system of equations (1.17). Assume that these solutions are deﬁned for t0 t t0 C T and kx0 y0 k < ı:

(1.19)

Theorem 4 ([142]). If the functions that determine the systems of Eqs. (1.10) and (1.17) satisfy inequalities (1.16) and (1.18), then the solutions x.t; x0 / and y.t; y0 / of systems (1.10) and (1.17) whose initial values satisfy inequality (1.19) admit the following estimate for all t0 t t0 C T : kx.t; x0 / y.t; y0 /k < ı C .1 C L/i.t0 ;t/ e L.tt0 / : L L

(1.20)

Proof. The solution x.t; x0 /, x.t0 ; x0 / D x0 , of system (1.10) for t 2 .i ; iC1 coincides with one of solutions of the system of ordinary differential equations dx D f .t; x/: dt Since every solution x D '.t / of the latter system can be represented in the form Z '.t / D '. / C

t

f .s; '.s//ds;

for t 2 .i ; i C1 we have Z x.t; x0 / D x.i ; x0 / C Ii .x.i ; x0 // C

t i

f .s; x.s; x0 //ds:

Hence, for t 2 Œt0 ; t0 C T , the solution x.t; x0 / admits the representation x.t; x0 / D x0 C

Z

X

Ii .x.i ; x0 // C

t0 i
t t0

f .s; x.s; x0 //ds:

(1.21)

An analogous representation is valid for the solution y.t; y0 / of the system of equations (1.17), i.e., X ŒIi .y.i ; y0 // C Ri .y.i ; y0 // y.t; y0 / D y0 C Z C

t0 i
Œf .s; y.s; y0 // C R.s; y.s; y0 //ds:

(1.22)

13

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

Therefore, for the norm of the difference of solutions x.t; x0 / y.t; y0 /, we have kx.t; x0 / y.t; y0 /k X ŒkIi .x.i ; x0 // Ii .y.i ; y0 //k C kRi .y.i ; y0 //k kx0 y0 k C Z C

t0 i
Œkf .s; x.s; x0 // f .s; y.s; y0 //k C kR.s; y.s; y0 //kds:

Taking inequalities (1.16) and (1.17) into account, we obtain X kx.t; x0 / y.t; y0 /k kx0 y0 k C Œ C Lkx.i ; x0 / y.i ; y0 /k Z C

t0 i
Œ C Lkx.s; x0 / y.s; y0 /kds:

In other words, the norm of the difference kx.t; x0 /y.t; y0 /k satisﬁes the conditions of Lemma 3 if we set u.t / D kx.t; x0 / y.t; y0 /k;

a D kx0 y0 k;

ˇ D ;

D L:

By virtue of Lemma 3, we have kx.t; x0 / y.t; y0 /k kx0 y0 k C .1 C L/i.t0 ;t/ e L.tt0 / : L L

(1.23)

The required inequality (1.20) follows directly from (1.23) because the initial values x0 and y0 satisfy condition (1.19). Theorem 4 is proved. We note some special cases of Theorem 4. Let D 0. Then x.t; x0 / and y.t; y0 / are solutions of the same system of equations (1.10), but with different initial conditions. For these solutions, estimate (1.20) takes the form kx.t; x0 / y.t; y0 /k ı.1 C L/i.t0 ;t/ e L.tt0 / ; i.e., for all t 2 Œt0 ; t0 C T we have kx.t; x0 / y.t; y0 /k ı.1 C L/p e LT ;

(1.24)

where p is the number of points i on the interval Œt0 ; t0 C T . It follows from inequality (1.24) that, for any " > 0, one can ﬁnd a number ı D ı."/ D ".1 C L/p e LT such that if kx0 y0 k < ı, then kx.t; x0 / y.t; y0 /k < " for all t 2 Œt0 ; t0 C T . This means that if inequalities (1.16) are satisﬁed, then the solutions of system (1.10) depend continuously on the initial conditions. Moreover,

14

Chapter 1 Impulsive Differential Equations

it follows from estimate (1.23) that the solutions of the system of equations (1.10) satisfy the Lipschitz condition with respect to x0 uniformly in t 2 Œt0 ; t0 C T : kx.t; x0 / y.t; y0 /k .1 C L/p e LT kx0 y0 k:

(1.25)

If ı D 0 and ¤ 0, then we have the case of permanently acting perturbations. Therefore, estimate (1.20) takes the form kx.t; x0 / y.t; y0 /k ..1 C L/i.t0 ;t/ e L.tt0 / 1/; L i.e., kx.t; x0 / y.t; y0 /k ..1 C L/p e LT 1/ (1.26) L for all t 2 Œt0 ; t0 C T . It follows from (1.26) that, for any number " > 0, one can ﬁnd D ."/ D "L..1 C L/p e LT 1/1 such that if inequalities (1.18) are satisﬁed, then kx.t; x0 / y.t; y0 /k < " for all t 2 Œt0 ; t0 C T : This statement expresses the property of continuity of solutions of the impulsive system (1.10) in some functional space of right-hand sides. In particular, if the righthand sides of system (1.10) depend continuously on a certain parameter , then the obtained estimates yield the continuity of the solutions with respect to this parameter. Systems with Nonﬁxed Times of Pulse Action. The systems subjected to pulse action at the times when the integral curve crosses given surfaces of the extended phase space are more complicated than the systems subjected to pulse action at ﬁxed times. If the integral curve is subjected to pulse action when it hits the surface deﬁned by the equation ˆ.t; x/ D 0 in the extended phase space, then the indicated impulsive system of differential equations can be written in the form dx D f .t; x/; ˆ.t; x/ ¤ 0; dt xjˆ.t;x/D0 D I.t; x/jˆ.t;x/D0 :

(1.27)

In this case, the sets t and t0 used in the deﬁnition of impulsive system are deﬁned as t D ¹.t; x/W ˆ.t; x/ D 0º; t0 D ¹.t; x/W ˆ.t; A1 t .t; x// D 0º; and the operator A t W t ! t0 acts according to the rule A t W .t; x/ ! .t; x C I.t; x//: With regard for the form of the function ˆ.t; x/, we can distinguish the following cases:

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

15

(I) ˆ.t; x/ does not explicitly depend on x, i.e., the equation of the impulsive surface takes the form ˆ.t / D 0. In this case, the times of pulse action are ﬁxed and correspond to the roots of the equation ˆ.t / D 0. The following qualitatively different cases are possible in this case: (1) ﬁnitely many pulses on a ﬁnite interval, e.g., ˆ1 .t / D sin t ; (2) countably many pulses with one accumulation point, e.g., ´ sin T01t ; t ¤ T0 ; ˆ2 .t / D 0; t D T0 I (3) countably many pulses with several accumulation points, e.g., ´Q k 1 iD1 sin Ti t ; t ¤ Ti ; i D 1; k; ˆ3 .t / D 0; t D Ti ; i D 1; kI (4) countably many pulses with an accumulation point that is the limit point of accumulation points, e.g., ´ sin ˆ21.t/ ; ˆ2 .t / ¤ 0; ˆ4 .t / D 0; ˆ2 .t / D 0: (II) ˆ.t; x/ does not explicitly depend on t , i.e., the equation of the impulsive surface has the form ˆ.x/ D 0. (III) ˆ.t; x/ depends explicitly on t and x. The simplest case here is the case where the equation ˆ.t; x/ D 0 has ﬁnitely many simple roots i .x/, though all situations analogous to those in case (I) are also possible. In what follows, we assume that the equation ˆ.t; x/ D 0 is solvable with respect to t ; furthermore, we assume that it has countably many solutions if the system is studied on the entire real axis or for t t0 , and ﬁnitely many solutions if the system is considered on a ﬁnite time interval. We denote these solutions by t D i .x/ and enumerate them by the set of integers (or its subset) so that i .x/ ! 1 as i ! 1, and i .x/ ! 1 as i ! 1. The restriction of the operator A0t to the hypersurface t D i .x/ deﬁnes an operator acting according to the rule x ! Ai .x/ x x C D x C I.i .x/; x/ x C Ii .x/:

(1.28)

Therefore, system (1.27) can also be written in the form dx D f .t; x/; dt xj tDi .x/ D Ii .x/:

t ¤ i .x/;

(1.29)

16

Chapter 1 Impulsive Differential Equations

In contrast to systems of the form (1.10), despite the fact that the solutions of system (1.29) are piecewise-continuous functions, the points of discontinuity depend on a solution, i.e., every solution has its own points of discontinuity in this case. This substantially complicates the investigation of these systems of equations. One of the difﬁculties consists of beating of solutions against the surface t D i .x/. Exactly due to beating, a solution quite often cannot be extended to a sufﬁcient interval. This phenomenon has already been mentioned in Example 1. Moreover, beating is quite often the main reason for a solution to leave the domain of deﬁnition of an impulsive system or the domain where it is studied. As an illustration, one may consider the following impulsive system: dx D x; dt

t ¤ i .x/;

xj t Di .x/ D ˛x;

where ˛ 2 R and i .x/ D arctan x C i , i D 0; 1; 2; : : : . If ˛ > 0, then the integral curve of any solution x D '.t /, '.0/ D x0 0, of this impulsive system of equations crosses every curve t D i .x/ only once (Figure 3); the integral curve of the solution x D '.t /, '.0/ D x0 > 0, hits the curve t D arctan x countably many times. In other words, the solution x D '.t / beats against impulsive surfaces. In the case under consideration, the integral curve of each solution goes to inﬁnity in time less than 2 .

Figure 3. Case ˛ > 0.

However, if 1 < ˛ < 0 in this example, then the integral curve of the solutions x D '.t / for which '.0/ D x0 0 hits each line t D i .x/ once (Figure 4). The solutions x D '.t / for which '.0/ D x0 < 0 undergo beating: the integral curve of each of these solutions hits the curve t D C arctan x countably many times and approaches the straight line x D 0 as t " . In this case, each solution x D '.t /,

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

17

Figure 4. Case 1 < ˛ < 0.

'.0/ D x0 < 0, cannot be extended to the interval Œ0; a, where a , and the integral curve of each solution of this type tends to the point .; 0/ as t " . In what follows, we consider impulsive systems of differential equations without beating. In other words, we consider equations whose solutions cross each hypersurface t D i .x/ only once. A sufﬁcient condition for the absence of beating is given by the following lemma: Lemma 4 ([142]). Let the functions f .t; x/, Ii .x/, and i .x/ that determine the system of equations (1.29) be continuous for .t; x/ 2 I D, let i .x/ be continuously differentiable with respect to x 2 D, and let @i .x/ N; max kf .t; x/k C; max (1.30) x2D @x .t;x/2I D where D is a certain compact set of the phase space and C and N are positive numbers. Also assume that the inequality @i .x C Ii .x// ; Ii .x/ 0 (1.31) max 01 @x holds for all x 2 D. Then one can ﬁnd a positive number N0 such that, for all N N0 , the integral curve of any solution x.t / of the system of equations (1.29) belonging to the domain D for t0 t t0 C T .Œt0 ; t0 C T I / crosses each hypersurface t D i .x/ from the interval Œt0 ; t0 C T only once.

18

Chapter 1 Impulsive Differential Equations

Proof. To prove the lemma, it sufﬁces to show that, for sufﬁciently small values of the constant N , any solution x.t / of the system of equations (1.29) that leaves the point x0 C Ii .x0 / at t D i .x0 / C 0 and lies in the domain D at least for i .x0 / < t < t i , where t i D max i .x/; x2D

does not hit the surface t D i .x/ for t > i .x0 /. Assume the contrary. Suppose that a solution x.t / leaves the point x0 C Ii .x0 / at t D i .x0 / C 0 and crosses the surface t D i .x/ at a certain point .ti ; x /, ti D i .x /. It is obvious that ti > i .x0 /, and, furthermore, the solution x.t / is continuous on the interval i .x0 / < t < ti . Moreover, Z

x D x0 C Ii .x0 / C

ti

f .s; '.s//ds:

(1.32)

i .x0 /

Consider the difference ti i .x0 /. We have ti i .x0 / D i .x / i .x0 / D i .x / i .x0 C Ii .x0 // C i .x0 C Ii .x0 // i .x0 / Z 1 @i .x0 C Ii .x0 / C h/; h d D @x 0 Z 1 @i C .x0 C Ii .x0 //; Ii .x0 / d; (1.33) @x 0 where

Z hD

ti

f .s; '.s//ds:

i .x0 /

Inequalities (1.30) imply that the ﬁrst term on the right-hand side of equality (1.33) admits the following estimate by virtue of the Cauchy–Schwarz inequality: Z 0

1

@i .x0 C Ii .x0 / C h/; h d NC.ti i .x0 //: @x

(1.34)

@i .x0 C Ii .x0 //; Ii .x0 / d: @x

(1.35)

Therefore, .1

NC /.ti

1

Z i .x0 //

0

To complete the proof the lemma, it sufﬁces to choose N0 from the condition CN0 1 because inequality (1.35) becomes contradictory in this case by virtue of condition (1.31). Lemma 4 is proved.

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

19

The lemma below gives conditions under which there is no beating of solutions of the system of equations (1.29) against the surfaces t D i .x/ in the case where the functions i .x/ are not continuously differentiable. Lemma 5 ([142]). Suppose that the functions f .t; x/ and Ii .x/ in the system of equations (1.29) satisfy the conditions of the previous lemma and the functions i .x/ satisfy the Lipschitz condition ji .x 0 / i .x 00 /j N kx 0 x 00 k;

x 0 ; x 00 2 D:

(1.36)

Also assume that the following inequalities hold for all x 2 D and i : i .x/ i .x C Ii .x//:

(1.37)

Then one can ﬁnd N0 > 0 such that, for all N N0 , the integral curve of any solution x.t / of the system of equations (1.29) that belongs to the domain D for t0 t t0 C T crosses each hypersurface t D i .x/ from the interval Œt0 ; t0 C T only once. Proof. The proof of this lemma is carried out by analogy with the proof of the previous lemma with relation (1.34) replaced by the following:

Z

ji .x / i .x0 C Ii .x0 //j N

i .x / i .x0 /

kf .s; x.s//kds

NC.i .x / i .x0 //: Therefore, ti i .x0 / D i .x / i .x0 / D i .x / i .x0 C Ii .x0 // C i .x0 C Ii .x0 // i .x0 / NC.i .x / i .x0 // C i .x0 C Ii .x0 // i .x0 /; i.e.,

.1 NC /.ti i .x0 // i .x0 C Ii .x0 // i .x0 /:

(1.38)

If N is so small that 1 NC > 0, then the last inequality becomes contradictory by virtue of (1.37), which completes the proof of the lemma. We have proved earlier that the solutions of the system of equations (1.10) subjected to pulse action at ﬁxed times depend continuously on the initial conditions if the functions f .t; x/ and Ii .x/ satisfy the Lipschitz condition. Furthermore, the continuous dependence of the solutions on the initial conditions is uniform in t 2 Œt0 ; t0 C T .

20

Chapter 1 Impulsive Differential Equations

The solutions of the impulsive systems (1.29) do not possess this property, which is well illustrated by the following example: dx D 0; dt

t ¤ i .x/;

xj tDi .x/ D b;

i .x/ D 2i x;

b > 0:

Consider two solutions of this system of equations that leave the points x D 1 and x D 1 C ˛ at t D 0, where ˛ is a number arbitrarily small in modulus. Denote these solutions by '.t / and .t /, respectively, i.e., '.0/ D 1 and .0/ D 1 C ˛. No matter how small ˛ is on the interval Œ0; 1, there is no continuous dependence on x0 because this interval contains the interval Œ1 ˛; 1 if ˛ > 0 (or Œ1; 1 C j˛j if ˛ < 0), on which j'.t / .t /j D b C j˛j, although the difference j'.t / .t /j is arbitrarily small outside this interval whenever ˛ is sufﬁciently small. However, if sufﬁciently small neighborhoods of the times when the integral curve crosses the surfaces t D i .x/ are removed from the interval Œt0 ; t0 C T , then the continuous dependence of solutions on the initial data is uniform with respect to the remaining values of the independent variable. Assume that the functions f .t; x/, Ii .x/, and i .x/ in Eqs. (1.29) are continuous for .t; x/ 2 I D and the following inequalities hold for all t 2 I and x; x 0 ; x 00 2 D: kf .t; x 0 / f .t; x 00 /k Lkx 0 x 00 k;

kf .t; x/k C; kIi .x 0 / Ii .x 00 /k Lkx 0 x 00 k;

ji .x 0 / i .x 00 /j N kx 0 x 00 k: (1.39)

Let x.t; x0 / and x.t; y0 / be two solutions of Eqs. (1.29) that belong to the domain D for all t 2 Œt0 ; t0 C T . Assume that each of these solutions crosses each hypersurface t D i .x/ only once, and denote the times when these solutions cross the surfaces t D i .x/ by ix0 and iy0 , respectively. The following lemma is true: Lemma 6 ([142]). If the conditions presented above are satisﬁed and NC < 1, then p L e LT kx0 y0 k (1.40) kx.t; x0 / x.t; y0 /k 1 C 1 NC y y for all t 2 . i ; iC1 , where i D min.ix0 ; i 0 / and i D max.ix0 ; i 0 /.

Proof. Without loss of generality, we assume that the hyperplanes t D t0 and t D t0 C T do not cross the hypersurfaces t D i .x/. Let us represent the solutions x.t; x0 / and x.t; y0 / indicated in the conditions of the lemma in the integral form: Z t X x.t; x0 / D x0 C f .s; x.s; x0 //ds C Ii .x.ix0 ; x0 //; t0

Z x.t; y0 / D y0 C

t t0

x

t0
f .s; x.s; y0 //ds C

X y

t0
y

Ii .x.i 0 ; y0 //:

21

Section 1.1 General Characterization of Systems of Impulsive Differential Equations

Estimating the difference of these solutions, we get kx.t; x0 / x.t; y0 /k Z t kx.s; x0 / x.s; y0 /kds kx0 y0 k C L C

X x t0
t0

Ii .x.ix0 ; x0 //

X y t0
Ii .x.iy0 ; y0 // :

(1.41)

S If t 2 Œt0 ; t0 C T n i Œ i ; i , where the union is taken over i for which ix0 ; iy0 2 Œt0 ; t0 C T , then X X y0 x0 Ii .x.i ; x0 // Ii .x.i ; y0 // x

y

t0
L

X

t0
.kx. i ; x0 / x. i ; y0 /k C kx.ix0 ; z0 / x.i 0 ; z0 /k/;

t0 < i
where z0 D x0 if i D iy0 , and z0 D y0 if i D ix0 . The second term on the right-hand side of the last inequality admits the estimate Z i kf .s; x.s; z0 //kds kx.ix0 ; z0 / x.iy0 ; z0 /k i

C jix0 iy0 j

NC kx. i ; x0 / x. i ; y0 /k 1 NC

because y

y

jix0 i 0 j N kx.ix0 ; x0 / x.i 0 ; y0 /k N kx. i ; x0 / x. i ; y0 /k C N kx.ix0 ; z0 / x.iy0 ; z0 /k N kx. i ; x0 / x. i ; y0 /k C NC jix0 iy0 j: Relation (1.41) yields

Z

kx.t; x0 / x.t; y0 /k kx0 y0 k C L C

L 1 NC

Hence, by virtue of Lemma 2, we get kx.t; x0 / x.t; y0 /k 1 C

X

t t0

kx.s; x0 / x.s; y0 /kds kx. i ; x0 / x. i ; y0 /k:

t0 < i
L 1 NC

p

e LT kx0 y0 k

S for all t 2 Œt0 ; t0 C T n i Œ i ; i , where p is the number of the points i (or, which is the same, i ) on the interval Œt0 ; t0 C T . Lemma 6 is proved.

22

Chapter 1 Impulsive Differential Equations

Theorem 5 ([142]). Suppose that the functions f .t; x/, Ii .x/, and i .x/ from the system of equations (1.29) satisfy inequalities (1.39), relation (1.37) is true, and NC < 1. If a solution x.t; x0 / of system (1.29) is determined for t 2 Œt0 ; t0 C T , then the continuous dependence of this solution on the initial condition x0 holds in the following sense: for any " > 0, one can ﬁnd ı D ı."/ > 0 such that, for any other solution x.t; y0 / of system (1.29), the inequality kx0 y0 k < ı implies that kx.t; x0 / x.t; y0 /k < "

(1.42)

for all t 2 Œt0 ; t0 C T that satisfy the condition jt ix0 j > ", where ix0 are the times when the integral curve of the solution x.t; x0 / crosses the hypersurfaces t D i .x/. Proof. The conditions of Theorem 5 rule out beating of solutions of system (1.29) against the surfaces t D i .x/ and guarantee that the conditions of Lemma 6 are satisﬁed. According to Lemma 6, estimate (1.40) holds for two solutions x.t; x0 / and x.t; y0 / of Eqs. (1.29). We take an arbitrary " > 0 and choose ı1 > 0 so small that max

kx0 y0 kı1

y

jix0 i 0 j < "

for all i such that ix0 2 Œt0 ; t0 C T . We choose ı as follows: 1 L LT ı D min ı1 ; " 1 C e ; 1 NC where p is the number of points ix0 that belong to the interval Œt0 ; t0 C T . If y0 is such that kx0 y0 k < ı, then, according to (1.40), the solutions x.t; x0 / and x.t; y0 / satisfy inequality (1.42) for all t 2 Œt0 ; t0 C T such that jt i .x0 /j > ". Theorem 5 is proved. We indicate one more general property of solutions of the system of equations (1.29) that is not inherent in the solutions of equations subjected to pulse action at ﬁxed times. In Eqs. (1.10), two different solutions can merge into a single one beginning with a certain time of pulse action t D i because the mapping gi W x ! x C Ii .x/ is not bijective. If a point y is the preimage of at least two points x1 and x2 under the indicated mapping, then the integral curves that hit the points x1 and x2 at time t D i merge into a single integral curve for t > i . For the system of equations (1.29), the described phenomenon is also possible in the case where the mapping gi is bijective. Let i .x/ D i .gi1 .x//, where gi1 is the mapping inverse to gi . Assume that i .x/ < i .x/ for all x from the domain of deﬁnition of system (1.29). Each solution of Eqs. (1.29) that starts at a point .t0 ; x0 / from the domain i .x/ < t < i .x/ and crosses the surface t D i .x/ for t > t0 , say, at a point .i .x/; x/, merges for t > i .x/ with the solution that hits the point gi1 .x/ at t D i .x/.

23

Section 1.2 Linear Systems

Analyzing the location of the surfaces t D i .x/ and t D i .x/, one can establish the presence or absence of beating in the system of equations (1.29). It is known that the beating of solutions against the surface t D i .x/ occurs if the inequality i .x/ < i .x/ has solutions in the domain of deﬁnition of the impulsive system (1.29). Discontinuous Dynamical Systems. A discontinuous dynamical system is deﬁned by relations (a)–(c) in which the differential Eq. (1.1) does not explicitly depend on t , the sets t and t0 are subsets of the phase space, and A t D A for all t 2 R. Thus, a discontinuous dynamical system can be represented in the form dx D f .x/; .t; x/ … ; dt xjx2 D Ax x D I.x/:

(1.43)

To obtain interesting effects in the systems under consideration, it is necessary that a moving point hit the set fairly often. Some examples of these systems were considered in detail in [142].

1.2

Linear Systems

General Properties of Solutions of Linear Systems. We now establish some properties of solutions of homogeneous linear systems of differential equations with pulse action dx D A.t /x; t ¤ i ; xj tDi D Bi x; (1.44) dt and inhomogeneous linear systems. Here, A.t / is an nn matrix continuous on the interval I , Bi are constant matrices, and i 2 I are ﬁxed times enumerated in the increasing order .i < iC1 / by the set of integers or its subset. One of the main results of the theory of linear systems of the form (1.44) is the following one: Theorem 6 ([142]). Suppose that the interval Œt0 ; t0 C h I contains ﬁnitely many points i . Then, for any x0 2 Rn , the solution x.t; x0 /, x.t0 ; x0 / D x0 , of the system of equations (1.44) exists for all t 2 Œt0 ; t0 C h. Furthermore, if the matrices E C Bi are not degenerate for all i such that i 2 Œt0 ; t0 C h, then x.t; x0 / ¤ x.t; y0 / for all t 2 Œt0 ; t0 C h whenever x0 ¤ y0 . Proof. Let t0 < j < j C1 < < j Ck t0 C h. By virtue of the Picard–Cauchy theorem, a solution x D 'j .t / of the system dx D A.t /x dt

(1.45)

24

Chapter 1 Impulsive Differential Equations

with initial condition 'j .t0 / D x0 exists and is unique on the interval Œt0 ; j for any x0 2 Rn . For t 2 Œt0 ; j , we set x.t; x0 / D 'j .t /. By virtue of system (1.44), for t D j we have x.j C 0; x0 / D .E C Bj /x.j ; x0 / D xjC : According to the Picard–Cauchy theorem, there exists a unique solution x D 'j C1 .t /, 'j C1 .j / D xjC , of the system of equations (1.45) on the interval Œj ; j C1 . Therefore, one can extend the solution x.t; x0 / of the original impulsive system (1.44) to the time t D j C1 by setting x.t; x0 / D 'j C1 .t / for j < t j C1 : For t D j C1 C 0, we have x.j C1 C 0; x0 / D .E C Bj C1 /x.j C1 ; x0 / D xjCC1 : Denoting the solution of system (1.45) with initial condition 'j C2 .j C1 / D xjCC1 by 'j C2 .t /, we can extend the solution x.t; x0 / of system (1.44) to the time j C2 by setting x.t; x0 / D 'j C2 .t / for j C1 < t j C2 , etc. Since, according to the conditions of the theorem, the interval Œt0 ; t0 C h contains ﬁnitely many points j , we can thus construct a solution x.t; x0 / of system (1.44) on the entire interval. We have indicated a method for the construction of a solution x.t; x0 / under the assumption that t0 < j . If t0 D j , then the construction of x.t; x0 / is the same as for t0 < j with the difference that, by setting x.t0 C 0/ D xjC D .E C Bj /x0 ; we construct the function 'j C1 .t / as a solution of system (1.45) satisfying the initial condition 'j C1 .j / D xjC . To prove the second assertion of the theorem, we note that if x.i C0; x0 / ¤ x.i C 0; y0 /, then, by virtue of the Picard–Cauchy theorem, one has x.t; x0 / ¤ x.t; y0 / for all i < t iC1 ; i D j; j C k. Since x.i C 0; x0 / x.i C 0; y0 / D .E C Bi /.x.i ; x0 / x.i ; y0 //; the relation x.i ; x0 / ¤ x.i ; y0 / yields x.i C 0; x0 / ¤ x.i C 0; y0 / under the condition of nondegeneracy of the matrix E C Bi . Hence, x.t; x0 / ¤ x.t; y0 / for all t 2 Œt0 ; t0 Ch whenever x0 ¤ y0 and det.E C Bi / ¤ 0 for i D j; j C k. Theorem 6 is proved. If Œt0 h; t0 I and the matrices E C Bi are nondegenerate for all i such that i 2 Œt0 h; t0 , then the solution x.t; x0 / can be uniquely extended to the interval Œt0 h; t0 . If the matrix E C Bi is degenerate at one of the indicated points i D i1 ,

Section 1.2 Linear Systems

25

then the solution x.t; x0 / can be uniquely extended to the left only up to the time i1 . The extension of this solution to the interval Œt0 h; t0 is either impossible, or possible, but in an ambiguous way. Indeed, let i1 be the time closest from the left to t0 such that the matrix E C Bi with its number is degenerate. Let r be the rank of the matrix E C Bi1 . The solution x.t; x0 / can be uniquely extended to the interval .i1 ; t0 . Denote x C D x.i1 C 0; x0 /. The linear operator in Rn determined by the matrix E C Bi1 projects the space Rn to a linear subspace Rr of this space. If x C is contained in the set of images of the indicated operator, then the solution x.t; x0 / can be extended to the left of the point i1 . Moreover, this extension is not unique because the equation .E C Bi1 /x D x C has an inﬁnite set of solutions. If x C does not belong to the set of images of the indicated linear operator, then the equation .E C Bi1 /x D x C does not have solutions. Hence, the solution x.t; x0 / cannot be extended to the left of the point i1 . Thus, the degeneracy of a certain matrix E C Bi with index i such that i 2 Œt0 h; t0 leads to the situation where a part of solutions x.t; x0 /, x.t0 ; x0 / D x0 , cannot be extended to the left of the point i , and each of the remaining solutions splits into a set of solutions at time i . Let i be the time closest from the right to t0 such that the matrix E C Bi is degenerate. Each of the solutions x.t; x0 / of the original system with initial condition x.t0 ; x0 / D x0 can be extended to the entire interval Œt0 ; t0 C h, but many of these solutions merge into a single one at time t D i . The set of points x.i ; x0 /, x0 2 Rn , forms the space Rn , and the set of points .E C Bi /x.i ; x0 / forms a subspace of this space with dimension n r, where r is the rank of the matrix E C Bi . Furthermore, new solutions deﬁned for t > i are “created” at time i , namely, the solutions x.t; y/, x.i C 0; y/ D y, for which the initial point is such that the algebraic system of equations .E C Bi /x D y is unsolvable. In what follows, we restrict ourselves to the investigation of systems (1.44) for which the following conditions are satisﬁed: (1) any compact interval Œa; b I contains ﬁnitely many points i ; (2) for all i such that i 2 I , the matrices E C Bi are not degenerate. Under these assumptions, the following statement is true: Theorem 7 ([142]). The set of all solutions ‡ of the linear homogeneous system of differential equations with pulse action (1.44) on the interval Œa; b forms an ndimensional vector space. Deﬁnition 2. A basis of the linear space of solutions ‡ is called a fundamental system of solutions of system (1.44).

26

Chapter 1 Impulsive Differential Equations

Theorem 7 yields the following important corollaries: (1) the system of equations (1.44) has a fundamental system of n solutions '1 .t /; '2 .t /; : : : ; 'n .t /; (2) any solution of the system of equations (1.44) is a linear combination of solutions of the fundamental system; (3) any n C 1 solutions of Eqs. (1.44) are linearly dependent. Let X.t / denote a matrix whose columns are solutions of system (1.44) that form a fundamental system of solutions. The matrix X.t / is called a fundamental matrix of system (1.44). It is obvious that, for any constant vector c, the function x.t / D X.t /c

(1.46)

is a solution of system (1.44). If c passes through the entire space Rn , then the family of functions (1.46) forms a space. It follows from the deﬁnition of the matrix X.t / that it satisﬁes the following matrix equation with pulse action: dX D A.t /X; dt

t ¤ i ;

Xj t Di D Bi X:

(1.47)

It is also obvious that any nondegenerate solution of the matrix system (1.47) is a fundamental matrix of the system of equations (1.44). All nondegenerate solutions of system (1.47) are given by the formula X.t / D X0 .t /C , where X0 .t / is a nondegenerate solution of system (1.47) and C is an arbitrary nondegenerate matrix. The nondegenerate solution X.t / of system (1.47) that satisﬁes the condition X.t0 / D E is called the matrizant of system (1.44) and is denoted by X.t; t0 /. Let U.t; s/ be a solution of the Cauchy matrix problem dU D A.t /U; dt

U.t; s/ D E;

(1.48)

i.e., the matrizant of system (1.45). Then any solution X.t / of the matrix system (1.47) admits the representation X.t / D U.t; j Ck /.E C Bj Ck /U.j Ck ; j Ck1 / .E C Bj /U.j ; t0 /X.t0 /; j 1 < t0 j < j Ck < t j CkC1 :

(1.49)

In particular, for the matrizant X.t; t0 /, we have X.t; t0 / D U.t; j Ck /.E C Bj Ck /U.j Ck ; j Ck1 / .E C Bj /U.j ; t0 /; j 1 < t0 j < j Ck < t j CkC1 ;

27

Section 1.2 Linear Systems

or X.t; t0 / D U.t; j Ck /.E C Bj Ck /

1 Y

U.j C ; j C1 /.E C Bj C1 /U.j ; t0 /:

(1.50)

Dk

By virtue of the Liouville–Ostrogradskii formula, relation (1.49) yields 1 Y

det X.t / D det U.t; j Ck / det.E C Bj Ck /

det U.j C ; j C1 /

Dk

det.E C Bj C1 / det U.j ; t0 / det X.t0 / De

Rt

j Ck

1 Y Dk R j

e

t0

Sp A.s/ds

e

R j C

j C1

Sp A.s/ds

det.E C Bj Ck / Sp A.s/ds

det.E C Bj C1 /

det X.t0 /;

i.e., det X.t / D det X.t0 /e

Rt t0

Sp A.s/ds

kC1 Y

det.E C Bj C1 /;

(1.51)

D1

j 1 < t0 j < j Ck < t j CkC1 : The condition of nondegeneracy of the matrices E C Bi and relation (1.51) imply that the matrix X.t / is nondegenerate if the matrix X.t0 / is nondegenerate. If the matrix X.t / is nondegenerate, then the inverse matrix X 1 .t / is determined by the relation X 1 .t / D X 1 .t0 /U 1 .j ; t0 /.E C Bj /1 U 1 .j Ck ; j Ck1 /.E C Bj Ck /1 U 1 .t; j Ck / D X 1 .t0 /U 1 .j ; t0 /

k Y

.E C Bj C1 /1 U 1 .j C ; j C1 /

D1 1

.E C Bj Ck /

U

1

.t; j Ck /;

j 1 < t0 j < j Ck < t j CkC1 ;

28

Chapter 1 Impulsive Differential Equations

and X.t /X 1 .s/ D U.t; j Ck /

mC1 Y

.E CBj C /U.j C ; j C1 /.E CBj Cm /U.j Cm ; s/;

Dk

j Cm1 < s j Cm < j Ck < t j CkC1 : In particular, for the matrizant X.t; t0 /, we have X

1

.t; t0 / D U

1

.j ; t0 /

k Y

.E C Bj C1 /1 U 1 .j C ; j C1 /

D1

.E C Bj Ck /1 U 1 .t; j Ck /; X.t; t0 /X 1 .s; t0 / D U.t; j Ck /

mC1 Y

.E C Bj C /U.j C ; j C1 /

Dk

.E C Bj Cm /U.j Cm ; s/ D X.t; s/;

(1.52)

j 1 < t0 j j Cm1 < s j Cm < j Ck < t j CkC1 : If i < s t i C1 , then X.t; t0 /X 1 .s; t0 / D U.t; s/. Also note that any solution of system (1.44) x.t; x0 /, x.t0 ; x0 / D x0 , can be written with the help of the matrizant X.t; t0 / in the form x.t; x0 / D X.t; t0 /x0 :

(1.53)

The system of equations dx D A.t /x C f .t /; dt

t ¤ i ;

xj t Di D Bi x C ai ;

(1.54)

where the matrices A.t / and Bi and times i are the same as in system (1.44), f .t / is a function continuous (piecewise continuous) on the interval I , and ai are constant vectors, is called a linear inhomogeneous system of differential equations with pulse action. The relationship between solutions of the inhomogeneous system (1.54) and the corresponding homogeneous system (1.44) is described by the following theorem: Theorem 8 ([142]). If x D '.t / is a solution of system (1.44) and x D .t / is a solution of system (1.54), then the function x D '.t / C .t / is a solution of system (1.54). Conversely, if x D '1 .t / and x D '2 .t / are solutions of the inhomogeneous system (1.54), then the function x D '1 .t / '2 .t / is a solution of the system of equations (1.44). In what follows, we use a linear change of dependent variables in systems (1.44) and (1.54).

29

Section 1.2 Linear Systems

Theorem 9 ([142]). Let S.t / be a nondegenerate matrix continuously differentiable for t 2 Œa; b n ¹i º. Then the linear change x D S.t /y

(1.55)

reduces system (1.54) to the form dy dS D S 1 .t / A.t /S.t / y C S 1 .t /f .t /; dt dt

t ¤ i ;

yj tDi D S 1 .i C 0/.S C Bi S /yj t Di C S 1 .i C 0/ai :

(1.56)

In particular, if S.t / is a fundamental matrix X.t / of the system of equations (1.44), then the change of variables (1.55) is called a “variation of constants” because it is realized by the replacement of the constant vector c in (1.46) by a variable vector y.t /. Then system (1.54) reduces to the system dy D X 1 .t /f .t /; dt

t ¤ i ;

yj tDi D X 1 .i C 0/ai ;

(1.57)

which can easily be integrated. With regard for the relation X.i C 0/ D .E C Bi /X.i /, the condition of jump in Eqs. (1.57) can be written in the form y D X 1 .i /.E C Bi /1 ai : For t t0 , Eqs. (1.57) yield Z t X y.t / D c C X 1 .s/f .s/ds C X 1 .i /.E C Bi /1 ai ; t0

(1.58)

(1.59)

t0 i
where c D y.t0 / is a constant vector. Let X.t / be a fundamental matrix of system (1.44) in which the matrices E C Bi are nondegenerate. Then any solution of the system of equations (1.54) for t t0 is given by the relation Z t X 1 1 1 x.t / D X.t / c C X .s/f .s/ds C X .i /.E C Bi / ai : (1.60) t0

t0 i
In particular, if X.t / D X.t; t0 / is the matrizant of system (1.44), then any solution x.t; x0 /; x.t0 ; x0 / D x0 , of system (1.54) for t t0 can be represented in the form Z t X X.t; s/f .s/ds C X.t; i /ai : (1.61) x.t; x0 / D X.t; t0 /x0 C t0

t0 i
The last two relations show that if the solutions of the corresponding homogeneous system are known, then the solutions of system (1.54) are determined in quadratures.

30

Chapter 1 Impulsive Differential Equations

Stability of Solutions of Linear Impulsive Systems. We now study the problem of stability of solutions of the homogeneous system with pulse action dx D A.t /x; dt

t ¤ i ;

xj tDi D Bi x;

(1.62)

where the matrix A.t / is continuous (piecewise continuous) and bounded for t t0 , the matrices Bi , i D 1; 2; : : : , are uniformly bounded with respect to i 2 N , and the times i are enumerated by the set of natural numbers t0 1 < 2 < < i < iC1 < and are such that i ! 1 as i ! 1. Let X.t; t0 / be the matrizant of system (1.62). Since the difference of any two solutions x.t; x0 / x.t; y0 / of system (1.62) can be represented in the form x.t; x0 / x.t; y0 / D X.t; t0 /.x0 y0 /;

(1.63)

we conclude that the stability or instability of solutions of system (1.62) depends on the behavior of the matrizant X.t; t0 / as t ! 1. If the matrizant X.t; t0 / is bounded for t t0 , i.e., there exists a constant M 0 such that the estimate kX.t; t0 /k M < 1 holds for all t t0 , then, for all t t0 and any solution x.t; x0 / of system (1.62), the following inequality is true: kx.t; x0 / x.t; y0 /k kX.t; t0 /k kx0 y0 k M kx0 y0 k: By virtue of this inequality, we have kx.t; x0 / x.t; y0 /k < " for t t0 whenever kx0 y0 k < ı D

" : M

This means that the solution x.t; x0 / is stable. Assume that lim kX.t; t0 /k D 0: t!1

In this case, the matrix X.t; t0 / is bounded for t t0 , and, hence, the solution x.t; x0 / is stable. Furthermore, it follows from relation (1.63) that lim kx.t; x0 / x.t; y0 /k D 0

t!1

for any solution x.t; y0 /, i.e., the solution x.t; x0 / is asymptotically stable. Assume that the matrix X.t; t0 / is unbounded for t t0 . In other words, there exists an inﬁnite increasing sequence of numbers t0 t1 < t2 < such that lim kX.tk ; t0 /k D 1:

k!1

31

Section 1.2 Linear Systems

In this case, the elements of the matrix X.t; t0 / contain at least one element q˛ˇ .t / for which lim jq˛ˇ .tk /j D 1: k!1

x.t; x0 /

Consider the solution x0 with the coordinates

of system (1.62) that passes at t D t0 through the point

D x10 ; x10

x20 D x20 ;

xˇC1 0 D xˇ C1 0 ;

:::;

:::;

xˇ 0 ¤ xˇ 0 ;

xn0 D xn0 :

For this solution, we have x˛ .t; x0 / x˛ .t; x0 / D q˛ˇ .t /.xˇ 0 xˇ 0 /: Therefore,

lim jx˛ .tk ; x0 / x˛ .tk ; x0 /j D 1:

k!1

No matter how small the difference xˇ 0xˇ 0 is in modulus, the function x˛ .t; x0 / x˛ .t; x0 / is unbounded for t ! 1, and, hence, the difference x.t; x0 / x.t; x0 / is also unbounded. This means that the solution x.t; x0 / of system (1.62) is unstable. We have proved that the boundedness of the matrizant X.t; t0 / for all t t0 is a sufﬁcient condition for the stability, the equality lim t!1 X.t; t0 / D 0 is a sufﬁcient condition for the asymptotic stability, and the unboundedness of the matrix X.t; t0 / for t t0 is a sufﬁcient condition for the instability of any solution of the system of equations (1.62). One can also prove that the conditions presented above are not only sufﬁcient but also necessary for the stability, asymptotic stability, and instability of any solution of the linear system (1.62), respectively. Thus, the following theorem is true: Theorem 10 ([142]). For the stability of a solution x.t; x0 / of the linear impulsive system (1.62), it is necessary and sufﬁcient that the matrizant X.t; t0 / (and, hence, any fundamental matrix) of this system be bounded for t t0 ; for its asymptotic stability, it is necessary and sufﬁcient that the matrizant satisfy the condition lim X.t; t0 / D 0I

t !1

and for its instability, it is necessary and sufﬁcient that the matrizant be unbounded for t t0 . Since the matrix X.t; t0 / does not depend on the initial value of a solution x.t; x0 / of system (1.62), all solutions of the linear impulsive system (1.62) are simultaneously stable or unstable. Therefore, the linear system (1.62) is called stable, asymptotically

32

Chapter 1 Impulsive Differential Equations

stable, or unstable depending on whether its solutions are stable, asymptotically stable, or unstable. For example, the ﬁrst-order equation dx D a.t /x; dt

t ¤ i ;

xj tDi D bi x

is stable, asymptotically stable, or unstable depending on whether the expression Z t X a.s/ds C ln j1 C bi j t0

t0 i
is bounded, tends to 1 as t ! 1, or is unbounded for t t0 . Assume that the matrices A.t / and Bi in the system of equations (1.62) are representable in the form A.t / D A C P .t / and Bi D B C Ii , where A and B are constant matrices. Then system (1.62) can be rewritten in the form dx D Ax C P .t /x; dt

t ¤ i ;

xj t Di D Bx C Ii x:

(1.64)

Together with system (1.64), we consider the system dx D Ax; dt

t ¤ i ;

xj tDi D Bx:

(1.65)

The following theorem is true: Theorem 11 ([142]). If the solutions of the system of equations (1.65) are stable, then the solutions of the system of equations (1.64) are also stable, provided that Z 1 Y kP .t /kdt < 1 and .1 C kIi k/ < 1: (1.66) t0

i t0

Proof. The matrizant X.t; t0 / of the system with constant coefﬁcients (1.65) takes the form Y X.t; t0 / D e A.ti / .E C B/e A.j j 1 / ; 0 D t0 : (1.67) t0 j
By virtue of the nondegeneracy of the matrix E C B, the matrix X.t; t0 / is nondegenerate and X.t; t0 /X 1 .s; t0 / D e A.ti / Y .E C B/e A.j j 1 / .E C B/e A.kC1 s/ ; sj
i < t iC1 ;

k < s < kC1 ;

k < i:

(1.68)

33

Section 1.2 Linear Systems

It follows from the stability of solutions of system (1.65) and from representation (1.68) that there exists a positive number K such that kX.t; t0 /X 1 .s; t0 /k K;

kX.t; t0 /k K;

t0 s t:

(1.69)

Note that any solution x.t; x0 /, x.t0 ; x0 / D x0 , of system (1.64) admits the representation Z t X.t; s/P .s/x.s; x0 /ds (1.70) x.t; x0 / D X.t; t0 /x0 C C

t0

X

X.t; i /Ii x.i ; x0 /:

t0 i
Therefore, for any two solutions x.t; x0 / and x.t; y0 / of system (1.64), with regard for (1.69) we get kx.t; x0 / x.t; y0 /k Kkx0 y0 k Z t KkP .s/k kx.s; x0 / x.s; y0 /kds C t0

X

C

KkIi k kx.i ; x0 / x.i ; y0 /k:

t0 i
Hence, by virtue of Lemma 1, we obtain the following estimate for all t t0 : Rt Y .1 C KkIi k/e t0 KkP .s/kds kx0 y0 k: (1.71) kx.t; x0 / x.t; y0 /k K t0 i
Q The convergence of the product i t0 .1 C kIi k/ yields the convergence of the prodQ uct t0 i
Y

.1 C KkIi k/e

R1 t0

t t0 ;

KkP .s/kds

(1.72)

:

i t0

The stability of solutions of system (1.64) follows from inequality (1.72). Theorem 11 is proved. Criterion for Stability in the First Approximation. We now study the stability of solutions of the nonlinear impulsive system of differential equations dx D f .t; x/; dt

t ¤ i ;

xj tDi D Ii0 .x/:

(1.73)

34

Chapter 1 Impulsive Differential Equations

The problem of stability of a solution x D '.t / of Eqs. (1.73) reduces to the study of the stability of the trivial solution of some other system. To construct this system we change the variables in Eqs. (1.73) by setting x D y C'.t /. As a result, the system of equations (1.73) is transformed into the system dy D F .t; y/; dt

yj tDi D Ii1 .y/;

t ¤ i ;

(1.74)

where F .t; y/ D f .t; y C '.t // f .t; '.t //;

F .t; 0/ 0;

Ii1 .y/ D Ii0 .y C '.i // Ii0 .'.i //;

Ii1 .0/ D 0;

and and the solution x D '.t / is transformed into the solution y D 0. Thus, without loss of generality, we assume that system (1.73) has the trivial solution x D 0 and study its stability. We represent the functions f .t; x/ and Ii0 .x/ as follows: Ii0 .x/ D Bi x C Ii .x/;

f .t; x/ D A.t /x C g.t; x/;

where A.t / and Bi are matrices, and g.t; x/ and Ii .x/ satisfy the conditions g.t; 0/ D 0 and Ii .0/ D 0, respectively. We now rewrite the system of equations (1.73) in the form dx D A.t /x C g.t; x/; t ¤ i ; xj t Di D Bi x C Ii .x/: (1.75) dt Together with Eqs. (1.75), we consider the linear system of equations dx D A.t /x; t ¤ i ; xj t Di D Bi x: (1.76) dt We call it the system of the ﬁrst approximation with respect to system (1.75). Assume that the times of pulse action are enumerated by the set of natural numbers in natural order and (1.77) iC1 i for some > 0. Theorem 12 ([142]). Suppose that, for all t and s, t0 s t , the matrizant X.t; s/ of the system of equations (1.76) admits the estimate kX.t; s/k Ke .ts/ ;

K 1; > 0;

(1.78)

and the functions g.t; x/ and Ii .x/ satisfy the inequalities kg.t; x/k akxk;

kIi .x/k akxk

(1.79)

for all t t0 , i D 1; 2; : : : , and kxk h, h > 0. Then, for sufﬁciently small a, the trivial solution of Eq. (1.75) is asymptotically stable.

35

Section 1.2 Linear Systems

Proof. Each solution of Eq. (1.75) can be represented in the form Z t x.t; x0 / D X.t; t0 /x0 C X.t; s/g.s; x.s; x0 //ds C

t0

X

X.t; i /Ii .x.i ; s0 //:

(1.80)

t0 i
Hence, with regard for inequalities (1.78) and (1.79), we get Z t kx.t; x0 /k Ke .tt0 / kx0 k C Ke .ts/ akx.s; x0 /kds t0

X

C

Ke

.ti /

akx.i ; x0 /k;

t0 i
or Z e .tt0 / kx.t; x0 /k Kkx0 k C X

C

t t0

Kae .st0 / kx.s; x0 /kds

Kae .i t0 / kx.i ; x0 /k:

(1.81)

t0 i
By virtue of Lemma 2, we obtain e .tt0 / kx.t; x0 /k Kkx0 k.1 C Ka/i.t0 ;t/ e Ka.tt0 / : Since the times i satisfy inequality (1.77), relation (1.81) yields 1

kx.t; x0 /k Ke .Ka ln.1CKa//.t t0 / kx0 k:

(1.82)

Thus, if a is so small that 1 ln.1 C Ka/ > 0;

Ka then any solution x.t; x0 /; kx0 k <

h K,

of Eq. (1.75) is deﬁned for all t t0 , and

lim kx.t; x0 /k D 0:

t!1

In other words, the trivial solution of Eq. (1.75) is asymptotically stable. Linear Periodic Systems with Pulse Action. A linear homogeneous system of differential equations with pulse action dx D A.t /x; dt

t ¤ i ;

xj t Di D Bi x

(1.83)

36

Chapter 1 Impulsive Differential Equations

is called periodic with period T (or T -periodic) if the matrix A.t / is T -periodic and one can ﬁnd a natural number p such that BiCp D Bi ;

iCp D i C T

(1.84)

for all i 2 Z. We assume that the matrix A.t / is continuous (piecewise continuous with discontinuities of the ﬁrst kind at t D i /, the matrices E C Bi are nondegenerate, and the times i are enumerated by the set of integers Z so that 0 1 < < p < T . Let X.t / be the fundamental matrix of the periodic system (1.83) such that X.0/ D E, i.e., the matrizant of this system. By virtue of the periodicity of system (1.83), it is easy to prove that X.t C T / is also a fundamental matrix of system (1.83) and X.t C T / D X.t /X.T /;

(1.85)

where X.T / D U.T; p /

1 Y

.E C BC1 /U.C1 ; /.E C B1 /U.1 ; 0/

Dp1

is the matrix of monodromy and U.t; s/, U.s; s/ D E, is the matrizant of the differential system from (1.83). The eigenvalues of the matrix X.T / are called the multipliers of system (1.83). The following theorem is true: Theorem 13 ([142]). For any multiplier , there exists a nontrivial solution x D '.t / of the periodic system of Eqs. (1.83) that satisﬁes the condition '.t C T / D '.t /:

(1.86)

Conversely, if relation (1.86) is satisﬁed for some nontrivial solution x D '.t / and some number , then is a multiplier of this system of equations. Proof. As a solution '.t /, we take the solution of system (1.83) for which the vector '.0/ is the eigenvector of the matrix of monodromy corresponding to the eigenvalue . We have X.T /'.0/ D '.0/; '.t / D X.t /'.0/: Hence, '.t C T / D X.t C T /'.0/ D X.t /X.T /'.0/ D X.t /'.0/ D '.t /; i.e., condition (1.86) is satisﬁed. Assume that equality (1.86) is satisﬁed for some nontrivial solution '.t / DX.t /'.0/, i.e., X.t C T /'.0/ D X.t /'.0/; X.t /X.T /'.0/ D X.t /'.0/;

37

Section 1.2 Linear Systems

whence .X.T / E/'.0/ D 0: This implies that is a root of the characteristic equation det.X.T / E/ D 0, i.e., it is a multiplier of system (1.83). Theorem 13 is proved. Theorem 13 yields the following important corollary: The linear T -periodic system of equations (1.83) has a nontrivial kT -periodic solution if and only if the k-th power of at least one of its multipliers is equal to 1. We now consider conditions for the existence of periodic solutions of linear inhomogeneous periodic systems of differential equations with pulse action, i.e., systems of the form dx D A.t /x C f .t /; dt

t ¤ i ;

xj t Di D Bi x C ai ;

(1.87)

where A.t / and f .t / are continuous (piecewise continuous) T -periodic matrix and vector functions, respectively, Bi are constant matrices, ai are constant vectors, and the times i are such that BiCp D Bi ;

aiCp D ai ;

i Cp D i C T

(1.88)

for some natural p and all i 2 Z. It is also assumed that det.E C Bi / ¤ 0, and 0 1 < 2 < < p < T . Let X.t; s/, X.s; s/ D E, be the matrizant of the homogeneous system corresponding to (1.87). Any solution x.t; x0 /, x.0; x0 / D x0 , of the system of equations (1.87) can be represented in the form Z t X x.t; x0 / D X.t; 0/x0 C X.t; s/f .s/ds C X.t; i /ai : (1.89) 0

0i
Among these solutions, the T -periodic solutions are those for which x0 satisﬁes the equation Z .E X.T; 0//x0 D

T 0

X.T; s/f .s/ds C

p X

X.T; i /ai :

(1.90)

i D1

Assume that det.E X.T; 0// ¤ 0. This condition is equivalent to the assumption that the homogeneous system corresponding to Eqs. (1.87) does not have nontrivial T -periodic solutions, i.e., none of its multipliers is equal to 1. In this case, Eq. (1.90) has the unique solution 1

x0 D .E X.T; 0//

Z

T 0

X.T; s/f .s/ds C

p X iD1

X.T; i /ai :

(1.91)

38

Chapter 1 Impulsive Differential Equations

Hence, the system of equations (1.87) has the unique T -periodic solution

1

Z

x .t / D X.t; 0/.E X.T; 0// Z C

0

t

T 0

X.T; i /ai

iD1

X

X.t; s/f .s/ds C

X.T; s/f .s/ds C

p X

X.t; i /ai ;

t 0:

0i
Denoting G.t; s/ D

´ X.t; 0/.E X.T; 0//1 X 1 .s; 0/; X.t C T; 0/.E

0 s t T;

X.T; 0//1 X 1 .s; 0/;

0 t < s T;

(1.92)

we can represent the periodic solution in the form x .t / D

Z

T 0

G.t; s/f .s/ds C

p X

G.t; i /ai :

(1.93)

i D1

The function G.t; s/ is called the Green function of the problem of periodic solutions of system (1.87). Note the following properties of the function G.t; s/: (1) if t ¤ i , then G.s C 0; s/ G.s; s/ D E; (2) G.0; s/ D G.T; s/; (3) for t ¤ s, the function G.t; s/ satisﬁes the impulsive system of equations dG.t; s/ D A.t /G.t; s/; dt

t ¤ i ; Gj tDi D Bi G.i ; s/;

so that G.i C 0; s/ D .E C Bi /G.i ; s/I (4) G.t; i C 0/ D G.t; i /.E C Bi /1 . These properties of the function G.t; s/ can easily be veriﬁed with regard for properties of the function X.t; 0/. Also note that properties (1)–(4) uniquely deﬁne the Green function G.t; s/. Let max kG.t; s/k D K: t;s2Œ0;T

It follows from (1.93) that x .t / admits the estimate

kx .t /k K

Z

T 0

kf .t /kdt C

p X iD1

kai k :

(1.94)

39

Section 1.2 Linear Systems

Thus, we have proved the following statement: Theorem 14 ([142]). If the homogeneous system corresponding to (1.87) does not have nontrivial T -periodic solutions, then the system of equations (1.87) has the unique T -periodic solution x .t / for any T -periodic function f .t / and any periodic sequence ai .aiCp D ai , i 2 Z/. This solution satisﬁes estimate (1.94). Consider the case where the homogeneous system corresponding to system (1.87) has nontrivial T -periodic solutions. The initial values of these solutions are determined from the equations .E X.T; 0//x0 D 0:

(1.95)

Assume that the homogeneous system has k n linearly independent solutions. Then the algebraic system (1.95) has exactly k linearly independent solutions, i.e., the rank of the matrix E X.T; 0/ is equal to n k. The matrizant of the conjugate system dx D AT .t /x; dt

t ¤ i ;

xj tDi D .E C BiT /1 Bi x

(1.96)

is the matrix Y .t; s/ D .X T .t; s//1 . Therefore, the initial conditions of nontrivial T -periodic solutions of the conjugate system must satisfy the condition .E .X T .T; 0//1 /y0 D 0 or

.X.T; 0/ E/T y0 D 0:

(1.97)

The rank of the matrix .X.T; 0/E/T is equal to the rank of the matrix X.T; 0/E, i.e., it is equal to n k. Hence, the system of algebraic Eqs. (1.97) has k linearly independent solutions, which correspond to k linearly independent T -periodic solutions of the conjugate system (1.96). Theorem 15 ([142]). Suppose that the linear homogeneous T -periodic system corresponding to Eqs. (1.87) has k linearly independent T -periodic solutions '1 .t /; : : : ; 'k .t /, k D 1; n. The system of equations (1.87) has T -periodic solutions if and only if the conditions Z

T

. 0

j .t /; f .t //dt C

p X .

j .i /; ai /

D 0;

j D 1; k;

(1.98)

iD1

are satisﬁed. Here, 1 .t /; : : : ; k .t / are linearly independent T -periodic solutions of the conjugate system of equations (1.96). In this case, the T -periodic solutions of system (1.87) form a k-parameter family of solutions.

40

Chapter 1 Impulsive Differential Equations

Proof. Let x.t / be a T -periodic solution of the inhomogeneous system of equations (1.87). It follows from the condition of T -periodicity that the initial data x.0/ D x0 satisfy the condition Z .E X.T; 0//x0 D

T

X.T; 0/X 1 .s; 0/f .s/ds

0

C

p X

X.T; 0/X 1 .i ; 0/ai :

(1.99)

iD1

Let .t / be a nontrivial T -periodic solution of the conjugate system of equations (1.96). Then .E X.T; 0//T .0/ D 0 and, hence, 0 D ..E X.T; 0//T .0/; x0 / D . .0/; .E X.T; 0//x0 / Z T p X D X T .T; 0/ .0/; X 1 .s; 0/f .s/ds C X 1 .i ; 0/ai 0

Z D

T

0

Z D

iD1

..X T .s; 0//1 .0/; f .s//ds C

p X

..X T .i ; 0//1 .0/; ai /

iD1 T

0

. .s/; f .s//ds C

p X . .i /; ai /: iD1

The necessity is proved. Let us prove the sufﬁciency of conditions (1.98). Assume that equalities (1.98) are true. If y0 is the eigenvector of the matrix X T .T; 0/ that corresponds to the multiplier D 1, then the solution y.t /, y.0/ D y0 , of the conjugate system (1.96) is T -periodic and k X y.t / D Y .t; 0/y0 D cj j .t /: j D1

Therefore, Z 0D

T 0

Z D

0

X k

cj

p X k X .cj j .t /; f .t / dt C

j D1 T

j .i /; ai /

i D1 j D1

p X .Y .t; 0/y0 ; f .t //dt C .Y .i ; 0/y0 ; ai / i D1

41

Section 1.2 Linear Systems

Z D

T 0

.X T .T; 0/y0 ; X 1 .t; 0/f .t //dt

p X C .X T .T; 0/y0 ; X 1 .i ; 0/ai /

iD1

Z

D y0 ;

T

0

X.T; 0/X 1 .t; 0/f .t /dt C

p X

X.T; 0/X 1 .i ; 0/ai :

iD1

Thus, the system of equations .X.T; 0/ E/T y0 D 0 is equivalent to the system Z y0 ;

T 0

.X.T; 0/ E/y0 D 0; p X 1 1 X.T; 0/X .t; 0/f .t /dt C X.T; 0/X .i ; 0/ai D 0; iD1

and, hence, the ranks of the matrices of these systems are identical. Therefore, denoting the row vector Z bD

T 0

X.T; 0/X 1 .t; 0/f .t /dt C

p X

X.T; 0/X 1 .i ; 0/ai

T ;

i D1

for the rank of the system of equations (1.99) we get .X.T; 0/ E/T T rang..X.T; 0/ E/b / D rang b D rang .X.T; 0/ E/T D rang .X.T; 0/ E/ D n k: By virtue of the Kronecker–Capelli theorem, the system of equations (1.99), which determines the initial conditions for T -periodic solutions of the inhomogeneous system (1.87), is consistent and has exactly k linearly independent solutions. Theorem 15 is proved.

Chapter 2

Impulsive Differential Inclusions

2.1

Differential Inclusions with Fixed Times of Pulse Action

Consider a differential inclusion with pulse action at ﬁxed times: xP 2 F .t; x/; xj tDi 2 Ii .x/;

t ¤ i ; x.t0 / D x0 2 X0 ; i D 1; k:

(2.1) (2.2)

Here, t 2 I D Œt0 ; T is time, x 2 Rn is a phase vector, F W I Rn ! conv.Rn /, i 2 I are the times of pulse action, and Ii W Rn ! conv.Rn /. Deﬁnition 1 ([23]). A vector function x.t / deﬁned on an interval or a segment J I is called a solution of the impulsive differential inclusion (2.1), (2.2) if it is absolutely continuous, satisﬁes inclusion (2.1) almost everywhere on intervals that do not contain i , and satisﬁes the condition of jump (2.2) for t D i . Deﬁnition 2 ([23]). The integral funnel of a point .t0 ; x0 / (or of the set K) is the set of points lying on the graphs of all solutions that pass through this point (respectively, through the points of the set K/. The cross-section t D t1 of the funnel of a point .t0 ; x0 / is the set of attainability at time t1 , i.e., the set of points ¹x.t1 /º that can be reached at time t1 by moving along various solutions leaving the point x0 at time t0 . If we omit the assumption that the set F .t; x/ is convex, then the funnel and the set of attainability can be nonclosed (see Appendix 2). Now consider the problem of the connectedness of the integral funnel and the set of attainability of system (2.1), (2.2), depending on properties of the functions F .t; x/ and Ii .x/ and the set K. Assume that Ii .x/ map any connected compact set onto a connected compact set. Then, in view of [23], the following statement is true: Theorem 1 ([113]). Suppose that the following conditions are satisﬁed in the domain Q D I D, where D is a compact set in Rn : (1) a set-valued mapping F .t; x/ is upper semicontinuous in x and measurable in t , and jF .t; x/j g.t /.1 C kxk/, where g.t / is summable on I ;

Section 2.1 Differential Inclusions with Fixed Times of Pulse Action

43

(2) Ii .x/ map a connected compact set onto a connected compact set; (3) the set K is compact and connected. Then the set H.K/ of all solutions of system (2.1), (2.2) with various initial conditions .t0 ; x0 / 2 K is compact in the metric of C.I / and is connected on the time interval between pulses. The set of attainability is connected and compact. By analogy with the notion of R-solution for differential inclusions without pulses [23, 101], we introduce the notion of R-solution for impulsive differential inclusions and present conditions for the existence and uniqueness of R-solutions. Deﬁnition 3. A set-valued mapping RW I ! comp.Rn / is called an R-solution of the differential inclusion (2.1), (2.2) if R.t / satisﬁes the initial condition R.t0 / D X0 and is absolutely continuous on the time interval between two successive pulses, for almost all t one has ³ Z t C [ ² 1 h R.t C /; F .s; x/ds ! 0 . # 0/; xC t x2R.t/

and the following relation holds at the times of pulse action: R.i C 0/ D

[

¹x C Ii .x/º:

x2R.i /

Here and in what follows, the integral of a set-valued mapping is understood in the sense of Aumann [10] (unless otherwise stated). Theorem 2. Suppose that the following conditions are satisﬁed in the domain Q D I D; D 2 comp.Rn /: (1) a set-valued mapping F .t; x/ is continuous in x and measurable in t , and jF .t; x/j g.t /.1 C kxk/, where g.t / is summable on I ; (2) Ii .x/ map a connected compact set onto a connected compact set. Then, for any compact set X0 Rn , there exists an R-solution of the differential inclusion (2.1), (2.2). The integral funnel is the graph of the R-solution R.t /. Assume, in addition, that the following condition is satisﬁed for any r > 0 and almost all t .t ¤ i /: If kx yk r, then h.F .t; x/; F .t; y// !.t; r/; where !.t; r/ is the Kamke function. Then the R-solution is unique.

44

Chapter 2 Impulsive Differential Inclusions

Now consider the problem of continuous dependence of solutions of impulsive differential inclusions on the initial data and right-hand sides. Consider differential inclusion with multivalued pulses: xP 2 F1 .t; x/;

t ¤ i ; x.t0 / 2 X0 ;

xj t Di 2 I1i .x/

(2.3) (2.4)

and t ¤ i ; y.t0 / 2 Y0 ;

yP 2 F2 .t; y/; yj tDi 2 I2i .y/;

i D 1; k;

(2.5) (2.6)

where Fj W I Rn ! conv.Rn / and Iji W Rn ! conv.Rn /, j D 1; 2. Theorem 3 ([114]). Suppose that the following conditions are satisﬁed in the domain Q D I D, D 2 comp.Rn /: (1) set-valued mappings Fj .t; x/, j D 1; 2, are measurable in t and satisfy the Lipschitz condition with respect to x with summable function l.t /; (2) set-valued mappings Iji .x/ satisfy the Lipschitz condition with constant ; (3) the following estimates are true: h.F1 .t; x/; F2 .t; x// ;

h.I1i .x/; I2i .x// ;

h.X0 ; Y0 / ı0 :

(2.7)

Then, for any solution y.t / of inclusion (2.3), (2.4), there exists a solution x.t / of inclusion (2.5), (2.6) such that kx.t / y.t /k Aı0 C B ; and vice versa. The constants A and B are independent of and ı0 . Proof. Let y.t / be a solution of inclusion (2.5), (2.6) and let 1 > t0 be the time of the ﬁrst pulse. Then, by virtue of (2.7), we have .y.t P /; F1 .t; y.t /// h.F2 .t; y.t //;

F1 .t; y.t // < :

According to the Filippov theorem [51], a solution x.t / of inclusion (2.3), (2.4) exists on the interval Œt0 ; 1 and is such that Z t kx.t / y.t /k ı0 e m.t/ C e m.t/m.s/ ds; t0

where

Z m.t / D

t

l.s/ds: t0

45

Section 2.1 Differential Inclusions with Fixed Times of Pulse Action

Hence, ı1

Z D kx.1 / y.1 /k ı0 e

m.1 /

C

1

e m.1 /m.s/ ds:

t0

According to (2.6), we have y.1 C 0/ 2 y.1 / C I21 .y.1 //. We choose x.1 C 0/ so that ı1C D kx.1 C 0/ y.1 C 0/k D

min

x2x.1 /CI11 .x/

kx y.1 C 0/k

h.x.1 / C I11 .x.1 //; y.1 / C I21 .y.1 /// kx.1 / y.1 /k C h.I11 .x.1 //; I21 .y.1 /// D ı1 C h.I11 .x.1 //; I21 .y.1 /// ı1 C h.I11 .x.1 //; I21 .x.1 /// C h.I21 .x.1 //; I21 .y.1 /// ı1 C kx.1 / y.1 /k C D ı1 C ı1 C : Hence, ı1C ı1 .1 C / C . By analogy, for any i D 0; k and t 2 .i ; iC1 , we get C ıiC1 ıiC1 .1 C / C ; ıiC1 ıiC e m.i C1 /m.i / C

kx.t / y.t /k ıiC e m.t/m.i / C

Z

Z

t i

ıiC e m.iC1 /m.i / C

iC1

e m.iC1 /m.s/ ds;

i

e m.t/m.s/ ds Z

i C1

e m.i C1 /m.s/ ds

i

.ıi .1 C / C / e m.iC1 /m.i / C e m.iC1 /

Z C m.i /m.i1 / m.i / .1 C / ıi1 e C e Z e

m.iC1 /m.i /

C e

m.i C1 /

i C1

Z

iC1

e m.s/ ds

i

i

e i 1

m.s/

ds C

e m.s/ ds

i

C e m.i C1 /m.i 1 / C e m.i C1 /m.i / .1 C /ıi1 Z i Z iC1 m.iC1 / m.s/ m.s/ C e e ds C e ds .1 C / i1

i

46

Chapter 2 Impulsive Differential Inclusions i1 X

.1 C /i ı0 e m.iC1 / C

.1 C /j e m.iC1 /m.ij /

j D0 i X

C e m.i C1 /

Z .1 C /j

j D0

i j C1

e m.s/ ds

ij

.1 C /k ı0 e m.T / C e

m.T /

k X

.1 C /

j

Z 1C

kj C1

e

m.s/

ds :

kj

j D0

Denoting A D .1 C /k e m.T / and BDe

m.T /

k X

Z .1 C / 1 C

kj C1

j

j D0

e

m.s/

ds ;

kj

we ﬁnally obtain kx.t / y.t /k Aı0 C B ;

t 2 I:

By analogy, we can prove the following generalization of Theorem 3: Theorem 4 ([114]). Suppose that the following conditions are satisﬁed in the domain Q D I D, D 2 comp.Rn /: (1) set-valued mappings Fj .t; x/, j D 1; 2, are measurable in t, and the following inequality holds for kx yk r and almost all t : h.Fj .t; x/; Fj .t; y// !.t; r/; where !.t; r/ is the Kamke function; (2) set-valued mappings Iji .x/ satisfy the Lipschitz condition with constant ; (3) the following estimates are true: h.F1 .t; x/; F2 .t; x// .t /; h.I1i .x/; I2i .x// ; N

h.X0 ; Y0 / ı0 ;

where .t / is summable on I . Then, for any solution y.t / of inclusion (2.5), (2.6), a solution x.t / of inclusion (2.3), (2.4) exists and is such that kx.t / y.t /k r.t /;

t 2 I;

and vice versa. Here, r.t / is the upper (maximum) solution of the problem r.t P / D !.t; r.t // C .t /; r.i C 0/ D .1 C / r.i / C ; N

t 2 .i ; i C1 ; 0 D t0 ; kC1 D T; r.t0 / D ı0 ; i D 0; k:

Section 2.1 Differential Inclusions with Fixed Times of Pulse Action

47

Deﬁnition 4 ([114]). We say that a solution of the differential inclusion (2.1), (2.2) is extendable (weakly extendable) to I if any solution (at least one solution) x.t / of the differential inclusion is extendable to I . Example 1. Consider the control problem x.t P / D u.t /.1 C x 2 .t //;

x.0/ D 0;

(2.8)

where t 0 is time, x 2 R is a phase vector, and the control u.t / 2 Œ0; 1 is a measurable function. It is obvious that Eq. (2.8) can be represented in the form of a differential inclusion: x.t P / 2 Œ0; 1 .1 C x 2 .t //;

x.0/ D 0:

(2.9)

If u.t / is an arbitrary given measurable function, then solution (2.8) takes the form Z x.t / D tan

t

u.s/ ds:

(2.10)

0

Note that solution (2.10) is deﬁned on Œ0; 1/ if and only if Z

t

u.s/ ds < 0

2

for any ﬁxed t < 1. If u.t / is such that Z

t 0

u.s/ ds D

2

for some t , then the corresponding solution cannot be extended to the interval Œ0; 1/. It is obvious that the set of solutions is weakly extendable. The set of solutions of (2.8) is not bounded on the interval Œ0; 1/. Using a pulse control, one can obtain the set of solutions that are strongly extendable and bounded. Let a pulse action be given in the form xj tD k D x; 2

k D 1; 2; : : : :

(2.11)

The set of solutions of (2.9) and (2.11) has the form ³ ² Z t .k 1/ k ; u.s/ ds; u.s/ 2 Œ0; 1; t 2 X.t / D tan ; k D 1; 2; : : : ; .k1/ 4 4 4 i.e., it is strongly extendable and bounded.

48

2.2

Chapter 2 Impulsive Differential Inclusions

Differential Inclusions with Nonﬁxed Times of Pulse Action

Consider differential inclusions with pulse actions at nonﬁxed times: xP 2 F .t; x/;

t ¤ i .x/; x.t0 / D x0 ;

xj tDi .x/ 2 Ii .x/;

(2.12) (2.13)

where t 2 I D Œt0 ; T , x 2 D (D is a domain in Rn ), F W I D ! conv.Rn /, i W D ! R, i D 1; k, and Ii W D ! comp.Rn /. Deﬁnition 5 ([114]). A piecewise absolutely continuous vector function x.t / deﬁned on a segment or interval J I is called a solution of inclusion (2.12), (2.13) if it satisﬁes (2.12) almost everywhere for t 2 J , t ¤ i .x.t //; i D 1; k, and has discontinuities of the ﬁrst kind at the points t D i .x.t // with jumps x.t / 2 Ii .x.t //: Deﬁnition 6 ([114]). Solutions of the differential inclusion (2.12), (2.13) are called extendable (weakly extendable) to I if every solution (at least one solution) x.t / of the differential inclusion is extendable to I . Deﬁnition 7 ([36, 128]). A tangent cone TD .x/ to a nonempty set D Rn at a point x 2 D is deﬁned as follows: ° ± TD .x/ D y 2 Rn j lim s 1 inf jx C sy zj D 0 : s!C0

z2D

If D 2 conv.Rn /, then the set TD .x/ is closed and convex, and 0 2 TD .x/. Consider sufﬁcient conditions for the absence of beating of solutions against the surface t D i .x/, i.e., conditions guaranteeing that any solution of a differential inclusion crosses the impulsive surface at most once. Theorem 5 ([114]). Suppose that the following conditions are satisﬁed in the domain Q D I D: (1) a set-valued mapping F .t; x/ is bounded by a constant C .jF .t; x/j C /, is continuous in t, satisﬁes the Lipschitz condition with respect to x with constant

, and is such that F .t; x/ TD .x/; (2) set-valued mappings x C Ii .x/W D ! conv.D/ satisfy the Lipschitz condition with constant , i D 1; k;

Section 2.2 Differential Inclusions with Nonﬁxed Times of Pulse Action

49

(3) surfaces i .x/ are smooth, disjoint, and such that, for every zi 2 Ii .x/, one has @i .x/ @i .x C zi / NI ; zi 0; max (2.14) 0 1 @x @x (4) NC < 1. Then every solution x.t /, x0 2 D, of the differential inclusion (2.12), (2.13) is extendable to I and crosses the surfaces t D i .x/ at most once. Proof. Conditions (1) and (2) guarantee that the solutions remain in the domain D. Assume that there exists a solution x.t / that hits the surface t D i .x/ at time t D i .x.t // and crosses the same surface at another point .t ; x.t //, where t D i .x.t //. First, we consider the case where .t ; t / is the interval of continuity of x.t /. Then x.t / D x.t / C zi C

Z

t

u.s/ ds; t

where zi 2 Ii .x.t // and u.s/ 2 F .s; x.s// is a measurable selector. Consider the difference t t D i .x.t // i .x.t // D i .x.t // i .x.t / C zi / C i .x.t / C zi / i .x.t // Z t Z 1 @i .x.s// @i .x.t / C zi / D ; u.s/ ds C ; zi d: (2.15) @x @x 0 t By virtue of the Cauchy–Schwarz inequality and conditions (1) and (3) of Theorem 5, we obtain the following estimate: Z

t t

@i .x.s// ; u.s/ ds NC.t t /: @x

(2.16)

Using (2.15) and (2.16), we get

.1 NC /.t t /

1

Z 0

@i .x.t / C zi / ; zi d: @x

(2.17)

Since NC < 1 and t > t , inequality (2.17) contradicts condition (3) of Theorem 5. Therefore, in the case under consideration, each solution crosses each surface t D i .x/ at most once. It follows from conditions (1) and (2) of Theorem 5 that every local solution can be extended to the interval I . Now consider the case where a solution x.t / crosses other surfaces on the interval .t ; t /, i.e., there exist the moments of time tij D ij .x.tij //, i1 D i, j D 1; p,

50

Chapter 2 Impulsive Differential Inclusions

p k. We have proved that, on the interval .tij 1 ; tij , which is the interval of continuity of the function x.t /, one has ij 1 .x.t // < t, and, obviously, tij D ij .x.tij // > ij 1 .x.tij //: We now prove that ij .x.t // > ij 1 .x.t // on the interval .tij 1 ; tij . Assume the contrary. Then there exists a continuous function y.t /W Œ0; 1 ! D, y.0/ D x.t /, y.1/ D x.tij /, such that ij .y.0// ij 1 .y.0//;

tij D ij .y.1// > ij 1 .y.1//:

It follows from these inequalities that there exists 2 Œ0; 1/ such that ij .y.// D ij 1 .y.//, which contradicts the fact that the surfaces t D j .x/ are disjoint. Since ij C1 .x.tij // > ij .x.tij // > ij 1 .x.tij //, we conclude, by analogy, that ij C1 .x.t // > ij 1 .x.t // on the interval .tij ; tij C1 . Using the method of complete mathematical induction, one can show that ip .x.t // > i .x.t // on the interval .tip1 ; tip . The interval .tip ; t is the interval of continuity of the solution x.t /. Taking the inequality tip > i .xip / into account and setting t D x.tip / and zi D zip , we conclude that inequality (2.17) is not true. Theorem 5 is proved. Remark 1. The condition F .t; x/ TD .x/ can be replaced by the following one: For every x0 2 D 0 D, there exists a constant > 0 for which all solutions of (2.12), (2.13) belong to the domain D on the interval Œt0 ; T together with their -neighborhoods. In this case, the conditions x C Ii .x/ D are redundant. Note that the aforementioned conditions are not equivalent. Remark 1 is also valid for the subsequent theorems. Remark 2. If the conditions of Theorem 5 F .t; x/ TD .x/, x C Ii .x/W D ! conv.D/, and, for every z 2 Ii .x/, inequalities (2.14) are true are replaced by the conditions T T F .t; x/ TD .x/ 6D ¿, Ii .x/W D ! conv.Rn /, x C Ii .x/ D 6D ¿, and there exists z 2 Ii .x/, x C z 2 D, for which inequalities (2.14) are true, then one can show that there exists a solution that crosses the impulsive surfaces at most once and that the set of solutions of (2.12), (2.13) is weakly extendable. The theorem below establishes conditions for the absence of beating of solutions of system (2.12), (2.13) against the surfaces t D i .x/ in the case where the functions i .x/ are not continuously differentiable.

Section 2.2 Differential Inclusions with Nonﬁxed Times of Pulse Action

51

Theorem 6 ([114]). Suppose that conditions .1/, .2/, and .4/ of Theorem 5 are satisﬁed for the system of equations (2.12), (2.13), and condition .3/ is replaced by the following one: (30 ) the surfaces t D i .x/ are disjoint, the functions i .x/ satisfy the Lipschitz condition with constant N , and, for all x 2 D and zi 2 Ii .x/, one has i .x/ .x C zi /:

(2.18)

Then every solution x.t /, x0 2 D, of the differential inclusion (2.12), (2.13) can be extended to I and crosses the surfaces t D i .x/ at most once. Proof. This theorem is proved by analogy with the previous theorem. The difference is that (2.15) is replaced by t t D i .x.t // i .x.t // D i .x.t // i .x.t / C zi / C i .x.t / C zi / i .x.t // Z t C i .x.t / C zi / i .x.t // N u.s/ds Z N

t

t t

ku.s/kds C i .x.t / C zi / i .x.t //

NC.t t / C i .x.t / C zi / i .x.t //; whence .1 NC /.t t / i .x.t / C zi / i .x.t //: By virtue of conditions (30 ) and (4) of Theorem 6, the last inequality becomes contradictory. Remark 3. It is obvious that if i .x/ possesses the Lipschitz property, then inequalities (2.14) can be written with the use of the subdifferential @i .x/ [128]. Consider the problem of continuous dependence of solutions on initial data. Let x.t; x0 / and x.t; y0 / be two solutions of (2.12), (2.13) with initial conditions x.t0 ; x0 / D x0 and x.t0 ; y0 / D y0 . Let i .x0 / and i .y0 / be the corresponding times when the surface t D i .x/ is crossed and let i D min¹i .x0 /; i .y0 /º;

iC D max¹i .x0 /; i .y0 /º:

Theorem 7 ([114]). Suppose that the conditions of Theorem 5 are satisﬁed in the domain Q.

52

Chapter 2 Impulsive Differential Inclusions

Then, for every solution x.t; x0 /, there exist a constant ı > 0 and a solution x.t; y0 / such that, if kx0 y0 k < ı, then the following inequality holds for t 2 I , t … Œi ; iC , i D 1; k: kx.t; x0 / x.t; y0 /k 1 C

1 NC

k

e .T t0 / kx0 y0 k:

(2.19)

Proof. Assume that the solution x.t; y0 / crosses the surfaces t D i .x/ in the same order as the solution x.t; x0 /. In what follows, we prove that this is indeed the case for all sufﬁciently small ı > 0. We represent the solutions x.t; x0 / and x.t; y0 / as follows: Z t X x.t; x0 / D x0 C u.s/ ds C zi ; t0

Z x.t; y0 / D y0 C

t t0

t0 i .x0 /
X

v.s/ ds C

wi ;

ti .y0 /
where u.s/ 2 F .s; x.s; x0 // and v.s/ 2 F .s; x.s; y0 // are measurable functions, z i 2 Ii .x.i .x0 /; x0 //, and w i 2 Ii .x.i .y0 /; y0 //. Let us estimate the distance between the indicated solutions: Z t kx.t; x0 / x.t; y0 /k kx0 y0 k C ku.s/ v.s/k ds C

X t0 i .x0 /
t0

i

z

X t0 i .y0 /
w : i

(2.20)

S If t 2 Œt0 ; T n i Œi ; iC , then, for every measurable function u.s/ 2 F .s; x.s; x0 //, there exists a measurable function v.s/ 2 F .s; x.s; y0 // such that Z t Z t ku.s/ v.s/k ds h.F .s; x.s; x0 //; F .s; x.s; y0 /// ds t0

t0

Z

t t0

kx.s; x0 / x.s; y0 /k ds:

Similarly, for every vector z i 2 Ii .x.i .x0 /; x0 //, there exists a vector w i 2 Ii .x.i .y0 /; y0 // such that X X X i i z w kz i w i k t0 i .x0 /
t0 i .y0 /
i

X i

h.Ii .x.i .x0 /; x0 //; Ii .x.i .y0 /; y0 ///

53

Section 2.2 Differential Inclusions with Nonﬁxed Times of Pulse Action

X

kx.i .x0 /; x0 / x.i .y0 /; y0 /k

i

X .kx.i .x0 /; z0 / x.i .y0 /; z0 /k i

C kx.i ; x0 / x.i ; y0 /k/; where z0 D x0 if i D i .y0 /, and z0 D y0 if i D i .x0 /. If r.s/ 2 F .s; x.s; z0 //, then Z kx.i .x0 /; z0 / x.i .y0 /; z0 /k

iC i

kr.s/k ds C.iC i /:

Since iC i D ji .x0 / i .y0 /j D ji .x.i .x0 /; x0 // i .x.i .y0 /; y0 //j N kx.i .x0 /; x0 / x.i .y0 /; y0 /k N.kx.i .x0 /; z0 / x.i .y0 /; z0 /k C kx.i ; x0 / x.i ; y0 /k/ NC.iC i / C N kx.i ; x0 / x.i ; y0 /k; we have

N kx.i ; x0 / x.i ; y0 /k: 1 NC Hence, inequality (2.20) takes the form Z t kx.s; x0 / x.s; y0 /k ds kx.t; x0 / x.t; y0 /k kx0 y0 k C

iC i

t0

C 1 NC

X

t0 i
kx.i ; x0 / x.i ; y0 /k;

and, according to Lemma 2 in Chapter 1, the following estimate is true: k

e .T t0 / kx0 y0 k; 1 NC [ t 2 Œt0 ; T n Œi ; iC :

kx.t; x0 / x.t; y0 /k 1 C

(2.21)

i

To complete the proof, we show that, for all sufﬁciently small ı > 0, the solution x.t; y0 / crosses the surfaces t D i .x/ in the same order as the solution x.t; x0 /. If .t ; t / is the interval without pulses for the solutions x.t; x0 / and x.t; y0 /, and

54

Chapter 2 Impulsive Differential Inclusions

t D i .x.t; x0 //, then the condition NC < 1 yields t < i .x.t ; x0 //. According to estimate (2.21), we can choose a sufﬁciently small ı so that t < i .x.t ; y0 //. Since the surfaces i .x/ are disjoint, the proper choice of ı > 0 and the condition NC < 1 guarantee that x.t; y0 / crosses t D i .x/ prior to crossing another surface. Then, by induction, we establish that the solution x.t; y0 / crosses the surfaces t D i .x/ in the same order as the solution x.t; x0 /. Theorem 7 is proved. Remark 4. The statement of Theorem 7 remains true if we assume that the functions i .x/ satisfy the Lipschitz condition with constant N , NC < 1, and condition (2.18) is satisﬁed. Consider the problem of continuous dependence of solutions of impulsive differential inclusions upon the right-hand sides, impulsive surfaces, and initial data. Lemma 1 ([114]). Let C ıiC1 a1 ıiC1 C a2 ; a3 ıiC C a4 ; ıiC1

where aj 0, j D 1; 4, and i D 1; 2; : : : . Then ıiC1 .a2 a3 C a4 /

.a1 a3 /i 1 C .a1 a3 /i ı0C a1 a3 1

ıiC1 .a2 a3 C a4 /i C ı0C

if a1 a3 ¤ 1;

if a1 a3 D 1:

(2.22) (2.23)

Proof. Indeed, C ıiC1 a3 a1 ıi C a3 a2 C a4 a1 a32 ıi1 C a3 a1 a4 C a3 a2 C a4

Œ1 C a1 a3 C C .a1 a3 /i1 .a2 a3 C a4 / C .a1 a3 /i ı0C :

(2.24)

If a1 a3 ¤ 1, then (2.24) yields (2.22), and if a1 a3 D 1, then we obtain (2.23). Lemma 1 is proved. Consider the impulsive differential inclusions xP 2 F1 .t; x/; xj tD i .x/ 2 I1i .x/; 1

yP 2 F2 .t; y/; yj tD i .y/ 2 I2i .y/; 2

t ¤ 1i .x/; x.t0 / D x0 2 X0 ; i D 1; 2; : : : ; t ¤ 2i .y/; y.t0 / D y0 2 Y0 ; i D 1; 2; : : : :

(2.25) (2.26) (2.27) (2.28)

Let i (iC ) denote the times when the left-continuous solution x.t / of inclusion (2.25), (2.26) (solution y.t / of inclusion (2.27), (2.28)) hits the surfaces t D 1i .x/ (t D 2i .x/).

55

Section 2.2 Differential Inclusions with Nonﬁxed Times of Pulse Action

Theorem 8 ([114]). Suppose that, in the domain Q, the conditions of Theorem 6 are satisﬁed for inclusions (2.25), (2.26) and (2.27), (2.28), and the following inequalities are true: D max¹ ; N º;

C < 1;

h.I1i .x/; I2i .x// ;

h.F1 .t; x/; F2 .t; x// ; j1i .x/ 2i .x/j ;

h.X0 ; Y0 / ı:

Then, for any > 0 and every solution x.t / of inclusion (2.25), (2.26), there exist > 0, ı > 0, and a solution y.t / of inclusion (2.27), (2.28) such that [ kx.t / y.t /k ; t 2 Œt0 ; T n Œi ; iC : (2.29) i

Proof. Without loss of generality, we can assume that i iC . Let x.t / be an arbitrary solution of the impulsive differential inclusion (2.25), (2.26) with initial condition x.t0 / D x0 2 X0 . According to the Filippov theorem [51], there exist a solution y.t / of the differential inclusion (2.27) and constants C1 and C2 such that ky.t / x.t /k C1 ıiC C C2 ; ; t 2 .iC ; iC1

0C D t0 ;

i D 0; 1; : : : ;

where ıi D kx.i / y.i /k and ıiC D kx.iC C 0/ y.iC C 0/k .ı0C D ı/. Note that ıiC1 D kx.iC1 / y.iC1 /k C1 ıiC C C2 : (2.30) The following estimates are true: kx.i / y.iC /k kx.i / y.i /k C ky.i / y.iC /k ıi C C ji iC j; ji iC j D j1i .x.i // 2i .y.iC //j j1i .x.i // 2i .x.i //j C j2i .x.i // 2i .y.iC //j C kx.i / y.iC /k C .ıi C C ji iC j/; i.e., ji iC j

C ıi : 1 C

(2.31)

We now determine y.iC C 0/ from the following problem of minimization: ıiC D kx.iC C 0/ y.iC C 0/k D

min

z2I2i .y.iC //

kx.iC C 0/ y.iC / zk:

56

Chapter 2 Impulsive Differential Inclusions

Then ıiC

Z i h x.i / C I1 .x.i // C

ıi

Z C

iC i

iC i

u.s/ ds ;

y.i /

Z C

.u.s/ v.s// ds

iC

i

v.s/ ds C

I2i .y.iC //

C h.I1i .x.i //; I2i .x.i /// C h.I2i .x.i //; I2i .y.iC /// ıi C 2C ji iC j C C .ıi C C ji iC j/

1 C C C 2C C 1 ıi C : 1 C 1 C

Taking inequality (2.30) into account, we obtain ıiC1 D kx.iC1 / y.iC1 /k C1 ıiC C C2 ;

ıiC

1 C C C 2C C 1 ıi C : 1 C 1 C

By virtue of Lemma 1, the above inequalities yield ıi K1 C K2 ı;

i D 1; 2; : : : ;

where K1 and K2 are constants independent of and ı. Properly choosing > 0 and ı > 0, we establish that the solution y.t / crosses the surfaces t D 2i .x/ in the same order as the solution x.t / crosses the surfaces t D 1i .x/. Thus, inequality (2.29) is true. Theorem 8 is proved.

2.3

Examples

In this section, we consider several examples illustrating various cases of beating of solutions (strong and weak beating) and various types of extendability (strong and weak extendability), as well as the property of continuous dependence of solutions, depending on whether the conditions of the theorems from the previous section are satisﬁed. Example 2. This example shows that the condition NC < 1 in the theorems on continuous dependence cannot be omitted. Consider the system 1 x.0/ D ˛ > 0; ju.t /j C; C ; t ¤ x 2 .x/; 2 D x I.x/; 0 t 1:

xP D u.t /; xj tDx 2

57

Section 2.3 Examples

If u.t / D C and ˛C D 14 , then the solution x.t / D ˛ C C t crosses the curve t D x 2 only at the point t D 4; x D 2. If ˛C > 14 , then the solution does not cross the curve t D x 2 because the system t D x 2 , x D C t C ˛ has a unique solution for ˛C D 14 and does not have solutions for ˛C > 14 . Let ˛C D 14 . By varying the initial data ˛ or the right-hand sides, we establish that the continuous dependence is violated because the solutions are not subject to pulse action for ˛C > 14 and have a discontinuity for ˛C D 14 . It remains to verify that, for ˛C D 14 , we have NC 1. Indeed, ˇ ˇ ˇ @ .x.t // ˇ ˇ ˇ D C sup 2.˛ C C t / NC D C sup ˇ @x ˇ t2Œ0;1 t2Œ0;1 D 2C.˛ C C / D

1 C 2C 2 1: 2

Example 3. Consider one more example in which the condition i .x C zi / i .x/, zi 2 Ii .x/ is violated and the phenomenon of beating is observed (see Figure 1): xP D 0;

x.0/ D ˛ > 0; t 6D x .x/;

xj tDx D x 1 I.x/: For ˛ < 1, the solutions undergo a single pulse action. If ˛ D 1, then the solution is x.t / 1. Finally, if ˛ > 1, then the solution undergoes pulse action inﬁnitely many times (in this case, the condition .x C z/ D x C x 1 x D .x/ is not satisﬁed because the inequality x > 1 holds for all t ).

Figure 1.

58

Chapter 2 Impulsive Differential Inclusions

Example 4. Consider the following differential equation with multivalued pulses: P D xP 0 ; xR C x D 0 for xP ¤ 0; x.0/ D x0 ; x.0/ xjxD0 D 0; P

(2.32)

xj P xD0 2 W D Œw ; w C ; w ; w C 2 R: P

Equation (2.32) can be interpreted as the equation of oscillation of a swing: at the uppermost position, the swing gets a push, which increases the swinging amplitude. We introduce the auxiliary variables x D x1 and x2 D xP 1 and rewrite (2.32) in the form of a system: xP 1 D x2 ;

xP 2 D x1

x1 jx2 D0 D 0; x1 .0/ D x0 ;

for x2 ¤ 0;

(2.33)

x2 jx2 D0 2 W;

(2.34)

x2 .0/ D xP 0 :

(2.35)

We introduce new variables by the relations x1 D y1 cos.t C y2 /;

x2 D y1 sin.t C y2 /;

y1 0:

(2.36)

Substituting them in (2.33), we get yP1 cos.t C y2 / y1 sin.t C y2 /.1 C yP2 / D y1 sin.t C y2 /; yP1 sin.t C y2 / y1 cos.t C y2 /.1 C yP2 / D y1 cos.t C y2 /: Hence, yP1 cos.t C y2 / y1 yP2 sin.t C y2 / D 0;

(2.37)

yP1 sin.t C y2 / C y1 yP2 cos.t C y2 / D 0:

(2.38)

Multiplying (2.37) and (2.38) by cos.t Cy2 / and sin.t Cy2 /, respectively, performing summation, and reducing similar terms, we obtain yP1 D 0. Thus, on the intervals between pulses, we have y1 .t / const. By analogy, multiplying (2.37) and (2.38) by sin.t C y2 / and cos.t C y2 /, respectively, performing summation, and reducing similar terms, we obtain y1 yP2 D 0, whence yP2 D 0 because, according to the nature of the problem, y1 .t / may vanish only on a set of measure zero (otherwise, by virtue of the change of variables, we have x2 .t / D x.t P / 0 on a certain interval I /. Consequently, the change of variables (2.36) reduces system (2.33), (2.35) to the form yP1 D 0; yP2 D 0; y1 .0/ D y10 ; y2 .0/ D y20 ; (2.39) where the initial data y10 and y20 satisfy the relations y10 cos y20 D x0 ;

y10 sin y20 D xP 0 ;

59

Section 2.3 Examples

q i.e., y10 D x02 C xP 02 , and y20 is not uniquely determined (it is determined to within a value multiple of 2/. We assume that y20 2 .; . Let us determine the values of the pulses y1 and y2 and the times at which they appear. To this end, we denote the values of y1 and y2 before and after a pulse by y1 ; y2 and y1C ; y2C , respectively. At the time of pulse action, we have x2 D 0 H) sin.ti C y2 / D 0 H) ti D y2 C i and cos.ti C y2 / D ˙1. By assumption, there is no pulse in the variable x1 at the time when x2 D 0, i.e., x1 D 0 H) y1C cos.ti C y2C / y1 cos.ti C y2 / D 0 H) y1C cos.ti C y2 C y2 / D y1 cos.ti C y2 / H) y1C Œcos.ti C y2 / cos y2 sin.ti C y2 / sin y2 D y1 cos.ti C y2 / H) y1C cos.ti C y2 / cos y2 D y1 cos.ti C y2 /: Since cos.ti C y2 / ¤ 0, we have y1C cos y2 D y1 :

(2.40)

For the value of the pulse in the variable x2 at the time when x2 D 0, we have x2 D y1C sin.ti C y2C / C y1 sin.ti C y2 / D wi 2 W H) y1C sin.ti C y2C / D wi H) y1C sin.ti C y2 C y2 / D wi H) y1C Œsin.ti C y2 / cos y2 C cos.ti C y2 / sin y2 D wi H) y1C cos.ti C y2 / sin y2 D wi H) y1C sign.cos.ti C y2 // sin y2 D wi :

(2.41)

Raising (2.40) and (2.41) to the second power and performing summation, we obtain q .y1C /2 D .y1 /2 C wi2 H) y1C D .y1 /2 C wi2 .so that y1 0/ q (2.42) H) y1 D y1C y1 D .y1 /2 C wi2 y1 : Let us determine y2 . Using (2.40) and (2.42), we obtain cos y2 D

y1

y1C

D

y1 y1 D > 0: q y1 C y1 .y1 /2 C wi2

(2.43)

60

Chapter 2 Impulsive Differential Inclusions

Relations (2.41) and (2.42) yield sin y2 D q

wi .y1 /2

C

wi2

sign.cos.t C y2 //:

(2.44)

Note that y2 is not uniquely determined (it is determined to within a value multiple of 2). Moreover, by virtue of (2.36), different choices of y2 lead the same variation in x1 and x2 . Therefore, we assume that y2 2 .; . Then

wi sign.cos.t C y2 // : y2 D arctan y1

(2.45)

In this case, system (2.33)–(2.35) takes the form yP1 D 0;

yP2 D 0 at ti ¤ y2 C i ; i D 0; 1; : : : ; q y1 j ti Dy2 Ci 2 .y1 /2 C W 2 y1 ; (2.46) W y2 j ti Dy2 Ci 2 arctan sign.cos.t C y2 // ; W D Œw ; w C ; y1 y1 .0/ D y10 ;

y2 .0/ D y20 :

We now study the behavior of solutions of system (2.46), depending on the restrictions imposed on w and w C . The following three cases are possible: a) w < w C 0; b) w < 0 < w C ; c) 0 w < w C . All conditions of Lemma 1 except condition 3) are obviously satisﬁed. Since .0; 1/T , condition 3) is satisﬁed if w sign.cos.t C y2 // 0 for any w 2 W:

@i @y

D

(2.47)

a1 / w < w C < 0. First, assume that y10 > 0 and 0 < y20 . The solution of system (2.46) before the ﬁrst pulse is given by y1 .t / y10 and y2 .t / y20 . The solution y2 .t / hits the surface 1 W t D y2 C . In this case, we have cos.t C y2 / D 1 < 0, and, hence, y2 < 0 by virtue of (2.45). Thus, “beating” takes place, and the solution cannot cross the surface 1 W t D y2 C (Figure 2).

61

Section 2.3 Examples

Figure 2.

We now show that y1 .t / ! 1 as t ! 1. Assume the contrary, i.e., let there exist a constant C > 0 such that y1 .t / C for any t . Then, for any i, we get y1i

q D .y1 /2 C wi2 y1 D q q

wi2 C 2 C wi2 C C

wi2 .y1 /2 C wi2 C y1

c1 > 0:

Since the “beating” of a solution occurs, we have y1 .t / ! y10 C

1 X

y1i D 1

as t ! 0:

iD1

In other words, we arrive at a contradiction with the assumption that y1 .t / is bounded. It follows from (2.42) and (2.45) that y1 ! 0 and y2 ! 0 as t ! 1. In the case where y10 > 0 and < y20 0, the solution y2 .t / hits the surface 0 W t D y2 , and no beating is observed because inequality (2.47) is satisﬁed. The subsequent behavior is analogous to that in the previous case. a2 / w < w C 0. In this case, there exist solutions with “beating” and without “beating” (corresponding to wi D 0/, i.e., the weak extendability of solutions takes place (Figure 3). b) 0 w < w C . This case is analogous to case a). The solutions cross the ﬁrst impulsive surface, and “beating” occurs on the second surface (Figure 4).

62

Chapter 2 Impulsive Differential Inclusions

Figure 3.

Figure 4.

c) w < 0 < w C . It is obvious that y1 is always positive for w ¤ 0. If y10 0, then y1 > 0 for all t . Since, in this case, the set W contains both positive and negative values of w 2 W , we conclude that, for each impulsive surface, there exist solutions that cross it and solutions with “beating” (Figure 5).

63

Section 2.3 Examples

Figure 5.

Example 5. Consider the following system with multivalued pulses: xR C x D 0 for x ¤ 0; x.0/ D x0 ; x.0/ P D xP 0 ; xjxD0 D 0;

xj P xD0 2 W; W D Œw ; w C ; w ; w C 2 R:

(2.48) (2.49)

By analogy with the previous example, we reduce this inclusion to a system and change variables according to (2.36). As a result, we obtain yP1 D 0;

yP2 D 0;

y1 .0/ D y10 ;

y2 .0/ D y20 ;

(2.50)

where the initial data y10 and y20 satisfy the conditions q

y10 cos y20 D x0 ;

y10 sin y20 D xP 0 ;

i.e., y10 D x02 C xP 02 and y20 is chosen from the interval .; . We now determine the values of the pulses y1 and y2 and the times at which they occur. To this end, we denote the values of y1 and y2 before and after a pulse by y1 ; y2 and y1C ; y2C , respectively. By assumption, there is no pulse in the variable x1 , i.e., x1 D 0 H) y1C cos.ti C y2C / D y1 cos.ti C y2 /:

(2.51)

The relation x2 D wi 2 W yields y1C sin.ti C y2C / C y1 sin.ti C y2 / D wi :

(2.52)

At the time of pulse action, we have x1 D 0. Since y1 > 0, this means that cos.ti Cy2 / D 0, i.e., ti Cy2 D 2 Ci , i D 0; 1; 2; : : : . Furthermore, sin.ti Cy2 / D ˙1. By virtue of (2.51), we conclude that cos.ti C y2C / D 0 .sin.ti C y2C / D ˙1/.

64

Chapter 2 Impulsive Differential Inclusions

We also exclude the cases where the velocity changes its direction at the time of pulse action (these cases are improbable from the viewpoint of the physical interpretation of the problem as the motion of a swing): a1 / x2 > 0; wi 0; x2 jwi j

and

a2 / x2 < 0; wi > 0; jx2 j wi :

If the velocity does not change its direction, then one of the following situations is realized: b1 / x2 > 0; wi 0I

b2 / x2 > 0; wi < 0; x2 > jwi jI

b3 / x2 < 0; wi 0I

b4 / x2 < 0; wi > 0; jx2 j > wi :

For example, consider the case b1 /. Since x2 > 0, we have y1 sin.ti C y2 / > 0 H) sin.ti C y2 / < 0 H) sin.ti C y2 / D 1:

(2.53)

By virtue of (2.52), we obtain y1C sin.ti C y2C / D y1 sin.ti C y2 / wi < 0 H) sin.ti C y2C / < 0 H) sin.ti C y2C / D 1:

(2.54)

Relations (2.53) and (2.54) yield y2 D y2C H) y2 D 0. In cases b2 /, b3 /, and b4 /, we establish by analogy that y2 D y2C H) y2 D 0;

(2.55)

i.e., there is no pulse in the variable y2 . It follows from relations (2.52) and (2.55) that .y1C C y1 / sin.ti C y2˙ / D wi . Since sin.ti C y2˙ / D ˙1 (depending on which of cases b1 /–b4 / is considered), we obtain y1 D y1C y1 D wi sign.sin.ti C y2˙ //:

(2.56)

Thus, the original system is reduced to the form yP1 D 0;

C i ; 2 2 W sign.sin.ti C y2 //;

yP2 D 0 for ti ¤ y2 C

y1 j ti Dy2 C 2 Ci

(2.57)

y2 j ti Dy2 C 2 Ci D 0; y1 .0/ D y10 ;

y2 .0/ D y20 :

Pulses appear at the times ti D i .y2 / D y2 C 2 C i , i D 0; 1; : : : . Since y2 .t / y20 D const by virtue of system (2.57), the times of pulses in y1 are ﬁxed

65

Section 2.3 Examples

the points of intersection of the curve y2 D y20 and the surfaces (curves) ti D y20 C 2 C i . In other words, the system yP1 D 0

C i ; y1 .0/ D y10 ; 2 2 W sign.sin.ti C y20 //

for ti ¤ y20 C

y1 j ti Dy20 C 2 Ci

is a system with pulse action at ﬁxed times. Thus, there is no “beating” in this system, and its solutions are extendable and depend continuously on the initial data and righthand sides.

Chapter 3

Linear Impulsive Differential Inclusions

3.1

Statement of the Problem. Theorem on Existence and Uniqueness

Consider the linear impulsive differential inclusion xP 2 A.t /x C F .t /;

t ¤ i ;

xj tDi 2 Bi x C Pi ;

(3.1) (3.2)

where x 2 Rn is a phase vector, t 2 I D Œt0 ; T is time, T C1, AW I ! comp.Rnn / is a measurable set-valued mapping, jA.t /j ˛.t /, F W I ! comp.Rn / is a measurable set-valued mapping, jF .t /j .t /, the functions ˛.t / and .t / are measurable and summable on any ﬁnite segment I I , i 2 I , i D 1; 2; : : : , are ﬁxed times numbered in the increasing order .i < iC1 /, the set ¹i º does not have points of accumulation, Bi are compact sets of n n matrices, and Pi 2 comp.Rn /. Deﬁnition 1. A function xW I ! Rn is called a solution of inclusion (3.1), (3.2) if it is absolutely continuous, satisﬁes (3.1) almost everywhere on intervals that do not contain i , and has discontinuities of the ﬁrst kind at points t D i with jumps x.i / satisfying inclusion (3.2). Deﬁnition 2. A set-valued function RW I ! comp.Rn / is called an R-solution generated by the impulsive differential inclusion (3.1), (3.2) if R.t / is absolutely continuous on intervals that do not contain i , for almost all t ¤ i one has ³ Z tC [ ² 1 lim h R.t C /; .A.s/x C F .s//ds D 0; xC #0 t x2R.t/

and, for t D i , the function R.t / satisﬁes the condition of jump [ ¹x C Bi x C Pi º: R.i C 0/ D

(3.3)

x2R.i /

Let A.t / 2 A.t /, let Bi 2 Bi , and let ˆABi .t; t0 / be the matrizant of the system xP D A.t /x; xj tDi D Bi x:

t ¤ i ;

Section 3.1 Statement of the Problem. Theorem on Existence and Uniqueness

67

Using the formula for a solution of an impulsive differential equation obtained in Chapter 1, we represent an arbitrary solution of inclusion (3.1), (3.2) with the initial condition x.t0 / D x0 for t 2 I in the form x.t; x0 / D ˆABi .t; t0 /x0 Z t X C ˆABi .t; /f . /d C ˆABi .t; i /pi ; t0

t0 i
where f .t / is a measurable branch of the mapping F .t / and pi 2 Pi . The integral funnel X.t; X0 /, X.t0 ; X0 / D X0 , of inclusion (3.1), (3.2) (bundle of solutions) is determined by the relation Z t [ ² X.t; X0 / D ˆABi .t; /F . / d ˆABi .t; t0 /X0 C t0

A.t/2A.t / Bi 2Bi

C

X t0 i
³ ˆABi .t; i /Pi :

Lemma 1. The bundle of solutions ³ Z t [ ² ˆA .t; s/F .s/ ds X.t; X0 / D ˆA .t; t0 /X0 C

(3.4)

(3.5)

t0

A.t/2A.t/

of the linear inhomogeneous differential inclusion xP 2 A.t /x C F .t /

(3.6)

is a nonempty compact set for any ﬁxed t 2 I . Proof. The set X.t; X0 / is nonempty because, by virtue of the Filippov theorem [49], there exist summable branches of the set-valued mappings A.t / and F .t /, and, furthermore, the set X0 is nonempty. Any element x 2 X.t; x0 / can be represented in the form Z x D x.t; x0 / D ˆA .t; t0 /x0 C

t

t0

ˆA .t; s/f .s/ds;

where x0 2 X0 , A.s/ 2 A.s/, and f .s/ 2 F .s/. Therefore, Z t Rt Rt kxk e t0 ˛.s/ds jX0 j C .s/e s ˛./d ds D K < 1;

(3.7)

t0

i.e., the set X.t; x0 / is bounded. We now prove its closedness, i.e., we prove that the limit of any convergent sequence of points xk 2 X.t; X0 / also belongs to the set X.t; X0 /.

68

Chapter 3 Linear Impulsive Differential Inclusions

Since xk 2 X.t; X0 /, there exist x0k 2 X0 , Ak .s/ 2 A.s/, and fk .s/ 2 F .s/ such that xk is the value of a solution of the Cauchy problem xP D Ak .s/x C fk .s/;

x.t0 / D x0k

at time t . By virtue of the equivalence of a differential equation and a Volterra integral equation, the following representation is true: Z t Z t k Ak .s/xk .s/ds C fk .s/ds: (3.8) xk D xk .t / D x0 C t0

t0

By analogy with (3.7), we get kxk .s/k K. Since Z s2 kxk .s2 / xk .s1 /k D ŒAk .s/xk .s/ C fk .s/ds ˇZ ˇ ˇˇ

s1 s2

s1

Z

where '. / D

ˇ ˇ Œk˛.s/ C .s/ds ˇˇ D j'.s2 / '.s1 /j; t0

Œk˛.s/ C .s/ds

is a function absolutely continuous and, hence, uniformly continuous on Œt0 ; t , for any " > 0 there exists ı."/ > 0 such that, for any s1 , s2 2 I W js2 s1 j < ı, one has j'.s2 / '.s1 /j <

" ; K

whence kxk .s2 / xk .s1 /k < ": Thus, the sequence of functions xk .s/ is uniformly bounded and equicontinuous on Œt0 ; t . Therefore, by virtue of the Arzelà theorem, one can choose its subsequence that converges uniformly to a continuous function x .s/. This means that, for any " > 0, there exists k0 such that the following inequality holds for all k > k0 and s 2 Œt0 ; t : kxk .s/ x .s/k < where

Z .t; s/ D

Since Z

t

t0

Z Ak .s/xk .s/ds D

t t0

" ; .t; t0 /

t

˛./d: s

Z Ak .s/Œxk .s/ x .s/ds C

t t0

Ak .s/x .s/ds;

Section 3.1 Statement of the Problem. Theorem on Existence and Uniqueness

where

69

Z t Z t Ak .s/Œxk .s/ x .s/ds ˛.s/kxk .s/ x .s/kds < "; t0

t0

and, by virtue of the Lyapunov theorem [78], there exists a subsequence Ak1 .s/ of the sequence Ak .s/ that converges weakly to a matrix A .s/ 2 A.s/ on Œt0 ; t , we conclude that Z t Z t Ak1 .s/xk1 .s/ds ! A .s/x .s/ds as k1 ! 1: t0

t0

Furthermore, x0k1 2 X0 2 comp.Rn /. Hence, there exists a subsequence ¹x0k2 º that converges to a certain vector x0 2 X0 . Moreover, by virtue of the Lyapunov theorem, there exists a subsequence of the sequence fk2 .s/ that converges weakly to f .s/ 2 F .s/ on Œt0 ; t . Passing to the limit in (3.8), we obtain Z t Z t x D x .t / D x0 C A .s/x .s/ds C f .s/ds: t0

t0

In other words, x is the value of a solution of the differential inclusion (3.6) at time t , i.e., x 2 X.t; X0 /. Thus, the compactness of the set X.t; X0 / is proved. By virtue of Lemma 1 and the compactness of the sets Bi and Pi , the integral funnel X.t; X0 / of inclusion (3.1), (3.2) is a compact set for every ﬁxed t 2 I . Moreover, since the right-hand side of inclusion (3.1) possesses the Lipschitz property with respect to x, with regard for [101] and the condition of jump (3.3) we conclude that the set X.t; X0 / is the unique R-solution of inclusion (3.1), (3.2). Any ﬁnite interval Œt0 ; T ; T T , contains ﬁnitely many points i . By virtue of Theorem 6 in Chapter 1, the solutions x.t; x0 /, x.t0 ; x0 / D x0 , of inclusion (3.1), (3.2) exist for all t 2 Œt0 ; T and any x0 2 Rn . Therefore, for any X0 2 comp.Rn /, the R-solution X.t; X0 /, X.t0 ; X0 / D X0 , exists for t 2 Œt0 ; T . It is obvious that, in the case T D C1, the solution x.t; x0 / and the R-solution X.t; x0 / can be extended to the inﬁnite interval. Note that, in this case, R-solutions that start from different points can coincide at a certain moment of time. Example 1. Consider the impulsive differential inclusion xP 2 Œ0; 1;

t ¤ 1;

xj t D1 2 ¹2; 0ºx;

x.0/ D x0 :

Then X.1; x0 / D Œx0 ; x0 C 1, and, for t D 1 C 0, we have X.1 C 0; x0 / D Œx0 ; x0 C 1 [ Œ1 x0 ; x0 :

70

Chapter 3 Linear Impulsive Differential Inclusions

Hence, X.1 C 0; 0/ D X.1 C 0; 1/ D Œ1; 1: We now consider the question of the possibility of coincidence of R-solutions that start from different points for linear impulsive differential inclusions in the special case xP 2 A.t /x C F .t /;

t ¤ i ;

(3.9)

xj tDi 2 Bi x C Pi ;

(3.10)

where A.t / is an n n matrix measurable on the interval I , and Bi are constant n n matrices. Theorem 1 ([117]). If the matrices E C Bi are nondegenerate for all i such that i 2 I , then X.t; x0 / ¤ X.t; y0 / for all t 2 I whenever x0 ¤ y0 , where X.t; x0 / and X.t; y0 / are R-solutions of inclusion (3.9), (3.10). Proof. The sets X.t; x0 / and X.t; y0 / are nonempty compact sets. For t 2 .i ; iC1 , the following representation is true: Z t X.t; x0 / D ˆABi .t; i C 0/X.i C 0; x0 / C ˆABi .t; /F . /d : (3.11) i C0

Therefore, by virtue of the nondegeneracy of the matrizant, we have co X.t; x0 / ¤ co X.t; y0 /

for all i < t iC1 ; i D 1; 2; : : : ;

provided that co X.i C 0; x0 / ¤ co X.i C 0; y0 /. Indeed, assume the contrary, i.e., assume that the following equality holds for some t 2 .i ; iC1 : co X.t; x0 / D co X.t; y0 /: By virtue of a property of support functions [23], for all

2 Rn we have

c.co X.t; x0 /; / D c.co X.t; y0 /; / ) c.X.t; x0 /; / D c.X.t; y0 /; / Z ) c ˆABi .t; i C 0/X.i C 0; x0 / C

t

i C0

Z D c ˆABi .t; i C 0/X.i C 0; y0 / C

ˆABi .t; /F . /d ;

i C0

) c.ˆABi .t; i C 0/X.i C 0; x0 /; / D c.ˆABi .t; i C 0/X.i C 0; y0 /; / T .t; i C 0/ / ) c.X.i C 0; x0 /; ˆAB i T D c.X.i C 0; y0 /; ˆAB .t; i C 0/ /: i

t

ˆABi .t; /F . /d ;

Section 3.1 Statement of the Problem. Theorem on Existence and Uniqueness

71

Since the matrices ˆABi .t; i C 0/ are nondegenerate, the set of vectors T .t; C0/ ; ¹ˆAB 2 Rn º coincides with Rn , whence co X.i C0; x0 / D co X.i C0; i i y0 /, which contradicts the condition. We now show that the equality co X.i C 0; x0 / D co X.i C 0; y0 / yields co X.i ; x0 / D co X.i ; y0 /. Indeed, by virtue of a property of support functions [23], for all 2 Rn we get c.co X.i C 0; x0 /; / D c.co X.i C 0; y0 /; / ) c.X.i C 0; x0 /; / D c.X.i C 0; y0 /; /; i.e., c..E C Bi /X.i ; x0 / C Pi ; / D c..E C Bi /X.i ; y0 / C Pi ; / ) c..E C Bi /X.i ; x0 /; / D c..E C Bi /X.i ; y0 /; / ) c.X.i ; x0 /; .E C Bi /T / D c.X.i ; y0 /; .E C Bi /T /: Since the matrices E C Bi are nondegenerate, the set of vectors ¹.E C Bi /T ; 2 Rn º coincides with Rn , whence co X.i ; x0 / D co X.i ; y0 /: Thus, we get co X.t0 ; x0 / D co X.t0 ; y0 /, i.e., x0 D y0 , whenever X.t; x0 / D X.t; y0 / for some t 2 I . Remark 1. If the initial condition takes the form X.t0 / D X0 2 comp.Rn /; then we can only prove that the equality X.t; X0 / D X.t; Y0 / for some t 2 I yields co X0 D co Y0 , i.e., the uniqueness may be violated. Example 2. Consider the differential inclusion xP D 0;

t ¤ 1;

xj tD1 2 S1 .0/: Then the solutions X.t; S1 .0// and X.t; B1 .0// that start from the sets S1 .0/ and B1 .0/, respectively, coincide at t D 1C 0 because S1 .0/C S1 .0/ D B1 .0/C S1 .0/ D B2 .0/.

72

Chapter 3 Linear Impulsive Differential Inclusions

Remark 2. Separate solutions x.t; x0 / and x.t; y0 / for x0 ¤ y0 may coincide for t t1 , where t1 2 I . Remark 3. In what follows, we assume that the matrices E C Bi are nondegenerate for all Bi 2 Bi , i D 1; 2; : : : . Remark 4. If Pi 2 conv.Rn / in (3.10), then X.t; x0 / 2 conv.Rn /. It should be noted that, in this case, the set of R-solutions of the impulsive differential inclusion (3.9), (3.10) is not a linear n-dimensional space because the space conv.Rn / is not linear. Of interest is the following special case: Let x0 2 X0 . Assume that the set X0 has a ﬁnite number m of corner points (i.e., it is a polyhedron). Then any point x0 2 X0 can be represented as a convex combination of corner points xi , i D 1; m. Consequently, any bundle of solutions X.t; x0 / can be represented as a convex combination of “basic bundles” X.t; xi /, i.e., X.t; x0 / D

m X

i X.t; xi /;

iD1

3.2

where i 0;

m X

i D 1;

iD1

m X

i xi D x0 :

iD1

Stability of Solutions of Linear Impulsive Differential Inclusions

By analogy with various notions of stability of solutions and R-solutions for differential inclusions (see [23] and [107]), we introduce analogous notions for impulsive differential inclusions. Deﬁnition 3. An R-solution R.t / .t0 t < C1/ of the impulsive differential inclusion (3.1), (3.2) is called stable in the sense of Lyapunov if, for any " > 0, one can ﬁnd ı."/ > 0 such that the following conditions are satisﬁed: (1) all R-solutions X.t / of inclusion (3.1), (3.2) that satisfy the condition h.X.t0 /; R.t0 // < ı

(3.12)

are deﬁned for all t t0 ; (2) for all solutions satisfying inequality (3.12), the following relation is true: h.X.t /; R.t // < ": Deﬁnition 4. An R-solution R.t / .t0 t < C1/ of the impulsive differential inclusion (3.1), (3.2) is called asymptotically stable if the following conditions are satisﬁed:

Section 3.2 Stability of Solutions of Linear Impulsive Differential Inclusions

73

(1) it is stable in the sense of Lyapunov; (2) it satisﬁes the following condition: lim h.X.t /; R.t // D 0:

t!1

Deﬁnition 5. A solution .t / .t0 t < C1/ of the impulsive differential inclusion (3.1), (3.2) is called stable if, for every " > 0, there exists ı."/ > 0 such that, for every xQ 0 such that kxQ 0 .t0 /k < ı, every solution x.t Q / with the initial condition x.t Q 0 / D xQ 0 exists and satisﬁes the following inequality for t0 t < C1: kx.t Q /

.t /k < ":

Deﬁnition 6. A solution .t / .t0 t < C1/ of the impulsive differential inclusion (3.1), (3.2) is called weakly stable if, for every " > 0, there exists ı."/ > 0 such that, Q / with the initial condition for every xQ 0 such that kxQ 0 .t0 /k < , some solution x.t x.t Q 0 / D xQ 0 exists and satisﬁes the following inequality for t0 t < C1: kx.t Q /

.t /k < ":

Deﬁnition 7. A solution .t / .t0 t < C1/ of the impulsive differential inclusion (3.1), (3.2) is called asymptotically stable if the following conditions are satisﬁed: (1) it is stable; (2) it satisﬁes the following condition: lim kx.t Q /

t!1

.t /k D 0:

Deﬁnition 8. A solution .t / .t0 t < C1/ of the impulsive differential inclusion (3.1), (3.2) is called weakly asymptotically stable if the following conditions are satisﬁed: (1) it is weakly stable; (2) it satisﬁes the following condition: lim kx.t Q /

t!1

.t /k D 0:

We now study the problem of stability of solutions of linear homogeneous impulsive inclusions of the form xP 2 A.t /x;

t ¤ i ;

xj tDi 2 Bi x;

(3.13)

where A.t / is a compact set of n n matrices measurable on Œt0 ; C1/, Bi are compact sets of n n matrices, and the times of pulse action i ! C1 as i ! 1.

74

Chapter 3 Linear Impulsive Differential Inclusions

Theorem 2 ([117]). For the linear homogeneous impulsive differential inclusion (3.13), the following assertions are true: (a) for the asymptotic stability of a solution x.t; x0 /, it is necessary and sufﬁcient that the matrizants ˆABi .t; t0 / satisfy the condition lim ˆABi .t; t0 / D 0

t !1

(3.14)

uniformly in all A.t / 2 A.t / and Bi 2 Bi , i D 1; 2; : : : ; (b) for the stability of the trivial solution, it is necessary and sufﬁcient that the matrizants ˆABi .t; t0 / be uniformly bounded for t t0 (i.e., there should exist a constant M > 0 such that kˆABi .t; t0 /k M for t t0 and all A.t / 2 A.t / and Bi 2 Bi , i D 1; 2; : : :/; (c) for the weak stability of a nontrivial solution x.t; x0 /, it is sufﬁcient that this solution be bounded for t t0 ; (d) for the weak stability of the trivial solution, it is sufﬁcient that there exist at least one matrizant ˆABi .t; t0 / bounded for t t0 ; (e) for the weak asymptotic stability of a nontrivial solution x.t; x0 /, it is sufﬁcient that lim x.t; x0 / D 0I t!1

(f) for the weak asymptotic stability of the trivial solution, it is sufﬁcient that there exist at least one matrizant ˆABi .t; t0 / ! 0 as t ! 1. Proof. (a) If condition (3.14) is satisﬁed, then lim kx.t; x0 / x.t; y0 /k

t!1

D lim kˆA1 B 1 .t; t0 /x0 ˆA2 B 2 .t; t0 /y0 k D 0; t !1

i

i

(3.15)

i.e., the solution is asymptotically stable. If condition (3.15) is satisﬁed for any y0 , then relation (3.14) is true. (c) Assume that a nontrivial solution x.t; x0 / is bounded for t t0 by a constant M0 and corresponds to the matrices A1 .t / 2 A.t / and Bi1 2 Bi . Choosing the matrices A.t / D A1 .t / and Bi D Bi1 for the solution that starts at the point y0 D ˛ x0 , j˛ 1j < ı, we obtain kx.t; x0 / x.t; y0 /k D kˆA1 B 1 .t; t0 /x0 ˆA1 B 1 .t; t0 /˛x0 k < ıM0 D " i

i

for all t t0 whenever ı D M"0 . This means that the solution x.t; x0 / is weakly stable. Moreover, it can be unstable.

Section 3.2 Stability of Solutions of Linear Impulsive Differential Inclusions

75

Indeed, the difference of solutions corresponding to different A.t / and Bi may not be small even for y0 D x0 : kx 1 .t; x0 / x 2 .t; x0 /k D kˆA1 B 1 .t; t0 /x0 ˆA2 B 2 .t; t0 /x0 k i

i

D k.ˆA1 B 1 .t; t0 / ˆA2 B 2 .t; t0 //x0 k: i

i

For example, if we consider the differential inclusion with zero pulses xP 2 Œ1; 0x;

x.0/ D x0 ¤ 0

and take A1 .t / 0 and A2 .t / 1, then we obtain ˆA1 ;0 .t; 0/ 1;

ˆA2 ;0 .t; 0/ D e t :

Then kx 1 .t; x0 / x 2 .t; x0 /k D kx0 k.1 e t / is an increasing function tending to kx0 k as t ! 1. (e) Let lim x.t; x0 / D lim ˆABi .t; t0 /x0 D 0: t!1

t!1

Consider the solutions x.t; y0 / D ˆABi .t; t0 /y0 ;

where y0 D ıx0 :

Then lim Œx.t; x0 / x.t; y0 / D lim ŒˆABi .t; t0 /x0 ˆABi .t; t0 /ıx0 D 0;

t!1

t!1

i.e., the solution x.t; x0 / is weakly asymptotically stable. Assertions (b), (d), and (f) are proved by analogy. Theorem 3 ([117]). For the linear homogeneous impulsive differential inclusion (3.13), the following assertions are true: (a) for the stability of an R-solution X.t; x0 /, it is necessary and sufﬁcient that the matrizants ˆABi .t; t0 / be uniformly bounded for t t0 ; (b) for the asymptotic stability of an R-solution, it is necessary and sufﬁcient that the matrizants ˆABi .t; t0 / uniformly satisfy condition (3.14); (c) for the instability of an R-solution, it is necessary and sufﬁcient that the matrizants ˆABi .t; t0 / be not uniformly bounded for t t0 .

76

Chapter 3 Linear Impulsive Differential Inclusions

Proof. (a) We prove the sufﬁciency. Let X.t; x0 / be an R-solution of inclusion (3.13) and let there exist a constant M0 > 0 such that kˆA;Bi .t; t0 /k M0 for t t0 and all A.t / 2 A.t / and Bi 2 Bi . Then h.X.t; x0 /; X.t; y0 // [ Dh ˆABi .t; t0 /x0 ; A.t/2A.t / Bi 2Bi

D

[

ˆABi .t; t0 /y0

A.t/2A.t / Bi 2Bi

sup ¹d1 .t /; d2 .t /º;

A.t/2A.t / Bi 2Bi

where

d1 .t / D ˆABi .t; t0 /x0 ;

[

ˆABi .t; t0 /y0

A.t/2A.t / Bi 2Bi

kˆABi .t; t0 /x0 ˆABi .t; t0 /y0 k M0 kx0 y0 k; [ d2 .t / D ˆABi .t; t0 /y0 ; ˆABi .t; t0 /x0 A.t/2A.t / Bi 2Bi

kˆABi .t; t0 /x0 ˆABi .t; t0 /y0 k M0 kx0 y0 k: Thus, h.X.t; x0 /; X.t; y0 // M0 kx0 y0 k < " for kx0 y0 k < ı, ı D M"0 , and t t0 . Hence, the R-solution X.t; x0 / is stable. Let us prove the necessity. Assume that the R-solution X.t; x0 / is stable, i.e., for any " > 0, there exists ı > 0 such that, for kx0 y0 k < ı, one has h.X.t; x0 /; X.t; y0 // [ Dh ˆABi .t; t0 /x0 ; A.t/2A.t / Bi 2Bi

[

ˆABi .t; t0 /y0

< ":

A.t/2A.t / Bi 2Bi

We ﬁx " > 0, select the corresponding ı > 0, and take y0 D ˛x0 , where j1 ˛jkx0 k < ı. Then [ [ h ˆABi .t; t0 /x0 ; ˆABi .t; t0 /y0 A.t/2A.t / Bi 2Bi

Dh

[ A.t/2A.t / Bi 2Bi

A.t/2A.t / Bi 2Bi

ˆABi .t; t0 /x0 ; ˛

[ A.t/2A.t / Bi 2Bi

ˆABi .t; t0 /x0

77

Section 3.2 Stability of Solutions of Linear Impulsive Differential Inclusions

ˇ ˇ D max ˇˇc 2S1 .0/

[

ˆABi .t; t0 /x0 ;

A.t/2A.t / Bi 2Bi

ˇ ˇ D j1 ˛j max ˇˇc 2S .0/ 1

D j1 ˛j h

ˇ ˇ ˇ ˆABi .t; t0 /x0 ; ˇ

[

˛ c

A.t/2A.t / Bi 2Bi

ˇ ˇ ˇ ˆABi .t; t0 /x0 ; ˇ

[ A.t/2A.t / Bi 2Bi

ˆABi .t; t0 /x0 ; ¹0º < ":

[ A.t/2A.t / Bi 2Bi

Hence,

[

h

ˆABi .t; t0 /x0 ; ¹0º <

A.t/2A.t / Bi 2Bi

" D M: j1 ˛j

Thus, the R-solution X.t; x0 / is bounded. Now assume that the matrizants ˆABi .t; t0 / are not uniformly bounded. Then, for any sequence Mk ! 1, k ! 1, there exists a matrizant ˆAk B k .t; t0 / such that i

kˆAk B k .t k ; t0 /k > Mk : i

Then there exists at least one element 'j k l k .t k ; t0 / of the matrix ˆAk B k .t k ; t0 / i

such that j'j k l k .t k ; t0 /j > Lk , where Lk ! 1 as k ! 1. By deﬁnition, we have h

[

ˆABi .t; t0 /x0 ;

A.t/2A.t / Bi 2Bi

D

[

ˆABi .t; t0 /y0

A.t/2A.t / Bi 2Bi

sup ¹d1 .t /; d2 .t /º;

A.t/2A.t / Bi 2Bi

where

d2 .t / D ˆABi .t; t0 /y0 ;

[

ˆABi .t; t0 /x0 :

A.t/2A.t / Bi 2Bi

As the matrices A.t / and Bi , we take the matrices Ak .t / and Bik , respectively. We also choose the initial value y0k as follows: k D x01 ; y01

:::;

ı k ; y0l k D x0l k C 2

:::;

k y0n D x0n :

78

Chapter 3 Linear Impulsive Differential Inclusions

In this case, we have k.ˆAk B k .t k ; t0 /y0k /j k i

k D k.ˆAk B k .t k ; t0 /x0 /j C 'j k l k .t k ; t0 /.y0l k x0l k /k i

k k j'j k l k .t k ; t0 /j jy0l k x0l k j k.ˆAk B k .t ; t0 /x0 /j k > Lk i

ı M: 2

Hence, h.X.t; x0 /; X.t; y0 //j tDtk

sup Ak .t /2A.t / B k 2Bi i

d2 .t k / D 1:

We arrive at a contradiction with the assertion that the R-solution X.t; x0 / is stable. Thus, we have shown that the requirement of the uniform boundedness of the matrizants ˆABi .t; t0 / is necessary and sufﬁcient for the stability of the R-solution X.t; x0 /. Assertion (c) of the theorem follows automatically from (a). The criterion for the asymptotic stability of an R-solution is proved by analogy. We now illustrate the obtained results by several examples. Example 3. Consider a linear impulsive differential inclusion of the form xP 2 Œ˛; ˇx;

t ¤ i; i 2 N; x.0/ D x0 0;

(3.16)

xj tDi 2 Œ0:1I 0x; where ˛ < ˇ are arbitrary constants. The matrizant corresponding to a measurable function a.t / 2 Œ˛; ˇ and a sequence bi 2 Œ0:1I 0 takes the form ˆa.t/;bi .t; t0 / D e

Rt 0

a.s/ ds

Y

.1 C bi /:

0
It is obvious that 0:9Œt e ˛t ˆa.t/;bi .t; t0 / e ˇ t . Also note that 0:9t e ˛t 0:9Œt e ˛t < 0:9t1 e ˛t . We now consider the problem of stability of the trivial solution x.t; 0/ 0. The trivial solution is asymptotically stable if and only if ˇ < 0. The trivial solution is stable if and only if ˇ 0. In order that the trivial solution (3.16) be weakly stable, it is sufﬁcient that 0:9e ˛ 1, i.e., ˛ ln 10 9 . For the weak asymptotic stability of the trivial solution, it is sufﬁcient that ˛ < ln 10 9 . The trivial R-solution is stable if and only if ˇ 0, asymptotically stable if and only if ˇ < 0, and unstable if and only if ˇ > 0.

79

Section 3.2 Stability of Solutions of Linear Impulsive Differential Inclusions

Example 4. Consider the impulsive differential inclusion xP 2 Ax;

t ¤ i; i 2 N;

xj tDi 2 Bi x; where x 2 R2 ; x.0/ D x0 2 R2 ; ² ³ 0 a ; a 2 Œa1 ; a2 ; AD a 0 ² ³ 0 0 Bi D ; bi 2 Œ1; 0 : 0 bi The matrizant corresponding to the matrices A and Bi has the form ˆABi .t; 0/ D e A.tk/

k Y

.E C Bi /e A :

iD1

For 0 t 1, we have ˆABi .t; 0/ D e

At

D

cos at sin at

!

sin at cos at

and kˆABi .t; 0/k1 D

p

2;

v uX u n kM k1 D t jmij j2 :

where

i;j D1

We now show that the function kˆABi .t; 0/k1 is nonincreasing for t 0. Assume that k < t k C 1. Let x11 x12 ˆABi .k C 0; 0/ D : x21 x22 Then ˆABi .t; 0/ D D

cos a.t k/ sin a.t k/ sin a.t k/ cos a.t k/

x11 x12 x21 x22

x11 cos a.t k/ C x21 sin a.t k/ x12 cos a.t k/ C x22 sin a.t k/ ; x11 sin a.t k/ C x21 cos a.t k/ x12 sin a.t k/ C x22 cos a.t k/

2 cos2 a.t k/ C 2x11 x21 sin a.t k/ cos a.t k/ kˆABi .t; 0/k21 D x11 2 2 C x21 sin2 a.t k/ C x12 cos2 a.t k/ 2 C 2x12 x22 sin a.t k/ cos a.t k/ C x22 sin2 a.t k/ 2 C x11 sin2 a.t k/ 2x11 x21 sin a.t k/ cos a.t k/ 2 2 C x21 cos2 a.t k/ C x12 sin2 a.t k/ 2 2x12 x22 sin a.t k/ cos a.t k/ C x22 cos2 a.t k/ 2 2 2 2 D x11 C x12 C x21 C x22 D kˆABi .k C 0; 0/k21 :

80

Chapter 3 Linear Impulsive Differential Inclusions

Thus, the function kˆABi .t; 0/k1 is constant on the semi-segments .k; k C 1. We determine its jump at the time of pulse action t D k. Let ! x12 x11 ˆABi .k; 0/ D : x x21 22 Then ˆABi .k C 0; 0/ D D

1 0 0 1 C bk

!

x x11 12 x x21 22

!

! x12 x11 ; .1 C b /x .1 C bk /x21 k 22

2 2 2 2 kˆABi .k C 0; 0/k21 D x11 C x12 C .1 C bk /2 x21 C .1 C bk /2 x22 2 2 2 2 2 2 D x11 C x12 C x21 C x22 C bk .2 C bk /.x21 C x22 /

kˆABi .k 0; 0/k21 because bk .2 C bk / 0. p Thus, the function kˆABi .t; 0/k1 is nonincreasing. Hence, kˆABi .t; 0/k1 2 for all matrices A 2 A and Bi 2 Bi , i D 1; 2; : : : , and all t 0. By virtue of the equivalence of the matrix norms k k and k k1 , there exist constants 0 < ˛ < ˇ < 1 such that ˛kM k1 kM k ˇkM k1 for any matrix M (Theoremp6.1.2 [77]). Thus, kˆABi .t; 0/k ˇ 2. Hence, all solutions of the original impulsive differential inclusion are weakly stable because they are bounded by virtue of the equality x.t; 0/ D ˆABi .t; 0/x0 . The jump of the function kˆABi .t; 0/k21 at t D k is equal to 2 2 C x22 /: k D bk .2 C bk /.x21

P Thus, if there exist a and bk such that 1 kD1 k D 2, then the solution corresponding to these A and Bk is weakly asymptotically stable because kˆABi .t; 0/k1 ! 0 as t ! 1. Consequently, x.t; 0/ ! 0 as t ! 1. Assume that, in the differential inclusion (3.13), we have A.t / D A C A0 .t /; where A and Bi are constant matrices.

Bi D Bi C Bi0 ;

Section 3.2 Stability of Solutions of Linear Impulsive Differential Inclusions

81

Then the differential inclusion (3.13) can be rewritten in the form xP 2 Ax C A0 .t /x

t ¤ i ;

xj t Di 2 Bi x C Bi0 x:

(3.17)

Consider the system of linear impulsive differential equations xP D Ax;

t ¤ i ;

xj t Di D Bi x:

(3.18)

Theorem 4 ([117]). If the solutions of the system of equations (3.18) are stable and there exist A0 .t / 2 A0 .t / and Bi0 2 Bi0 such that Z

1 t0

Y

kA0 .t /kdt < 1;

i t0

.1 C kBi0 k/ < 1;

(3.19)

then the solutions of the differential inclusion (3.17) that correspond to the matrices A0 .t / and Bi0 are weakly stable. Proof. In Theorem 11 of Chapter 1, we have considered the problem of stability of solutions of an impulsive differential equation of the type (3.17) under the condition of stability of solutions of an impulsive differential equation of the type (3.18), where Bi B. It follows from the proof of this theorem that the last requirement can be omitted. Thus, the stability of solutions of the system of equations (3.18) under conditions (3.19) yields the stability of solutions of the system xP D Ax C A0 .t /x

t ¤ i ;

xj tDi D Bi x C Bi0 x:

Thus, these solutions are weakly stable solutions of inclusion (3.17). Remark 5. The solutions of inclusion (3.17) can be unstable even if condition (3.19) is satisﬁed for all matrices A0 .t / 2 A0 .t / and Bi0 2 Bi0 . Example 5. Consider the linear homogeneous differential inclusion with zero pulses °a ± (3.20) xP 2 A0 .t /x; where A0 .t / D 2 ; a 2 Œ1; 2 ; x.1/ D x0 ¤ 0: t The solution of the truncated system x.t; x0 / x0 is stable in the sense of Lyapunov in the positive direction. Condition (3.19) is satisﬁed for all A0 .t / 2 A0 .t /. In this case, solution (3.20) that corresponds to the matrix A01 .t / D t12 is not stable because, by choosing y0 D x0 and A02 .t / D t22 , we obtain kx 1 .t; x0 / x 2 .t; x0 /k D kˆA0 ;0 .t; t0 /x0 ˆA0 ;0 .t; t0 /x0 k 1

D je

1 1t

2

e

2 2t

j kx0 k ! e.e 1/kx0 k ¤ 0 as t ! 1:

82

Chapter 3 Linear Impulsive Differential Inclusions

Theorem 5 ([117]). If the solutions of the system of equations (3.18) are stable, then an R-solutions of inclusion (3.17) are also stable under the condition that Z

1 t0

Y

jA0 .t /j dt < 1;

i t0

.1 C jBi0 j/ < 1;

(3.21)

where jA0 .t /j D

max

A0 .t/2A0 .t/

kA0 .t /k and

jBi0 j D max kBi0 k: Bi0 2Bi0

Proof. Let X.t; x0 / be an R-solution of (3.17). Then ² Z t ˆABi .t; /A0 . /x.; x0 / d X.t; x0 / D x.t; x0 / D ˆABi .t; t0 /x0 C t0

X

C

t0 i
ˆABi .t; i /Bi0 x.i ; x0 /;

³ 0

A .t / 2 A

0

.t /; Bi0

2

Bi0

Using the estimates presented in Theorem 11 of Chapter 1, we get Z t ˆABi .t; /A0 . /x.; x0 /d d1 .t / D ˆABi .t; t0 /x0 C C

t0

X t0 i

ˆABi .t; i /Bi0 x.i ; x0 /; X.t; y0 /

Z t ˆ .t; t /x C ˆABi .t; /A0 . /x.; x0 / d 0 0 AB i t0

X

C

t0 i
ˆABi .t; i /Bi0 x.i ; x0 /

Z t ˆABi .t; /A0 . /x.; y0 / d ˆABi .t; t0 /y0 C C

X t0 i
t0

Kkx0 y0 k;

ˆABi .t; i /Bi0 x.i ; y0 /

Z t ˆABi .t; /A0 . /x.; y0 / d d2 .t / D ˆABi .t; t0 /y0 C C

X t0 i
t0

ˆABi .t; i /Bi0 x.i ; y0 /; X.t; x0 /

:

Section 3.2 Stability of Solutions of Linear Impulsive Differential Inclusions

83

Z t ˆ .t; t /y C ˆABi .t; /A0 . /x.; y0 / d 0 0 ABi C

t0

X t0 i
ˆABi .t; i /Bi0 x.i ; y0 /

Z t ˆABi .t; t0 /x0 C ˆABi .t; /A0 . /x.; x0 / d t0

X

C

t0 i
where KD

Y

ˆABi .t; i /Bi0 x.i ; x0 /

.1 C

i t0

jBi0 j/

M e

M

R1 t0

Kkx0 y0 k;

jA0 .t/j dt

and M is a positive constant such that kˆABi .t; t0 /k M . Thus, h.X.t; x0 /; X.t; y0 // D

sup A0 .t/2A0 .t/;Bi0 2Bi0

for kx0 y0 k < ı, ı D stable.

" K.

¹d1 .t /; d2 .t /º Kkx0 y0 k < "

Hence, the R-solution X.t; x0 / of inclusion (3.17) is

Remark 6. Note that condition (3.21) implies that condition (3.19) is satisﬁed for all A0 .t / and Bi0 . We now examine the relationship between the stability of solutions of the linear homogeneous impulsive inclusion (3.13) and the stability of solutions of the linear inhomogeneous impulsive differential inclusion xP 2 A.t /x C F .t /;

t ¤ i ;

xj tDi 2 Bi x C Pi ;

(3.22) (3.23)

where F W Œt0 ; C1/ ! comp.Rn / is a measurable set-valued mapping and Pi 2 comp.Rn /. Theorem 6. If the solutions of (3.13) are stable, then the solutions of (3.22), (3.23) are weakly stable. Proof. Let x.t; x0 / D ˆABi .t; t0 /x0 Z t X C ˆABi .t; /f . / d C ˆABi .t; i /pi t0

t0 i
84

Chapter 3 Linear Impulsive Differential Inclusions

be a solution of (3.22), (3.23). By virtue of a criterion for the stability of solutions of (3.13), there exists the constant M D sup kˆABi .t; t0 /k: tt0

We set ky0 x0 k < ı D

" : M

Consider x.t; y0 / D ˆABi .t; t0 /y0 Z t X C ˆABi .t; /f . / d C ˆABi .t; i /pi : t0

t0 i
Then kx.t; x0 / x.t; y0 /k D kˆABi .t; t0 /.x0 y0 /k kˆABi .t; t0 /k kx0 y0 k M kx0 y0 k < "; i.e., x.t; t0 / is weakly stable. Note that the stability of solutions of inclusion (3.13) does not yield the stability of solutions of (3.22), (3.23) in the general case. Consider the inclusion with zero pulses xP 2 Œ0; 1:

(3.24)

A solution of the homogeneous system x.t; x0 / x0 is stable. In this case, the solution of (3.24) that corresponds to f .t / 0 2 Œ0; 1 is not stable because if we choose, say, y0 D x0 and f1 .t / 1 2 Œ0; 1, then kx 1 .t; x0 / x 2 .t; x0 /k D kx0 .t t0 C x0 /k D jt t0 j ! 1 as t ! 1: Theorem 7. If the R-solutions of (3.13) are stable, then the R-solutions of (3.22), (3.23) are also stable. Proof. Let X.t; x0 / be an R-solution of inclusion (3.22), (3.23) with initial condition X.t0 ; x0 / D x0 . Then Z t [ ² X.t; x0 / D ˆABi .t; /F . / d ˆABi .t; t0 /x0 C t0

A.t/2A.t / Bi 2Bi

C

X t0 i
³ ˆABi .t; i /Pi :

Section 3.2 Stability of Solutions of Linear Impulsive Differential Inclusions

85

It is necessary to prove that, for any " > 0, there exists ı D ı."/ > 0 such that, for all y0 such that kx0 y0 k < ı, one has h.X.t; x0 /; X.t; y0 // < ". We now estimate h.X.t; x0 /; X.t; y0 // D

sup A.t/2A.t/;Bi 2Bi

¹d1 .t /; d2 .t /º;

where Z t ˆABi .t; /f . / d d1 .t / D ˆABi .t; t0 /x0 C X

C

t0 i
t0

ˆABi .t; i /pi ; X.t; y0 /

Z t X ˆ .t; t /x C ˆABi .t; /f . / d C ˆABi .t; i /pi 0 0 ABi t0

t0 i
Z t X ˆABi .t; t0 /y0 C ˆABi .t; /f . / d C ˆABi .t; i /pi t0

t0 i
D kˆABi .t; t0 /.x0 y0 /k kˆABi .t; t0 /k kx0 y0 k M kx0 y0 k; the constant M > 0 exists by virtue of the criterion for stability of R-solutions for inclusion (3.13), and Z t ˆABi .t; /f . / d d2 .t / D ˆABi .t; t0 /y0 C X

C

t0 i
t0

ˆABi .t; i /pi ; X.t; x0 /

Z t X ˆABi .t; t0 /y0 C ˆABi .t; /f . / d C ˆABi .t; i /pi t0

ˆABi .t; t0 /x0 C

Z

t0 i
t t0

ˆABi .t; /f . / d C

D kˆABi .t; t0 /.x0 y0 /k M kx0 y0 k: We set ı D

" M

. Then, for kx0 y0 k < ı, we have h.X.t; x0 /; X.t; y0 // < ";

which was to be proved. We prove the converse statement in a special case.

X t0 i
ˆABi .t; i /pi

86

Chapter 3 Linear Impulsive Differential Inclusions

Consider linear inhomogeneous inclusions of the form xP 2 A.t /x C F .t /;

t ¤ i ;

xj tDi 2 Bi x C Pi ;

(3.25) (3.26)

where A.t / is an n n matrix measurable on the interval I D Œt0 ; C1/ and Bi are constant n n matrices. The following statement is true: Theorem 8. The R-solutions of (3.25), (3.26) are stable if and only if the solutions of xP D A.t /x;

t ¤ i ;

xj tDi D Bi x

(3.27) (3.28)

are stable. Proof. The sufﬁciency follows from Theorem 7. Let us prove the necessity. Assume that the R-solution of (3.25), (3.26) Z t X.t; x0 / D ˆABi .t; t0 /x0 C ˆABi .t; /F . / d C

X t0 i
t0

ˆABi .t; i /Pi

is stable. Then, for any " > 0, there exists ı D ı."/ > 0 such that, for all y0 W kx0 y0 k < ı, one has h.X.t; x0 /; X.t; y0 // D h.ˆABi .t; t0 /x0 ; ˆABi .t; t0 /y0 / < "; i.e., the solutions of (3.27), (3.28) are stable. Stability in the First Approximation. inclusion of the form

Consider a nonlinear impulsive differential

xP 2 A.t /x C F .t; x/; xj tDi 2 Bi x C Pi .x/;

t ¤ i ;

(3.29) (3.30)

where x 2 Rn is a phase vector, t 2 I D Œt0 ; C1 is time, A.t / is a compact set of n n matrices measurable on I , F W I Rn ! conv.Rn / is a set-valued mapping continuous in x for every ﬁxed t and measurable in t for every ﬁxed x, jF .t; x/j .t /, the function .t / is measurable and summable on any ﬁnite segment I I , i 2 I , i D 1; 2; : : : , are ﬁxed times enumerated in the increasing order .i < iC1 /, the set ¹i º does not have points of accumulation, Bi are compact sets of n n matrices, and Pi W Rn ! conv.Rn / is a continuous set-valued mapping.

87

Section 3.2 Stability of Solutions of Linear Impulsive Differential Inclusions

Assume that the set-valued mappings F .t; x/ and Pi .x/ satisfy the following conditions: (i) 0 2 F .t; 0/ for all t 2 I ; (ii) 0 2 Pi .0/ for all i 2 N . It is obvious that inclusion (3.29), (3.30) has the trivial solution. Let us study its stability. Together with inclusion (3.29), (3.30), we consider the linear inclusion xP 2 A.t /x;

t ¤ i ;

xj t Di 2 Bi x

(3.31) (3.32)

and call it the inclusion of the ﬁrst approximation with respect to inclusion (3.29), (3.30). Theorem 9. Suppose that the matrizants ˆABi .t; / of inclusion (3.31), (3.32) admit the following estimate for all t and , t0 t : kˆABi .t; /k Ke .t/ ;

K 1; > 0;

(3.33)

where the constants K and do not depend on the choice of matrices A.t / 2 A.t / and Bi 2 Bi , and the functions F .t; x/ and Pi .x/ satisfy the inequalities jPi .x/j akxk

jF .t; x/j akxk;

(3.34)

for all t t0 , i 2 N , kxk h, and h > 0. Then, for sufﬁciently small a, the trivial solution of inclusion (3.29), (3.30) is asymptotically stable. Proof. Every solution of inclusion (3.29), (3.30) can be represented in the form Z t x.t; x0 / D ˆABi .t; t0 /x0 C ˆABi .t; /f .; x.; x0 //d C

X t0 i
t0

ˆABi .t; i /pi .x.i ; x0 //;

where A.t / 2 A.t /, Bi 2 Bi , f .t; x/ is the branch of the mapping F .t; x/ measurable in t and continuous in x, and pi .x/ is the continuous branch of the mapping Pi .x/ [52]. Then, by analogy with Theorem 12 of Chapter 1, we obtain the estimate 1

kx.t; x0 /k Ke .Ka ln.1CKa//.tt0 / kx0 k: Thus, if a is so small that Ka

1 ln.1 C Ka/ > 0;

(3.35)

88

Chapter 3 Linear Impulsive Differential Inclusions

h then any solution x.t; x0 /; kx0 k K , of inclusion (3.29), (3.30) is deﬁned for all t t0 and lim kx.t; x0 /k D 0; t !1

i.e., the trivial solution of inclusion (3.29), (3.30) is asymptotically stable. Corollary 1. If inequality (3.33) in the conditions of Theorem 9 is satisﬁed only for some A.t / 2 A.t / and Bi 2 Bi and condition (3.34) is replaced by kf .t; x/k akxk;

kpi .x/k akxk

(3.36)

for some branch f .t; x/ of the mapping F .t; x/ measurable in t and continuous in x and the continuous branch pi .x/ of the mapping Pi .x/, then we can prove only the weak asymptotic stability of the trivial solution of inclusion (3.29), (3.30). Remark 7. By virtue of estimate (3.35), inclusion (3.29), (3.30) has a unique solution with initial condition x.0/ D 0. Hence, inclusion (3.29), (3.30) has the unique Rsolution .R.t / 0/ with initial condition R.0/ D 0. Corollary 2. If the conditions of the theorem are satisﬁed, then the trivial R-solution of inclusion (3.29), (3.30) is asymptotically stable.

3.3

Periodic Solutions of Linear Impulsive Differential Inclusions

Consider the problem of existence of periodic R-solutions and ordinary solutions of linear inhomogeneous periodic impulsive differential inclusions of the form xP 2 A.t /x C F .t /;

t ¤ i ;

xj tDi 2 Bi x C Pi ;

(3.37)

where A.t / is a continuous T -periodic matrix function, F W R ! comp.Rn / is a continuous T -periodic mapping, and the constant matrices Bi , sets Pi 2 conv.Rn /, and times i are such that, for a certain natural number r, one has BiCr D Bi ;

PiCr D Pi ;

iCr D i C T

for all i 2 Z. Also assume that 0 1 < < r < T and det.E C Bi / ¤ 0 for all i D 1; r. Let ˆ.t; s/ D ˆABi .t; s/ be the matrizant of the homogeneous system corresponding to (3.37). A solution x.t; x0 /, x.0; x0 / D x0 , of the linear impulsive differential inclusion (3.37) can be represented in the form Z t X x.t; x0 / D ˆ.t; 0/x0 C ˆ.t; /f . /d C ˆ.t; i /pi ; 0

where f .t / 2 F .t / and pi 2 Pi .

0i
89

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

By analogy with (3.4), we write an R-solution X.t; R0 /, X.t0 ; R0 / D R0 , of the impulsive differential inclusion (3.37) in the form Z t X ˆ.t; /F . /d C ˆ.t; i /Pi : X.t; R0 / D ˆ.t; 0/R0 C 0

0i
If R0 2 conv.Rn /, then, using the Lyapunov theorem, we establish that X.t; R0 / 2 conv.Rn / for any t . Thus, the existence of T -periodic R-solutions and ordinary solutions is directly related to the existence of solutions of the following equations in the space conv.Rn /: R0 D ˆ.T; 0/R0 C G; Z T X GD ˆ.T; /F . /d C ˆ.T; i /Pi ; 0

(3.38)

0i
x0 D ˆ.T; 0/x0 C g; Z T X gD ˆ.T; /f . /d C ˆ.T; i /pi 2 G: 0

(3.39)

0i
In Eq. (3.39), different f .t / 2 F .t / and pi 2 Pi are associated with points g 2 G 2 conv.Rn /. If the matrix E ˆ.T; 0/ is nondegenerate (i.e., the eigenvalues of the matrix ˆ.T; 0/ do not contain 1), then ordinary periodic solutions exist for all g 2 G, i.e., for all f .t / 2 F .t / and pi 2 Pi . If the matrix E ˆ.T; 0/ is degenerate, then ordinary periodic solutions exist for g 2 G for which the rank of the extended matrix .E ˆ.T; 0/; g/ is equal to the rank of the matrix E ˆ.T; 0/ [142]. Thus, if Eq. (3.39) has a solution for at least one g 2 G, then there exists an ordinary T -periodic solution of problem (3.37). In the general case, this solution may not be unique, i.e., a single point g 2 G (and, hence, a single initial point x0 ) may correspond to several T -periodic solutions. Example 6. Consider the impulsive differential inclusion xP 2 Œ0; 1;

t ¤ 2n 1; n 2 N;

xj t D2n1 D 2x: In this case, we have ´ 1; s 2 Œ0; 1; ˆ.2; s/ D 1; s 2 .1; 2;

Z and

GD

2 0

ˆ.2; s/Œ0; 1ds D Œ1; 1:

Thus, the initial points corresponding to 2-periodic solutions can be found from the equations x0 D x0 C g; jgj 1:

90

Chapter 3 Linear Impulsive Differential Inclusions

We take g0 D 0. Then x0 D 0. This initial condition corresponds to inﬁnitely many 2-periodic solutions, e.g., ´ k.t 2m/; t 2 Œ2m; 2m C 1; xk .t / D 2k C k.t 2m/; t 2 .2m C 1; 2.m C 1/; where k 2 Œ0; 1 and m 2 N. The initial points of ordinary periodic solutions (if periodic solutions exist) correspond to different points g 2 G. Let X0 denote the set of initial points x0 of ordinary periodic solutions, i.e., X0 D ¹x0 2 Rn W x0 D ˆ.T; 0/x0 C g; g 2 Gº: Consider Eq. (3.38). For the matrix ˆ.T; 0/, there exists a real nondegenerate matrix M such that 0

ˆ.T; 0/ D M 1 JM;

0 J2 :: : : : : 0 0

J1 B 0 B where J D B : @ ::

0 0 :: :

1 C C C A

(3.40)

Js

is the real canonical form [32]. Thus, Eq. (3.38) takes the form R0 D M 1 JMR0 C G , MR0 D JMR0 C M G , Z0 D J Z0 C D;

(3.41)

where Z0 D MR0 and D D M G 2 conv.Rn /. Let ƒ denote the spectral radius of the matrix ˆ.T; 0/. Consider the problem of existence of solutions of Eq. (3.41) in dependence on the value of ƒ. Case ƒ < 1 ([118]). For the matrix J , there exists a diagonal matrix L D diag¹1; "; : : : ; "n1 º such that J D L1 J."/L, where J."/ is a modiﬁed form [131], i.e., Ji ."/ D Ji if i is a simple root and 0

i B "1 B i Ji ."/ D B :: @ :

0

i :: :

:: :

0 0 :: :

0 "ki i 1 i

1 C C C A

91

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

or

0

Re i Im i B Im i Re i

B B B B Ji ."/ D B B B B @

0

"1i E2

Re i Im i Im i Re i

0

::

::

0

:

0

:

k 1

"i i

1

E2

C C C C 0 C C C :: C : C Re i Im i A Im i

Re i

j in the case of a multiple (respectively, real or complex) root, "i 2 ¹0; "º; " is an arbitrary positive number, and E2 is the 2 2 identity matrix. In this case, Eq. (3.41) can be transformed as follows:

Z0 ."/ D J."/Z0 ."/ C D."/;

(3.42)

where Z0 ."/ D LZ0 and D."/ D LD 2 conv.Rn /. The spectral radius of the matrix J T ."/J."/ is a continuous function of " at the point " D 0. Therefore, .J T ."/J."// D .J T .0/J.0// C ."/ D ƒ2 C ."/; where ."/ ! 0 as " ! 0. p We choose the constant " > 0 so that kJ."/k D ƒ2 C ."/ < 1. The space conv.Rn / is complete [23,28]. The existence of a T -periodic R-solution of Eq. (3.37), i.e., the existence of an initial set Z0 ."/ that satisﬁes Eq. (3.42), is related to the existence of a ﬁxed point of the mapping .Z/ D J."/Z C D."/:

(3.43)

Using the properties of the Hausdorff distance and support function, we obtain h..Z1 /; .Z2 // D h.J."/Z1 C D."/; J."/Z2 C D."// D h.J."/Z1 ; J."/Z2 / D sup jc.J."/Z1 ; / c.J."/Z2 ; /j 2S

T

D jc.Z1 ; J ."/

/c.Z2 ; J T ."/

/j kJ T ."/

k h.Z1 ; Z2 /

kJ."/k h.Z1 ; Z2 / < h.Z1 ; Z2 /; i.e., the mapping .Z/ is contracting. Hence, all conditions of the Banach theorem [14] are satisﬁed, and, therefore, mapping (3.43) has a unique ﬁxed point in conv.Rn /. Thus, if the spectral radius of the matrix ˆ.T; 0/ is less than 1, then, for any T periodic set-valued mapping F .t / and periodic sequence Pi , the linear impulsive differential inclusion (3.37) has a unique T -periodic R-solution R.t / whose initial value is determined by the formula R0 D M 1 L1 Z0 ."/, where Z0 ."/ is determined as a ﬁxed point of mapping (3.43).

92

Chapter 3 Linear Impulsive Differential Inclusions

Example 7. Consider the impulsive differential inclusion xP 1 1 0 x1 2 C ˛.1 C sin t /B1 .0/; t ¤ .2i 1/; 0 1 x2 xP 2 1 2 ˇ x1 x1 ˇˇ e 1 0 2 2 C pB1 .0/; 1 2 ˇ x2 tD.2i1/ x2 0 2e 1 where

4 12e 2 and p D : 3e C 1 3e C 1 The initial set R0 corresponding to a 2-periodic R-solution satisﬁes Eq. (3.38). In this example, we have T D 2, ´1 s 1 0 2 e 0 1 ; s ; ˆ.2; s/ D s > ; e s2 E; Z 2 GD ˆ.T; s/˛.1 C sin s/B1 .0/ds C ˆ.2; /pB1 .0/ ˛D

0

D G1 C G2 C G3 ; where 1 G1 D ˛ 2

Z

0

e s .1 C sin s/

1 0 0 1

B1 .0/ds

Z 1 e s .1 C sin s/B1 .0/ds D ˛ 2 0 Z 3 1 1 1 s e B1 .0/; e .1 C sin s/ds B1 .0/ D ˛ D ˛ 2 0 2 2 2 Z 2 1 3 2 s G2 D ˛e e e .1 C sin s/B1 .0/ds D ˛ B1 .0/; 2 2 1 1 1 0 G3 D pe B1 .0/ D pe B1 .0/: 0 1 2 2 Thus, GD

3 1 1 3 1 1 ˛ e C˛ e C pe B1 .0/ D B1 .0/: 2 2 2 2 2 2

Hence, Eq. (3.38) takes the form R0 D

1 2

0 0 12

R0 C B1 .0/:

(3.44)

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

93

By virtue of the uniqueness of a solution of Eq. (3.44) in the space conv.R2 /, we get R0 D B2 .0/. For the determination of the set X0 , we obtain the system ´ 0 x1 D 12 x10 C g1 ; x20 D 12 x20 C g2 ;

g12 C g22 1;

which implies that .x10 /2 .x 0 /2 C 9 2 1: 4 4 Thus, X0 is an ellipse and X0 R0 . Case ƒ > 1 ([118]). Since ƒ D max j i j > 1; iD1;n

where i are the eigenvalues of the matrix ˆ.T; 0/, there exists i0 W j i0 j > 1 (in what follows, we assume, for simplicity, that i0 D 1). In Eq. (3.41), we separately consider the cases where 1 is real (simple or multiple) and complex (simple or multiple). (1) Let 1 be simple and real. Then the Jordan cell has the form J1 D 1 : Let prx1 A be the projection of the set A to the axis x1 . Then mes.prx1 Z0 / D mes.prx1 .J Z0 C D// D mes.prx1 J Z0 C prx1 D/ D mes.prx1 J Z0 / C mes.prx1 D/ D mes. 1 prx1 Z0 / C mes.prx1 D/ D j 1 j mes.prx1 Z0 / C mes.prx1 D/: Consequently, Eq. (3.41) has solutions if and only if mes.prx1 D/ D 0 and mes.prx1 Z0 / D 0. In other words, the set D lies in the hyperplane x1 D d1 , and the set Z0 lies in the hyperplane x1 D x1 , where x1 is determined from the equation x1 D 1 x1 C d1 , i.e., d1 : x1 D 1 1 In this case, Z0 and D can be considered as sets of dimension n 1, and Eq. (3.41) reduces to the equation ! J2 0 Q where JQ D ZQ 0 D JQ ZQ 0 C D; ; : 0 :: ZQ 0 D ¹.x2 ; : : : ; xn /W .x1 ; x2 ; : : : ; xn / 2 Z0 º; DQ D ¹.d2 ; : : : ; dn /W .d1 ; d2 ; : : : ; dn / 2 Dº:

94

Chapter 3 Linear Impulsive Differential Inclusions

Thus, we have obtained a problem whose dimension is less by 1, which can be studied by analogy with the original problem. Example 8. Consider a linear impulsive differential inclusion of the form 10 1 0 1 0 1 x1 xP 1 2 0 0 @ xP 2 A 2 @ 0 1 0 A @ x2 A C ˛.1 C cos t /BQ 1 .0/; xP 3 x3 0 0 1 0 0 1 10 1 1ˇ 0 0 2e 1 x1 ˇˇ x1 1 2 @ x2 Aˇ @ A @ x2 A C p BQ 1 .0/; 0 2 e 1 0 ˇ 2 1 2 x3 ˇ tD.2i1/ x3 0 0 1 2e where BQ 1 .0/ D ¹.x1 ; x2 ; x3 /T W x1 D 0; .x2 ; x3 /T 2 B1 .0/º, ˛ D p D e . In this case, the fundamental matrix has the form 8 ! s ˆ 2e 2 0 0 ˆ 1 s ˆ 0 0 ˆ ; s 2e < 1 s 0 0 e 2 ˆ.2; s/ D 1 ˆ ˆ e 2 .2s/ 0 0 ˆ ˆ ; s> : 0 0 e 2s 0

0

2 2e e 3 ,

e 2s

We now determine the set G. We have Z 2 ˆ.T; s/˛.1 C sin s/BQ 1 .0/ds C X.2; /p BQ 1 .0/ GD 0

D G1 C G2 C G3 ; where 0

Z G1 D ˛

0

Z

s

2e 2 @ 0 0

0 1 s e 2 0

1 0 0 A .1 C cos s/BQ 1 .0/ds 1 s 2e

˛ s e .1 C cos s/BQ 1 .0/ds 2 0 Z ˛ ˛ s e .1 C cos s/ds BQ 1 .0/ D .3 e /BQ 1 .0/; D 2 0 4 0 2s 1 Z 2 e 2 0 0 @ 0 e 2s G2 D ˛ 0 A .1 C cos s/BQ 1 .0/ds 0 0 e 2s Z 2 e 2s .1 C cos s/BQ 1 .0/ds D˛ D

and

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

Z

95

2

1 e 2s .1 C cos s/ds BQ 1 .0/ D ˛e 2 .e 3e 2 /BQ 1 .0/; 2 0 1 1 2e 2 0 0 p G3 D p @ 0 12 e 0 A BQ 1 .0/ D e BQ 1 .0/: 2 0 0 12 e D˛

Hence, 1 1 2 p Q 2 GD ˛ .3 e / C e .e 3e / C e B1 .0/ 4 2 2 D BQ 1 .0/: Thus, Eq. (3.38) takes the form 0

1 2 0 0 R0 D @ 0 12 0 A R0 C BQ 1 .0/: 0 0 12

Since 1 D 2 and BQ 1 .0/ belongs to the hyperplane x1 D 0, we conclude that R0 must belong to the hyperplane x1 D 0, and Eq. (3.38) reduces to the equation 1 0 RQ 0 D 2 1 RQ 0 C B1 .0/; 0 2 whose solution is RQ 0 D B2 .0/. In other words, R0 D ¹.x1 ; x2 ; x3 /T W x1 D 0; .x2 ; x3 /T 2 B2 .0/º D BQ 2 .0/: Ordinary periodic solutions exist for all g 2 BQ 1 .0/, and the corresponding initial conditions are determined from the system 0 1 2 0 0 x0 D @ 0 12 0 A x0 C g: 0 0 12 Thus, the set of initial values for ordinary periodic solutions X0 has the form 0

11 1 0 0 X0 D @ 0 12 0 A BQ 1 .0/ D 2BQ 1 .0/ D R0 : 0 0 12 Example 9. Consider the linear impulsive differential inclusion from Example 8 with the mapping F .t / and set P chosen as follows: F .t / D ˛.1 C cos t /B1 .0/ and P D pB1 .0/. In this case, we have G D B1 .0/. Therefore, a 2-periodic R-solution

96

Chapter 3 Linear Impulsive Differential Inclusions

does not exist because 1 > 1 and G does not belong to the hyperplane x1 D g1 . Ordinary T -periodic solutions exist for any g 2 G because the eigenvalues of the matrix ˆ.T; 0/ do not contain 1. In this case, the set X0 is an ellipsoid. (2) Let 1 be of multiplicity k. Then the cell J1 takes the form 0

1 0 B e 1 1 B 1 J1 D B :: : : : : @ : : : k1 0 e1

1

0 0 :: :

C C C; A

1

where e1i 2 ¹0; 1º. In this case, by analogy with the previous one, we establish that Eq. (3.41) has solutions if and only if mes.prx1 D/ D 0 and mes.prx1 Z0 / D 0. In other words, the set D lies in the hyperplane x1 D d1 , and the set Z0 lies in the hyperplane x1 D x1 . Since mes.prx2 Z0 / D mes.prx2 .J Z0 C D// D mes.prx2 J Z0 C prx2 D/ D mes.prx2 J Z0 / C mes.prx2 D/ D mes. 1 prx2 Z0 C x1 / C mes.prx2 D/ D j 1 j mes.prx2 Z0 / C mes.prx2 D/; we conclude that Eq. (3.41) has solutions if and only if mes.prx2 D/ D 0 and mes.prx2 Z0 / D 0, i.e., the set D lies in the hyperplane x2 D d2 , and the set Z0 lies in the hyperplane x2 D x2 , etc. Thus, Eq. (3.41) has solutions if and only if the set D lies in the hyperplanes xi D di , i D 1; k, and the set Z0 lies in the hyperplanes xi D xi , i D 1; k. In this case, the constants xi are determined from the system 1 0 1 0 1 x1 x1 d1 B :: C B :: C B :: C @ : A D J 1 @ : A C @ : A; 0

xk

xk

(3.45)

dk

which is solved successively from the top downward and has a unique solution because the matrix E J1 is nondegenerate. (3) Let 1 be simple and complex. Then the Jordan cell has the form J1 D

Re 1 Im 1 : Im 1 Re 1

Let pr.x1 ;x2 / A be the projection of a set A to the hyperplane .x1 ; x2 /. Then pr.x1 ;x2 / Z0 D pr.x1 ;x2 / .J Z0 C D/:

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

97

The projection of the set Z0 to the hyperplane .x1 ; x2 / is a convex compact set. Therefore, either pr.x1 ;x2 / Z0 is a segment, or we can choose a disk with maximum area among the disks contained in pr.x1 ;x2 / Z0 (let this be Sr .a/, r > 0/. In the ﬁrst case, if mes.pr.x1 ;x2 / D/ > 0, then Eq. (3.41) does not have solutions because mes.pr.x1 ;x2 / .J Z0 C D// mes.pr.x1 ;x2 / D/ > 0 D mes.pr.x1 ;x2 / Z0 /: If mes.pr.x1 ;x2 / D/ D 0, i.e., pr.x1 ;x2 / D is a segment, then it is obvious that the segment pr.x1 ;x2 / J Z0 must be parallel to the given one, otherwise pr.x1 ;x2 / .J Z0 C D/ is a parallelogram by virtue of the deﬁnition of the sum of two sets and cannot coincide with the set pr.x1 ;x2 / Z0 . By .x11 ; x21 / and .x12 ; x22 / we denote the endpoints of the segment pr.x1 ;x2 / Z0 . Then length pr.x1 ;x2 / .J Z0 C D/ D length pr.x1 ;x2 / .J Z0 / C length pr.x1 ;x2 / D D Œ.Re 1 x11 Im 1 x21 .Re 1 x12 Im 1 x22 //2 1

C .Im 1 x11 C Re 1 x21 .Im 1 x12 C Re 1 x22 //2 2 C length pr.x1 ;x2 / D 1

D Œ.Re 21 C Im 21 /..x11 x12 /2 C .x21 x22 /2 / 2 C length pr.x1 ;x2 / D D j 1 jlength pr.x1 ;x2 / Z0 C length pr.x1 ;x2 / D: Thus, in this case, Eq. (3.41) has a solution if and only if length pr.x1 ;x2 / Z0 D 0 and length pr.x1 ;x2 / D D 0, i.e., pr.x1 ;x2 / Z0 and pr.x1 ;x2 / D are points. In the second case, since Sr .a/ pr.x1 ;x2 / Z0 , we have J1 Sr .a/ C pr.x1 ;x2 / D pr.x1 ;x2 / .J Z0 C D/; and the set J1 Sr .a/ C pr.x1 ;x2 / D contains the disk J1 Sr .a/ C d D j 1 j cos ' D

Re 1 ; j 1 j

cos ' sin ' sin ' cos '

sin ' D

Im 1 ; j 1 j

Sr .a/ C d;

d 2 pr.x1 ;x2 / D;

i.e., a disk of radius r1 D j 1 jr > r, which, by virtue of Eq. (3.41), contradicts the choice of the disk Sr .a/. Thus, a T -periodic R-solution may exist only in the case where the sets pr.x1 ;x2 / Z0 and pr.x1 ;x2 / D are singletons. For the determination of pr.x1 ;x2 / Z0 , we obtain the

98 system

or

Chapter 3 Linear Impulsive Differential Inclusions

x1 x2

D

Re 1 Im 1 Im 1 Re 1

1 Re 1 Im 1 Im 1 1 Re 1

x1 x2

x1 x2

C

D

d1 ; d2

(3.46)

d1 ; d2

whose determinant is .1 Re 1 /2 C Im 21 > 0. Hence, this system has a unique solution. The sets Z0 and D can be considered as sets in the space conv.Rn2 /. Thus, the dimension of the problem is decreased by 2, and the obtained problem Q ZQ 0 D JQ ZQ 0 C D; where JQ D

! J2 0 ; : 0 ::

ZQ 0 D ¹.x3 ; : : : ; xn /W .x1 ; x2 ; x3 ; : : : ; xn / 2 Z0 º, and DQ D ¹.d3 ; : : : ; dn /W .d1 ; d2 ; d3 ; : : : ; dn / 2 Dº, can be studied by analogy with the original one. Example 10. Consider the linear impulsive differential inclusion xP 1 D x2 ; xP 2 D x1 ; xP 3 2 x3 C ˛Œ1; 1;

t ¤ .2i 1/;

x1 j t D.2i1/ D x2 ; x2 j t D.2i1/ D x1 ; 1 2 e x3 j tD.2i1/ 2 1 x3 C pŒ1; 1; 2 where i 2 N , ˛ D 2e e1 1 , and p D e . The fundamental matrix of this system has the form

ˆ.2; s/ D

8 cos sCsin s ˆ ˆ < sin scos s s ˆ ˆ : cos sin s 0

0

cos ssin s cos sCsin s 0

sin s 0 cos s 0 0 e 2s

;

0 0 1 s 2e

; s ; s > :

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

99

In this case, we have 0 1 Z cos s C sin s cos s sin s 0 @ sin s cos s cos s C sin s 0 A B 1 .0/ds G1 D ˛ 1 s 0 0 0 2e Z ˛ ˛ e s ds B 1 .0/ D .1 e /B 1 .0/; D 2 0 2 0 1 Z 2 cos s sin s 0 @ sin s cos s 0 A B 1 .0/ds G2 D ˛ 0 0 e 2s Z ˛ 2 2s D e ds B 1 .0/ D ˛.e 1/B 1 .0/; 2 0 1 1 1 0 p G3 D ˆ.2; /P D p @ 1 1 0 A B 1 .0/ D e B 1 .0/; 2 0 0 12 e where B 1 .0/ D ¹.x1 ; x2 ; x3 /T W x1 D 0; x2 D 0; jx3 j 1º. Thus, 1 p .1 e / C .e 1/ C e GD ˛ B 1 .0/ D B 1 .0/: 2 2 Then Eq. (3.38) takes the form 0

1 1 1 0 R0 D @ 1 1 0 A R0 C B 1 .0/: 0 0 12

By virtue of the arguments presented above, R0 is a point in the plane .x1 ; x2 /, which is determined from the system 0 1 x1 0 D : 1 0 0 x2 Thus, x1 D 0 and x2 D 0. Equation (3.38) reduces to the following one: 1 RQ 0 D RQ 0 C Œ1; 1: 2 Therefore, RQ 0 D Œ2; 2, whence R0 D B 2 .0/. Considering the problem of existence of ordinary periodic solutions, we conclude that the matrix E ˆ.T; 0/ is nondegenerate, periodic solutions exist for all d 2 B 1 .0/, and 0 11 0 1 0 X0 D @ 1 0 0 A B 1 .0/ D B 2 .0/ D R0 : 0 0 12

100

Chapter 3 Linear Impulsive Differential Inclusions

(4) Let 1 be a complex root of multiplicity k. Then the cell J1 takes the form 1 0 Re 1 Im 1 0 0 C B Im 1 Re 1 C B C B Re 1 Im 1 1 C B e1 E2 0 C B Im 1 Re 1 C: B C B :: :: :: C B : : 0 : C B @ Re 1 Im 1 A k1 E 0 e1 2 Im 1 Re 1

Consequently, similarly to the case of a simple complex root, the sets Z0 and D must lie in the hyperplanes x1 D x1 and x2 D x2 , as well as in the hyperplanes x2i1 D x2i 1 , x2i D x2i , i D 2; k (which is established by successively decreasing the dimension of the problem by analogy with the case of a multiple real root). Thus, Z0 and D are sets of dimension n 2k. For the determination of xi , i D 1; 2k, we obtain the system 0 1 0 1 0 1 x1 x1 d1 B :: C B :: C B :: C (3.47) @ : A D J1 @ : A C @ : A: x2k x2k d2k Case ƒ D 1 ([118]). Without loss of generality, we assume that max j i j D j 1 j:

i D1;n

(1) Let 1 be simple and real, i.e., 1 D ˙1. By virtue of (3.41), we have prx1 Z0 D prx1 .J Z0 C D/ D prx1 J Z0 C prx1 D D 1 prx1 Z0 C prx1 D: If 1 D 1, then it is clear that a solution of Eq. (3.41) exists if and only if prx1 D D ¹d1 º D ¹0º. In this case, prx1 Z0 is an arbitrary segment in R. Let 1 D 1. Denote prx1 Z0 D Œx1 ; x1C and prx1 D D Œd1 ; d1C . Then x1 D x1C C d1 ;

x1C D x1 C d1C ;

whence d1 D d1C D d1 , i.e., D can be considered as a set of dimension n 1. d

In this case, we have prx1 Z0 D Œx1 ; d1 x1 , where x1 21 is an arbitrary constant. (2) Let 1 D ˙1 be a root of multiplicity k. Then, by analogy with the previous case, we consider projections to the axis x1 and establish that, for 1 D 1, the set D must lie in the hyperplane x1 D 0, and prx1 Z0 is an arbitrary segment. For 1 D 1, the set D must lie in the hyperplane x1 D d1 , and prx1 Z0 D Œx1 ; d1 x1 , where x1

d1 2

is an arbitrary constant.

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

101

Let e11 D 0. Then prx2 Z0 D prx2 .J Z0 C D/ D prx2 J Z0 C prx2 D D 1 prx2 Z0 C prx2 D: Similarly to the previous case, for 1 D 1 the set D must lie in the hyperplane x2 D 0, and prx2 Z0 is an arbitrary segment. For 1 D 1, the set D must lie in the d

hyperplane x2 D d2 , and prx2 Z0 D Œx2 ; d2 x2 , where x2 22 is an arbitrary constant. Now assume that e11 ¤ 0. Let !T

;p ; 0; : : : ; 0 2 Rn e; D p 2 2 2 2

C

C and let x; be the straight line in the hyperplane .x1 ; x2 / that passes through the point .0; 0/ and is determined by the vector e; . Then mes.prx2 Z0 / D mes.prx2 .J Z0 C D// D mes.prx2 J Z0 / C mes.prx2 D/ D c.J Z0 ; e0;1 / C c.J Z0 ; e0;1 / C mes.prx2 D/ D c.Z0 ; J T e0;1 / C c.Z0 ; J T e0;1 / C mes.prx2 D/ q p ® ¯ D J T e0;1 D .1; 1 ; 0; : : : ; 0/T D 1 C 21 e1; 1 D 2e1; 1 p D 2Œc.Z0 ; e1; 1 / C c.Z0 ; e1; 1 / C mes.prx2 D/ p D 2 mes.prx1; Z0 / C mes.prx2 D/ 1 p D 2.mes.prx1; J Z0 / C mes.prx1; D// C mes.prx2 D/ 1 1 p p D 2Œc.Z0 ; J T e1; 1 / C c.z0 ; J T e1; 1 / C 2 mes.prx1; D/ C mes.prx2 D/ 1 p ² 5 1 D J T e1; 1 D p .2 1 ; 21 ; 0; : : : ; 0/T D p e2 1 ; 2 I 1 2 2 ³ 1 mes.prx1; D/ D mes.prx2 D/j cos e0;1 ; e1; 1 j D p mes.prx2 D/ 1 2 p D 5 mes.prx Z0 / C 2 mes.prx2 D/ D 2

4

21 ;1

p D m2 C 1 mes.prx

mm1 ;m 1 1

Z0 / C m mes.prx2 D/:

102

Chapter 3 Linear Impulsive Differential Inclusions

As m ! 1, the straight line xm m1 ; m tends to the straight line x1 . Therefore, 1 1 the limit p lim m2 C 1 mes.prx m1 m Z0 / m!1

m 1

; 1

exists if and only if mes.prx1 Z0 / D 0, i.e., in the case where prx1 Z0 D ¹x1 º. Thus, Z0 can be considered as a set of dimension n 1. Furthermore, mes.prx2 D/ D 0, whence prx2 D D ¹d2 º. Then prx2 Z0 D prx2 .J Z0 C D/ D prx2 J Z0 C d2 D 1 prx2 Z0 C x1 C d2 : By analogy with the case of a simple root, we establish that, for 1 D 1, the set D must lie in the hyperplane d2 D x1 , and prx2 Z0 is an arbitrary segment in R. For

1 D 1, we have prx1 Z0 D ¹ 12 d1 º and prx2 Z0 D Œx2 ; .d2 C x1 / x2 , where d Cx

x2 2 2 1 is an arbitrary constant. Thus, prxi D D ¹di º for all i D 1; k. Let i1 ; : : : ; im1 be the set of indices for which e1i D 0 and im D k. Then prxi Z0 is an arbitrary segment that satisﬁes j

i 1

the equation prxi Z0 D 1 prxi Z0 C xij 1 C dij if e1j j

j

i 1 D 0 (we

1 prxi Z0 C dij if e1j j we have prxj Z0 D ¹xj º, where

xj D

8 d ˆ ˆ ˆ j C1 < ˆ ˆ ˆ :

assume here that

e10

D 1 and prxi Z0 D j

D 0). For the other indices,

if 1 D 1;

xj 1 Cdj

if 1 D 1 and j ¤ iq C 1; q D 1; m 1;

2 dj

if 1 D 1 and j D iq C 1; q D 1; m 1:

2

By virtue of Eq. (3.41), we get c.Z0 ; / D c.J Z0 C D; / D c.Z0 ; J T / C c.D; /

(3.48)

for all D . 1 ; : : : ; k ; 0; : : :/T 2 Rn . Let 1 D 1. We rewrite equality (3.48) as follows: c.Z0 ; / D max

x2Z0

k X

xi

i

D max.xi1

i1

C C xi m

im /

C

iD1

X i 21;k i¤iq

c.Z0 ; J T / C c.D; / D max.xi1 C

X i21;k i¤iq

i1

xi .

C C xi m i

C

iC1 / C

im / k X i D1

di

i:

xi

i;

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

103

Since d1 D 0; diq C1 D 0, q 2 1; m 1, and xi D diC1 for the other indices, it follows from equality (3.48) that the projection of the set Z0 in the space xi1 ; : : : ; xim is an arbitrary convex compact set. Let 1 D 1 and let the set Z be such that

Z0 D Z C .x1 ; : : : ; xk ; 0; : : : ; 0/T ; where .x1 ; : : : ; xk /T is a solution of system (3.45). Then Eq. (3.48) is equivalent to the equation c.Z; / D c.Z; J T /; or max..xi1 xi1 /

i1

C C .xim xim /

D max¹..xi1 xi1 /

i1

im /

C C .xim xim /

im /º:

Thus, the projection of the set Z0 in the space xi1 ; : : : ; xim is an arbitrary convex compact set symmetric with respect to the point .xi1 ; : : : ; xim /. (3) Let 1 be a simple complex root. Then, by virtue of Eq. (3.41), we obtain pr.x1 ;x2 / Z0 D pr.x1 ;x2 / .J Z0 C D/ D J1 pr.x1 ;x2 / Z0 C pr.x1 ;x2 / D:

(3.49)

In this case, J1 is the matrix of rotation by an angle ': cos ' D Re 1 , sin ' D Im 1 . It is obvious that mes.pr.x1 ;x2 / D/ D 0, i.e., pr.x1 ;x2 / D is a segment because mes.pr.x1 ;x2 / Z0 / D mes.J1 pr.x1 ;x2 / Z0 C pr.x1 ;x2 / D/ mes.J1 pr.x1 ;x2 / Z0 / C mes.pr.x1 ;x2 / D/ D mes.pr.x1 ;x2 / Z0 / C mes.pr.x1 ;x2 / D/: We have Z0 2 conv.Rn /. Therefore, pr.x1 ;x2 / Z0 2 conv.R2 /, i.e., pr.x1 ;x2 / Z0 is either a segment or a solid set. Hence, this set contains a segment AB of maximum length r. If pr.x1 ;x2 / D is not a singleton, then the set J1 pr.x1 ;x2 / Z0 C pr.x1 ;x2 / D contains a segment of length greater than r (the major diagonal of the parallelogram AB C pr.x1 ;x2 / D/, which is impossible by virtue of (3.49). Thus, pr.x1 ;x2 / D D .d1 ; d2 /T and the set pr.x1 ;x2 / Z0 is such that pr.x1 ;x2 / Z0 D J1 pr.x1 ;x2 / Z0 C .d1 ; d2 /T : Assume that a set Z 2 conv.R2 / is such that pr.x1 ;x2 / Z0 D Z C .x1 ; x2 /T ;

(3.50)

104

Chapter 3 Linear Impulsive Differential Inclusions

where .x1 ; x2 /T is a solution of the system of equations x1 d1 x1 D J1 C : x2 x2 d2

(3.51)

Then Eq. (3.50) takes the form Z D J1 Z. Let x 2 Z be such that kxk D h.¹0º; Z/. The point J1 x obtained from the point x as a result of rotation by an angle ' about the point .0; 0/T also belongs to the boundary of the set Z. k k Thus, if ' D 2 m , where m is an irreducible fraction, then the solutions of the equation are “pseudo-m-gons”. The term “pseudo-m-gon” stands for a convex compact set in R2 , which remains invariant under the rotations by the angle 2 m about the T origin. It is clear that a disk of any radius centered at the point .0; 0/ is a special case of “pseudo-m-gons”. In view of the convexity of the set Z, it is clear that Zm Z Bkxk .0/; where Zm is a regular m-gon with vertices at the points x; J1 x; : : : ; J1m1 x. For m D 6, the corresponding example is presented by the set depicted in Figure 1. k If ' ¤ 2 m , then the set of boundary points ¹J1i x; i 2 Nº is everywhere dense on a circle of radius kxk centered at the point .0; 0/T . Hence, Z D Bkxk .0/.

Figure 1.

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

105

(4) Let 1 be a complex root of multiplicity k. By analogy with the previous case, we now consider the projections onto the hyperplane .x1 ; x2 / and conclude that the set pr.x1 ;x2 / D must contain a single point. In this case, pr.x1 ;x2 / Z0 is an arbitrary convex compact set in R2 satisfying the equation pr.x1 ;x2 / Z0 D

cos ' sin ' sin ' cos '

pr.x1 ;x2 / Z0 C .d1 ; d2 /T ;

(3.52)

i.e., either a disk or a “pseudo-m-gon” whose center is located at the point .x1 ; x2 /T . Consider the projections onto the hyperplane .x3 ; x4 /. By virtue of Eq. (3.41), we get pr.x3 ;x4 / Z0 D pr.x3 ;x4 / .J Z0 C D/ D pr.x3 ;x4 / J Z0 C pr.x3 ;x4 / D: If e11 D 0, then, as in the previous case, we conclude that the set pr.x3 ;x4 / D must be one-point. In this case, pr.x3 ;x4 / Z0 is an arbitrary convex compact set in R2 satisfying an equation similar to (3.52), i.e., either a disk or a “pseudo-m-gon” centered at the point .x3 ; x4 /T . Let e11 D 1 and let prx1 ;:::;x4 A be the projection of the set A 2 conv.Rn / in the space x1 ; : : : ; x4 . We choose a set Z such that prx1 ;:::;x4 Z0 D Z C .x1 ; x2 ; 0; 0/T ; where .x1 ; x2 /T is a solution of system (3.51). Then Z D J Z C D1 ; where D1 W prx1 ;:::;x4 D D D1 C .d1 ; d2 ; 0; 0/T ; 0 1 cos ' sin ' 0 0 B sin ' cos ' 0 0 C C: J D B @ 1 0 cos ' sin ' A 0 1 sin ' cos ' If Z is a solution of this equation, then Z is also a solution of the equation Z D Jm Z C .Jm1 D1 C C D1 / for all m 2 N . The matrix Ji takes the form 0

1 cos i' sin i' 0 0 B C sin i' cos i' 0 0 C Ji D B @ i cos.i 1/' i sin.i 1/' cos i' sin i' A: i sin.i 1/' i cos.i 1/' sin i' cos i'

(3.53)

106

Chapter 3 Linear Impulsive Differential Inclusions

Furthermore, according to the properties of the operations in conv.Rn /, we conclude that Jm1 D1 C C D1 .Jm1 C C J C E/D1 P 0 Pm1 1 m1 0 0 iD0 cos i' iD1 sin i' B Pm1 C Pm1 B C 0 0 iD1 sin i' iD0 cos i' B C D BP CD : B m1 i cos.i 1/' Pm1 i sin.i 1/' Pm1 cos i' Pm1 sin i'C 1 iD1 i D0 iD1 @ i D1 A Pm1 Pm1 Pm1 Pm1 i D1 i sin.i 1/' iD1 i cos.i 1/' iD1 sin i' iD0 cos i' Thus, for any x D .x1 ; x2 ; x3 ; x4 /T 2 Z and d D .0; 0; d3 ; d4 /T 2 D1 , the point y D .y1 ; : : : ; y4 /T D Jm x C .Jm1 C C E/d 2 Z for all m 2 N. In this case, y3 D mx1 cos.m 1/' mx2 sin.m 1/' C x3 cos m' x4 sin m' C d3

m1 X

cos i' d4

iD0

m1 X

sin i';

iD1

y4 D mx1 sin.m 1/' C mx2 cos.m 1/' C x3 sin m' C x4 cos m' C d3

m1 X

sin i' C d4

iD1

m1 X

cos i':

iD0

Moreover, if x12 C x22 > 0, then

q y3 D m x12 C x22 cos..m 1/' C ˇ12 / C x3 cos m' x4 sin m' C d3

m m sin mC1 sin mC1 2 ' cos 2 ' 2 ' sin 2 ' d ; 4 sin '2 sin '2

q y4 D m x12 C x22 sin..m 1/' C ˇ12 / C x3 sin m' C x4 cos m' m m sin mC1 sin mC1 2 ' sin 2 ' 2 ' cos 2 ' C d4 ; ' ' sin 2 sin 2 x1 x2 Dq ; sin ˇ12 D q : x12 C x22 x12 C x22

C d3

cos ˇ12

In this case, y32 C y42 ! 1 as m ! 1, which is impossible because the set Z 2 conv.R4 /.

107

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

Hence, equality (3.53) is possible only in the case pr.x1 ;x2 / Z D .0; 0/T , i.e., pr.x1 ;x2 / Z0 D .x1 ; x2 /T . Thus, Eq. (3.52) is reduced to cos ' sin ' T pr.x3 ;x4 / Z0 C pr.x3 ;x4 / D: pr.x3 ;x4 / Z0 D .x1 ; x2 / C sin ' cos ' As in the case of a simple complex root, we obtain pr.x3 ;x4 / D D .d3 ; d4 /T , and the set pr.x3 ;x4 / Z0 is either a “pseudo-m-gon” or a disk centered at the point .x3 ; x4 /T . Further, we consider the projections onto the hyperplanes .x2i1 ; x2i /, i k, etc. T This yields pr.x2i1 ;x2i / D D .d2i 1 ; d2i / for all i D 1; k. Let i1 ; : : : ; im1 be the i set of indices such that e1 D 0 and im D k. Then pr.x2i 1 ;x2i / Z0 ; j 2 1; m, is an arbitrary convex compact set in R2 satisfyj j ing an equation similar to (3.52), i.e., a disk or a “pseudo-m-gon” centered at the point /T . For all remaining indices, we get pr T ; x2i .x2i .x2i1 ;x2i / Z0 D .x2i 1 ; x2i / . j 1 j Thus, we arrive at the following system for the determination of xi ; i D 1; 2k: 0 1 0 1 0 1 x1 x1 d1 B :: C B :: C B :: C @ : A D J1 @ : A C @ : A: x2k x2k d2k

Hence, by virtue of Eq. (3.41), c.Z0 ; / D c.J Z0 C D; / D c.Z0 ; J T / C c.D; /

(3.54)

for all D . 1 ; : : : ; k ; 0; : : :/T 2 Rn . Assume that the set Z is such that Z0 D Z C .x1 ; : : : ; x2k ; 0; : : : ; 0/T :

Then Eq. (3.54) is equivalent to the equation c.Z; / D c.Z; J T /

(3.55)

for all D . 1 ; : : : ; k ; 0; : : :/T . We rewrite this equation in a more detailed form: c.Z; / D max

2k X

Q x2Z i D1

xi

i

D max¹.x2i1 1 x2i / 1 1

2i1 1

C .x2im 1 x2i / m 1

C .x2i1 x2i / 1

2im 1

2i1

C

C .x2im x2i / m

2im º;

108

Chapter 3 Linear Impulsive Differential Inclusions

c.Z; J T / D max¹.

2i1 1 cos '

C . C.

C

C

2i1 1 sin '

2im 1 cos '

C .

2i1

C

2im 1 sin '

sin '/.x2i1 1 x2i / 1 1 2i1

cos '/.x2i1 x2i / C 1

2im

sin '/.x2im 1 x2i / m 1

C

2im

cos '/.x2im x2i /º: m

Thus, equality (3.55) means that the projection of the set Z0 in the space x2i1 1 ; x2i1 ; : : : ; x2is 1 ; x2is is an arbitrary convex compact set invariant under rotations by an angle ' in the hy T ; x2i / . perplanes .x2ij 1 ; x2ij /; j D 1; m, about the points .x2i j 1 j Consider the general case. Let 1 ; : : : ; m be different eigenvalues of the matrix ˆ.T; 0/ whose moduli are equal to 1 and let kj ; j D 1; m, be the dimensions of the corresponding Jordan cells Jj . now ﬁnd the projection Z1 of the set Z0 in the space x1 ; : : : ; xm0 , where m0 D PWe m j D1 kj . As shown above, the set Z1 “degenerates” into points in a part of variables. In the variables corresponding to D 1, the set Z1 is symmetric about the given point and, in the couples of variables corresponding to roots of the form D cos ' C i sin ', the set Z1 is invariant under rotations by the angle ' about the given point. from the set Z . Consider a section We now choose an arbitrary point x1 ; : : : ; xm 1 0 . By using the structure of of the set Z0 by the hyperplanes x1 D x1 , : : : , xm0 D xm 0 the matrix J and the set D, we arrive at the following equation for the given section: Q Z2 D JQ Z2 C D;

(3.56)

where DQ is a section of the set D [as shown above, the set DQ does not depend on / and the matrix JQ contains solely the Jordan the choice of the point .x1 ; : : : ; xm 0 cells of the matrix J corresponding to eigenvalues whose moduli are smaller than 1. As indicated above, since .JQ / < 1, Eq. (3.56) possesses a unique solution Z2 . Thus, Z D Z Z . independent of the choice of the point x1 ; : : : ; xm 0 1 2 0 Example 11. Consider a linear impulsive differential inclusion of the form 10 1 0 1 0 1 0 0 x1 xP 1 @ xP 2 A 2 @ 0 1 0 A @ x2 A C ˛.1 C cos t /B 1 .0/; 1 xP 3 x3 0 0 2 0 2 1ˇ 0 10 1 e 1 0 0 x1 ˇˇ x1 2 1 @ @ x2 Aˇ A @ x2 A C pB 1 .0/; 0 e 0 2 ˇ 1 1 x3 ˇ tD.2i1/ x3 0 0 1 2e where ˛D

1 C 4 2 3 2

16 3 .e e

12

; /

1

p D e2:

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

The fundamental matrix takes the form 8 es 0 0 ˆ 0 e s 0 ˆ ; ˆ < 0 0 1 e s 2 2 ˆ.2; s/ D 2s e 0 0 ˆ ˆ ˆ 0 0 e 2s : 1 0

0

109

s ; ; s > :

e 2 .2s/

In this case, 0

Z G1 D ˛ D

˛ 2

0

Z

0

e s 0 @ 0 e s 0 0

0 0 s 1 2 2e

1 A .1 C cos s/B 1 .0/ds

s

e 2 .1 C cos s/dsB 1 .0/

1 4 .2 2 .1 e 2 / C 1/B 1 .0/; 1 C 4 2 0 1 Z 2 e 2s 0 0 @ 0 e 2s 0 A .1 C cos s/B 1 .0/ds G2 D ˛ 2s 0 0 e 2 Z 2 2s e 2 .1 C cos s/dsB 1 .0/ D˛

D˛

1 4 .2 2 e 2 e 1 .1 C 2 2 //B 1 .0/; 2 1 C 4 0 2 1 e e 0 0 0 e 2 e 0 A B 1 .0/ G3 D ˆ.2; /P D p @ 1 12 0 0 2e e p 1 D e 2 B 1 .0/: 2

D ˛e

Hence, GD ˛

1 4 .2 2 .1 e 2 / C 1/ 1 C 4 2

Ce

p 1 4 2 12 1 2 2 B .0/ D B .0/: e .2 e e .1 C 2 // C 1 1 1 C 4 2 2

Therefore, Eq. (3.38) takes the form 0 1 1 0 0 R0 D @ 0 1 0 A R0 C B 1 .0/: 0 0 12

110

Chapter 3 Linear Impulsive Differential Inclusions

The eigenvalues 1;2 D 1 and the set B 1 .0/ belongs to the hyperplanes x1 D 0 and x2 D 0. Hence, pr.x1 ;x2 / R0 is an arbitrary convex compact set in R2 , and the analyzed equation is reduced to 1 R0 D R0 C Œ1; 1; 2

R0 2 R:

The solution of the last equation is R0 D Œ2; 2 and, thus, R0 D pr.x1 ;x2 / R0 Œ2; 2: As for the existence of ordinary periodic solutions, we note that the matrix E ˆ.T; 0/ is degenerate but periodic solutions exist for all d 2 B 1 .0/ because 0 1 0 1 0 0 0 0 0 0 0 rang@ 0 0 0 0 A D rang@ 0 0 0 A D 1: 0 0 12 0 0 12 d3 In this case, the system for x0 degenerates into the equation 1 x3 D d3 ; 2

d3 2 Œ1; 1

and, therefore, X0 D ¹.x1 ; x2 ; x3 /T W x1 ; x2 2 R; x3 2 Œ2; 2º: Remark 8. If ƒ 1 and there exists i D 1, then ordinary periodic solutions exist for all d 2 D provided that a periodic R-solution exists. Indeed, the rank of the extended matrix 0 1 0 0 0 0 d1 B e 1 0 0 0 d2 C B i C B :: : : : : :: :: :: C B : : : : : C .E J; d / D B : C B 0 e k1 0 0 dk C i @ A : : :: 0 0 0 : : is equal to the rank of the matrix E J in view of the fact that d1 D 0 and dj C1 D 0 j for all j 2 ¹1; : : : ; k 1º such that ei D 0. Otherwise, dj D dj D const. Relationship Between the Sets X0 and R0 . As shown above, it is clear that Eq. (3.38) may have more than one solution. Therefore, we consider a set R deﬁned as the union of the sets R0 speciﬁed by Eq. (3.38). We now show that X0 R provided that R exists.

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

111

By using representation (3.40) for the matrix ˆ.T; 0/, by analogy with (3.41), we conclude that Eq. (3.39) is equivalent to the equation y D Jy C d;

where d 2 D; y D M x0 :

In view of the “cellular” structure of the matrix J , this equation decomposes into s independent equations of the form y i D Ji y i C d i ;

i D 1; s:

(3.57)

For the numbers i such that j i j > 1, the vector d i is chosen from a one-point set. Moreover, by virtue of systems (3.45), (3.46), and (3.47), the sets Y0 D M X0 and Z D MR coincide in the variables corresponding to a given eigenvalue. In addition, they can be regarded as sets of lower dimensions. Let i be such that j i j D 1. Then the vector d i is also chosen from a one-point set, the component of the set Z0 in the corresponding variables is not unique, and the union of the sets Z0 covers the entire space in a part of variables (for which the property of uniqueness is absent). At the same time, Y0 is a one-point set in the case where i ¤ 1 and, hence, Y0 is a strict subset of Z in the variables corresponding to a given eigenvalue. However, if i D 1, then a part of variables in system (3.57) turns out to be free (these are the variables for which Z0 is not unique). Therefore, Y0 coincides with Z in the variables corresponding to the given eigenvalue. Consider the set I of all i such that j i j < 1. By analogy with (3.42), Eqs. (3.57) with i 2 I are equivalent to the equation y I ."/ D JI ."/y I ."/ C d I ."/;

(3.58)

where d I ."/ 2 D I ."/ D LD I , y I ."/ D Ly I , y I is a vector formed by the vectors y i ; i 2 I , and D I and JI are deﬁned similarly. In these variables, for any d I ."/ 2 D I ."/, the solution of Eq. (3.58) is unique because the matrix E JI ."/ is nondegenerate. We now show that Y ."/ D LY0 is a subset of the set Z ."/ D LZ in the corresponding variables. The set Z I ."/ is a nonempty convex compact set. Hence, this set is a complete metric space. A mapping .y/ D JI ."/y C d I ."/; d I ."/ 2 D I ."/ is contracting because ..y1 /; .y2 // D .JI ."/y1 C d I ."/; JI ."/y2 C d I ."// kJI ."/k.y1 ; y2 /; and kJI ."/k < 1. Therefore, has a unique ﬁxed point in Z I ."/. Thus, in the corresponding variables, Y ."/ Z ."/ and, hence, X0 R . As follows from Example 7, the sets Y ."/ and Z ."/ do not coincide in the general case.

112

Chapter 3 Linear Impulsive Differential Inclusions

Remark 9. The accumulated results can readily be extended to the case of measurable A.t / and F .t /. Remark 10. A linear differential inclusion .Bi D 0, Pi D 0 for all i / and a linear discrete inclusion .A.t / 0; F .t / 0/ are special cases of the impulsive differential inclusion (3.37). Remark 11. If F .t / and Pi are one-point sets [i.e., (3.37) is a linear impulsive differential equation], then Eqs. (3.38) and (3.39) specify different objects: Eq. (3.39) speciﬁes a periodic solution, whereas Eq. (3.38) speciﬁes a periodic bundle of solutions. Example 12. Consider an impulsive differential equation xP D 0;

t ¤ 2k 1;

xj t D2k1 D 2x: This equation possesses a unique 2-periodic ordinary solution x.t; 0/ 0 and a one-parameter family of 2-periodic bundles of solutions R.t; Œa; a/ D Œa; a, a 0. Remark 12. The accumulated results can be extended to the case where the segment Œ0; T contains inﬁnite many times of pulses. In this case, for the set R0 , we get an equation of the form (3.38) with the sole difference that the sum in the deﬁnition of the set G contains inﬁnitely many terms [95]. We now consider sufﬁcient conditions for the existence of periodic R-solutions of linear inhomogeneous periodic differential inclusions with pulse action of the form xP 2 A.t /x C F .t /;

t ¤ i ;

xj tDi 2 Bi x C Pi ;

(3.59) (3.60)

where t 2 R is time, x 2 Rn is the phase vector, AW R ! comp.Rnn / and F W R ! comp.Rn / are measurable T -periodic set-valued mappings, jA.t /j ˛.t /; jF .t /j .t /, where ˛.t / and .t / are summable on Œ0; T , and the compact sets Bi of .nn/matrices, sets Pi 2 conv.Rn /, and times i are such that BiCr D Bi ;

PiCr D Pi ;

i Cr D i C T

for all i 2 Z and some natural r. We also assume that 0 1 < < r < T and det.E C Bi / ¤ 0 for all Bi 2 Bi , i D 1; r. We now prove several auxiliary assertions.

113

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

Let ˆA .t; s/ be the matrizant of system (3.59) corresponding to a measurable branch A.t / of the set-valued mapping A.t /, i.e., the solution of the matrix Cauchy problem dX D A.t /X; dt

X.s; s/ D E:

(3.61)

Lemma 2. A set of matrizants ˆA .t; s/ D ¹ˆA .t; s/W A.t / 2 A.t /º is a nonempty compact set in the space Rnn for any ﬁxed t; s 2 R; t s. Proof. We choose arbitrary real t and s such that t s. The set ˆA .t; s/ is nonempty because, by virtue of the Filippov theorem, there exists a summable branch of the setvalued mapping A.t / and, by virtue of the Carathéodory theorem for linear systems, there exists a solution of the matrix problem (3.61) . We now show that the set ˆA .t; s/ is bounded. The matrix ˆA .t; s/ is represented in the form Z t Z t Z t1 A.t1 /dt C A.t1 / A.t2 /dt2 dt1 C : ˆA .t; s/ D E C s

s

s

As a result, we get the following sequence of estimates: Z t Z t Z t1 kA.t1 /kdt C kA.t1 /k kA.t2 /kdt2 dt1 C kˆA .t; s/k 1 C s

Z 1C

t

s

s

Z ˛.t1 /dt C

t s

˛.t1 /

s

Z

t1 s

˛.t2 /dt2 dt1 C :

The validity of the inequality Z

Z

tQ s

˛.t1 /

t1 s

Z ˛.t2 /

tk1 s

˛.tk /dtk : : : dt2 dt1

k .tQ; s/ kŠ

(3.62)

for all tQ 2 .s; t is proved by induction. For k D 1, this inequality is true. Assume that (3.62) holds for k D m. Then, for k D m C 1, we get Z

Z

tQ s

˛.t1 / Z

t1 s

tQ s

˛.t1 /

Z ˛.t2 /

tm s

˛.tmC1 /dtmC1 : : : dt2 dt1

m .t1 ; s/ mC1 .tQ; s/ dt1 D ; mŠ .m C 1/Š

Thus, kˆA .t; s/k

1 X k .t; s/ D e .t;s/ ; kŠ

kD0

and the boundedness of the set ˆA .t; s/ is proved.

(3.63)

114

Chapter 3 Linear Impulsive Differential Inclusions

We now show that the set ˆA .t; s/ is closed, i.e., the limit of any convergent sequence of matrices ˆAk .t; s/ 2 ˆA .t; s/ also belongs to the set ˆA .t; s/. In view of the equivalence of the differential equation to a Volterra integral equation, we get the following representation: Z ˆAk .t; s/ D E C

t

Ak ./ˆAk .; s/d

s

(3.64)

By analogy with (3.63), we get kˆAk .; s/k e .t;s/ for all 2 Œs; t . Since Z kˆAk .t2 ; s/ ˆAk .t1 ; s/k D ˇZ ˇ ˇˇ

t2 t1 t2

t1

Ak ./ˆAk .; s/d

ˇ ˇ ˛./e .t;s/ d ˇˇ D e .t;s/ j.t2 ; s/ .t1 ; s/j;

where the function .; s/ is absolutely continuous on Œs; t , for any " > 0, one can ﬁnd ı."/ > 0 such that the inequality j.t2 ; s/ .t1 ; s/j "e .t;s/ is true for any t1 ; t2 2 Œs; t W jt2 t1 j < ı and, hence, kˆAk .t2 ; s/ ˆAk .t1 ; s/k < ": Thus, the sequence of functions ˆAk .; s/ is uniformly bounded and equicontinuous on Œs; t . Hence, by the Arzelà theorem, this sequence contains a subsequence uniformly convergent to a continuous matrix function ˆ .; s/. This means that, for any " > 0, one can ﬁnd k0 such that the inequality kˆAk .; s/ ˆ .; s/k <

" .t; s/

holds for all k > k0 and 2 Œs; t . Since Z t Z t Ak ./ˆAk .; s/d D Ak ./ŒˆAk .; s/ ˆ .; s/d s

s

C

Z

t s

Ak ./ˆ .; s/d;

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

where

115

Z t Ak ./ŒˆAk .; s/ ˆ .; s/d s Z t ˛./kˆAk .; s/ ˆ .; s/kd < "; s

and, by virtue of the Lyapunov theorem, there exists a subsequence Ak1 ./ of the sequence Ak ./ weakly convergent to the matrix A ./ 2 A./ on Œs; t , we conclude that Z t Z t Ak1 ./ˆAk .; s/d ! A ./ˆ .; s/d as k1 ! 1: s

s

Passing to the limit in (3.64), we ﬁnd Z ˆ .t; s/ D E C

t s

A ./ˆ .; s/d;

i.e., ˆ .t; s/ D ˆA .t; s/ 2 ˆA .t; s/, as required. Thus, the set ˆA .t; s/ 2 comp.Rnn /. Let ˆABi .t; s/ be the matrizant of system (3.59), (3.60) corresponding to the matrices A.t / 2 A.t /, Bi 2 Bi , i.e., the solution of the matrix Cauchy problem for the system with pulse action XP D A.t /X; X j tDi D Bi X;

t ¤ i ; X.s; s/ D E:

By virtue of [143], for the matrizant ˆABi .t; s/, we get ˆABi .t; s/ D ˆA .t; k /.E C Bk /ˆA .k ; k1 / .E C Bp /ˆA .p ; s/; p < s pC1 ;

(3.65)

k < t kC1 :

Lemma 3. The set of matrizants ˆABi .t; s/ D ¹ˆABi .t; s/W A.t / 2 A.t /; Bi 2 Bi º is a nonempty compact set in the space Rnn for any ﬁxed t; s 2 R, t s. Proof. We choose arbitrary real t and s such that t s. For the sets F; G 2 comp.Rnn /, we deﬁne the operation of multiplication as follows: F G D ¹f gW f 2 F; g 2 Gº: It is clear that F G 2 comp.Rnn /. Indeed, F G is nonempty (in view of the fact that the sets F and G are nonempty) and bounded because, for any matrix M 2 F G, there exist f0 2 F and g0 2 G such that M D f0 g0 and, hence, kM k kf0 k kg0 k jF j jGj < 1; where jF j is the modulus of the set F .

116

Chapter 3 Linear Impulsive Differential Inclusions

We now show that the set F G is closed. We choose an arbitrary sequence of matrices Mk 2 F G convergent to a matrix M 2 Rnn . It is necessary to show that M 2 F G. By the deﬁnition of the operation of multiplication of sets, for any k, there exist fk 2 F and gk 2 G such that the following representation is true: Mk D fk gk . Since the sets F and G are compact, there exist subsequences of the sequences ¹fk º and ¹gk º convergent to f 2 F and g 2 G respectively. Then M D f g 2 F G. By using representation (3.65) for the matrizant, we represent the set ˆABi .t; s/ in the form ˆABi .t; s/ D ˆA .t; k /.E C Bk /ˆA .k ; k1 / .E C Bp /ˆA .p ; s/; p < s pC1 ;

k < t kC1 :

By virtue of Lemma 2 and the fact that the sets Bi are compact, the set ˆABi .t; s/ 2 comp.Rnn /. In view of the T -periodicity of the right-hand sides of inclusion (3.59), (3.60), the existence of T -periodic R-solutions is directly connected with the existence of solutions to the equation [

R0 D

¹ˆABi .T; 0/R0 C GABi º

(3.66)

A.t/2A.t / Bi 2Bi

in the space comp.Rn /, where Z GABi D

T 0

ˆABi .T; /F . /d C

X

ˆABi .T; i /Pi 2 comp.Rn /:

0i
Theorem 10. Assume that the inequality kˆABi .T; 0/k < 1 is true for any A.t / 2 A.t / and Bi 2 Bi . Then inclusions (3.59), (3.60) possess a unique T -periodic R-solution. Proof. Consider an operator .R/ D

mapping the space comp.Rn / into itself as follows: [ A.t/2A.t / Bi 2Bi

¹ˆABi .T; 0/R C GABi º:

Section 3.3 Periodic Solutions of Linear Impulsive Differential Inclusions

117

It is necessary to show that this operator is contracting. To do this, we choose arbitrary sets R1; R2 2 comp.Rn /. Then h. .R1 /, .R2 // D sup.d1 ; d2 /, where [ d1 D ˆABi .T; 0/r1 C gABi ; ¹ˆABi .T; 0/R2 C GABi º ; d2 D ˆABi .T; 0/r2 C gABi ;

A.t/2A.t / Bi 2Bi

[

¹ˆABi .T; 0/R1 C GABi º ;

A.t/2A.t / Bi 2Bi

r1 2 R1 ; r2 2 R2 . Further, we estimate d1 . We choose r2 2 R2 such that kr1 r2 k D .r1 ; R2 /. Then d1 kˆABi .T; 0/r1 C gABi ŒˆABi .T; 0/r2 C gABi k kˆABi .T; 0/k .r1 ; R2 / kˆABi .T; 0/k h.R1 ; R2 /: Similarly, d2 kˆABi .T; 0/k h.R1 ; R2 /. Hence, h. .R1 /; .R2 // h.R1 ; R2 /

sup A.t/2A.t / Bi 2Bi

kˆABi .T; 0/k:

We now show that sup A.t/2A.t / Bi 2Bi

kˆABi .T; 0/k < 1:

Since, by virtue of Lemma 2, the set of matrizants ˆABi .T; 0/ is compact in the space comp.Rnn / and k k is a continuous function, by the Weierstrass theorem, we conclude that sup A.t/2A.t / Bi 2Bi

kˆABi .T; 0/k D

max kˆABi .T; 0/k < 1:

A.t/2A.t / Bi 2Bi

Thus, .R/ is a contracting operator and, by the Banach theorem, has a single ﬁxed point R0 2 comp.Rn / [14], i.e., Eq. (3.66) is uniquely solvable. Hence, inclusion (3.59), (3.60) possesses a unique T -periodic R-solution. Example 13. Consider a controlled system xP D ux C 1;

u 2 Œ2; 1

which can be represented in the form of a differential inclusion xP 2 Ax C 1;

A D ¹uW u 2 Œ2; 1º:

(3.67)

118

Chapter 3 Linear Impulsive Differential Inclusions

The right-hand side is T -periodic, where T > 0 is an arbitrary constant. The initial sets of T -periodic R-solutions are determined by Eq. (3.66). In this case, Eq. (3.66) takes the form ³ [ ² e uT 1 uT R0 D : (3.68) e R0 C u u2Œ2;1

Since kˆA .T; 0/k D e uT < 1 for all u 2 Œ2; 1, by virtue of Theorem 10, Eq. (3.68) possesses a unique solution R0 2 comp.R/. In view of the fact that R0 is a connected set in R, we conclude that R0 D Œa; b. Then Eq. (3.68) is reduced to e 2T 1 T 2T T ; e b .e a 1/ Œa; b D e 2 ´ ´ 2T a D 12 ; a D e 2T a e 2 1 ; , , b D 1: b D e T b .e T 1/ Hence, R0 D Œ 12 ; 1. Thus, for any T > 0, inclusion (3.67) possesses a unique T -periodic R-solution R.t; Œ 12 ; 1/. We now establish sufﬁcient conditions for the existence of periodic ordinary solutions of nonlinear impulsive differential inclusions of the form xP 2 A.t /x C F .t; x/;

t ¤ i ;

(3.69)

xj t Di 2 Bi x C Ii .x/; where x 2 Rn is the phase vector, t 2 R is time, A.t / is a continuous T -periodic matrix, F W RRn ! conv.Rn / is a set-valued mapping continuous in its variables, T periodic in t , and bounded (i.e., there exists a set-valued mapping QW R ! conv.Rn / such that the inclusion F .t; x/ Q.t / is true for any ﬁxed t 2 R and all x 2 Rn /, the matrices Bi , the set-valued mappings Ii W Rn ! conv.Rn /, and the times of pulses i are such that BiCp D Bi ;

IiCp .x/ Ii .x/;

iCp D i C T

(3.70)

for all i 2 Z and some natural p, and the sets Ii .x/ are bounded, i.e., there exist sets Pi 2 conv.Rn / such that the inclusion Ii .x/ Pi is true for all i D 1; p. We also assume that 0 1 < < p < T and det.E C Bi / ¤ 0 for all i D 1; p. Let R.T; x/ be the set of attainability (R-solution) of inclusion (3.69) from the initial point .0; x/ at time t D T . Together with inclusion (3.69), we consider the inclusion xP 2 A.t /x C Q.t /; xj t Di 2 Bi x C Pi :

t ¤ i ;

(3.71)

119

Section 3.4 Linear Differential Equations with Pulse Action at Indeﬁnite Times

Assume that inclusion (3.71) possesses a T -periodic R-solution R.t; R0 /. Then the mapping R.x/ D R.T; x/ maps the set R0 into itself because R.x/ R.T; R0 / D R0 for all x 2 R0 . The set-valued mapping R.x/ is upper semicontinuous [23]. Hence, by the Kakutani theorem [14], there exists a ﬁxed point x0 2 R0 of the given mapping. Thus, there exists a periodic solution of the initial inclusion (3.69).

3.4

Linear Differential Equations with Pulse Action at Indeﬁnite Times

Consider a linear differential equation with pulses at indeﬁnite times: xP D A.t /x C f .t /;

t ¤ i ;

(3.72)

xj tDi D Bi x C pi ; where x 2 Rn is the phase vector, t 2 I D Œt0 ; T is time, A.t / is a matrix function continuous on I , f .t / is a vector function continuous on I , i 2 Œi ; iC I , i D 1; m, are the times of pulses, Œi ; iC are disjoint segments, Bi are .n n/matrices, and pi 2 Rn . An equation of the form (3.72) describes, e.g., the physical processes subjected to pulse actions at times known with certain errors. Let .t0 ; T / D .1 ; : : : ; m / and let ˆ.t; t0 ; .t0 ; T // be the matrizant of the homogeneous impulsive differential equation corresponding to (3.72): xP D A.t /x;

t ¤ i ;

xj tDi D Bi x: By virtue of relation (1.50) in Chapter 1, the following representation is true for k t < kC1 : ˆ.t; t0 ; .t0 ; T // D e

Rt k

A.s/ds

.E C Bk /

k1 Y R j C1

e

j

A.s/ds

.E C Bj /e

R 1 t0

A.s/ds

:

j D1

Thus, every solution x.t; x0 ; .t0 ; T //; x.t0 ; x0 ; .t0 ; T // D x0 of Eq. (3.72) for t 2 I can be represented in the form x.t; x0 ; .t0 ; T // D ˆ.t; t0 ; .t0 ; T //x0 Z t X C ˆ.t; s; .t0 ; T //f .s/ds C ˆ.t; i ; .t0 ; T //pi : t0

t0 i
(3.73)

120

Chapter 3 Linear Impulsive Differential Inclusions

By X.t; t0 ; x0 / we denote the bundle of solutions originating from the initial point .t0 ; x0 / and corresponding to different times of pulses 1 ; : : : ; m . Theorem 11 ([119]). X.t; t0 ; x0 / 2 comp.Rn / for any ﬁxed t 2 I . Proof. By virtue of [142], the set X.t; t0 ; x0 / is nonempty. Denote M1 D max kA.t /k; t 2I

M2 D max kf .t /k: t 2I

Then kx.t; x0 ; .t0 ; T //k kˆ.t; t0 ; .t0 ; T //k kx0 k Z t kˆ.t; s; .t0 ; T //k kf .s/kds C t0

X

C

kˆ.t; i ; .t0 ; T //k kpi k

t0 i

m Y

.1 C kBi k/e

M1 .T t0 /

² kx0 k C M2 .T t0 / C

iD1

m X

³ kpi k

iD1

D K: Hence, the set X.t; t0 ; x0 / is bounded. We now show that the set X.t; t0 ; x0 / is closed. Consider a convergent sequence xk .t; x0 ; k .t0 ; T // 2 X.t; t0 ; x0 /: It is necessary to show that its limit x.t / 2 X.t; t0 ; x0 /. Since 1k 2 Œ1 ; 1C , there exists a subsequence ¹k1 º of the sequence ¹kº such that 1k1 ! 10 2 Œ1 ; 1C . We choose a subsequence ¹k2 º of the sequence ¹k1 º such that 2k2 ! 20 2 Œ2 ; 2C . Further, we continue this process and, ﬁnally, construct a sequence ¹km º km 0 2 Œ ; C . ! m such that m m m Since ˆ.t; t0 ; .t0 ; T // is a continuous function of the last argument, we get ˆ.t; t0 ; km .t0 ; T // ! ˆ.t; t0 ; 0 .t0 ; T // as km ! 1. Thus, by virtue of relation (3.73), we ﬁnd x.t / D lim xk .t; x0 ; k .t0 ; T // D lim xkm .t; x0 ; km .t0 ; T // km !1

k!1

0

D x.t; x0 ; .t0 ; T // 2 X.t; t0 ; x0 /: Hence, the compactness of the set X.t; t0 ; x0 / is proved.

Section 3.4 Linear Differential Equations with Pulse Action at Indeﬁnite Times

121

Remark 13. The assertion of the theorem remains true if I D Œt0 ; C1/ and i˙ ! C1 as i ! C1. To prove this, we successively consider Eq. (3.72) on the segments Œt0 ; Tp , where Tp ! C1, and conclude that the set X.t; t0 ; x0 / is compact for any ﬁxed t 2 Œt0 ; Tp and all p 2 N . We now consider the problem of existence of an !-periodic solution x.t; x0 / and a periodic bundle of solutions X.t; 0; X0 /, X0 2 comp.Rn /, for equations of the form (3.72) in the case where I D Œ0; C1/ and the right-hand side is !-periodic, i.e., the functions A.t / and f .t / are assumed to be !-periodic and the matrices Bi , vectors pi , and times i˙ are such that BiCr D Bi ;

piCr D pi ;

˙ iCr D i˙ C !

for all i 2 N and a natural number r. The problem of existence of periodic solutions and periodic bundles of solutions is reduced to the problem of existence of solutions of the equations x0 D ˆ.!; 0; .0; !//x0 C d. .0; !//; where

Z

d. .0; !// D

! 0

ˆ.!; s; .0; !//f .s/ds C

X

(3.74)

ˆ.!; i ; .0; !//pi

t0 i
and

[

X0 D

¹ˆ.!; 0; .0; !//X0 C d. .0; !//º:

(3.75)

i 2Œi ;iC

Equation (3.74) is solvable if and only if either the matrix E ˆ.!; 0; .0; !// is nondegenerate for at least one collection i 2 Œi ; iC or the rank of the extended matrix .E ˆ.!; 0; .0; !//; d. .0; !/// is equal to the rank of the initial matrix. In this case, different i 2 Œi ; iC correspond to different matrices ˆ.!; 0; .0; !// and vectors d. .0; !// and, hence, a single initial point may correspond to several periodic solutions. Consider the problem of solvability of Eq. (3.75). Theorem 12 ([119]). If the inequality kˆ.!; 0; .0; !//k M < 1 holds for all i 2 Œi ; iC ; i D 1; r, then Eq. (3.72) possesses a single !-periodic bundle of solutions. Proof. Consider a mapping .X0 / D

[

¹ˆ.!; 0; .0; !//X0 C d. .0; !//º:

i 2Œi ;iC

122

Chapter 3 Linear Impulsive Differential Inclusions

By using the properties of the Hausdorff distance, we obtain [ ¹ˆ.!; 0; .0; !//X0 C d. .0; !//º; h..X0 /; .Y0 // D i 2Œi ;iC

[

¹ˆ.!; 0; .0; !//Y0 C d. .0; !//º D sup¹d1 ; d2 º;

i 2Œi ;iC

where

d1 D ˆ.!; 0; .0; !//x0 C d. .0; !//; [

¹ˆ.!; 0; .0; !//Y0 C d. .0; !//º

i 2Œi ;iC

.ˆ.!; 0; .0; !//x0 C d. .0; !//; ˆ.!; 0; .0; !//Y0 C d. .0; !/// kˆ.!; 0; .0; !//k.x0 ; Y0 / M h.X0 ; Y0 / < h.X0 ; Y0 /; d2 D ˆ.!; 0; .0; !//y0 C d. .0; !//; [

(3.76)

¹ˆ.!; 0; .0; !//X0 C d. .0; !//º

i 2Œi ;iC

.ˆ.!; 0; .0; !//y0 C d. .0; !//; ˆ.!; 0; .0; !//X0 C d. .0; !/// kˆ.!; 0; .0; !//k.y0 ; X0 / M h.X0 ; Y0 / < h.X0 ; Y0 /:

(3.77)

Thus, the mapping .X0 / is contracting. Hence, all conditions of the Banach theorem [14] are satisﬁed. This means that the mapping .X0 / has a single ﬁxed point in comp.Rn /. We now consider the problem of stability of a bundle of solutions of the impulsive differential Eq. (3.72) [119]. Theorem 13. Assume that the matrizants ˆ.t; t0 ; .0; C1// of Eq. (3.72) are uniformly bounded. Then the bundle of solutions X.t; t0 ; x0 / is Lyapunov stable in the positive direction. Proof. We now show that, for any " > 0, there exists ı."/ > 0 such that the estimate h.X.t; t0 ; x0 /; X.t; t0 ; y0 // < " is true for kx0 y0 k < ı.

Section 3.4 Linear Differential Equations with Pulse Action at Indeﬁnite Times

Further, we estimate h.X.t; t0 ; x0 /; X.t; t0 ; y0 // D sup¹d1 .t /; d2 .t /º; where, by analogy with estimates (3.76) and (3.77), d1 .t / D .x.t; x0 /; X.t; t0 ; y0 // M kx0 y0 k; d1 .t / D .x.t; y0 /; X.t; t0 ; x0 // M kx0 y0 k: Finally, we choose ı D

" M

and arrive at the required assertion.

123

Chapter 4

Linear Systems with Multivalued Trajectories

4.1

Differential Equations with Hukuhara Derivative

The spaces comp.Rn / and conv.Rn / are not linear because they do not contain the opposite element and, hence, the operation of subtraction. However, the operation of subtraction is urgently required in numerous cases and, hence, one can use several approaches for its deﬁnition. In what follows, we consider the notion of Hukuhara difference [58] in more detail. Deﬁnition 1 ([15]). Let X; Y 2 conv.Rn /. A set Z 2 conv.Rn / such that X D h Y C Z is called the difference of sets X and Y and denoted as follows: X Y. Remark 1. If the difference exists, then it is uniquely deﬁned. The operation of subtraction has following properties [25]: h h h h (1) .X C Y / .U C V / D .X U / C .Y V / if the differences X U and h Y V exist. h h h h h h (2) .X U/ .Y V / D .X Y / C .V U / if the differences X Y, h h V U and Y V exist. h h h h h h h h U/ .Y V / D .X Y/ .U V / if the differences .X Y/ (3) .X h h V / and Y V exist. .U h h h .~U CV / D . ~/X C~.X U /C./Y C C.Y V/ (4) . X CY / h h if ~ 0; 0, and the differences X U and Y V exist. h h h h h U D .X V / C .V U / if the differences X V and V U exist. (5) X h h h h (6) h.X U; Y V / h.X; Y / C h.U; V / if the differences X U and Y V exist. h h Y; 0/ if the difference X Y exists. (7) h. X; Y / D h.X

(8) h. X; Y / ˇh.X; Y / C j j.h.X; 0/ C h.Y; 0// provided that ˇ D max¹ ; º. (9) The operation of subtraction is continuous relative to the Hausdorff metric: If 1 sequences ¹Xn º1 nD1 and ¹Yn ºnD1 converge to X and Y and the differences h h Xn Yn exist for all n 2 N , then the difference X Y exists and the seh h 1 quence ¹Xn Yn ºnD1 converges to X Y .

Section 4.1 Differential Equations with Hukuhara Derivative

125

Deﬁnition 2 ([24]). A set-valued mapping X W Œ0; T ! conv.Rn / is differentiable in Hukuhara’s sense at a point t0 2 .0; T / if there exists Dh X.t0 / 2 conv.Rn / such that the limits h X.t0 C t /X.t 0/ t t#0

lim

and

h X.t0 /X.t 0 t / t t #0

lim

exist and are equal to Dh X.t0 /. Note that, in this deﬁnition, it is assumed that, for all sufﬁciently small t > 0, the h h differences X.t0 /X.t 0 t /, X.t0 C t /X.t0 / exist. It makes sense to speak about unilateral derivatives at the points t D 0 and t D T . A differential equation with Hukuhara derivative was considered for the ﬁrst time in [25]: (4.1) Dh X D F .t; X /; X.0/ D X0 ; where F W Œ0; T conv.Rn / ! conv.Rn / is a set-valued mapping, X0 2 conv.Rn / is an initial state, and Dh X is the Hukuhara derivative of a set-valued mapping X W Œ0; T ! conv.Rn /. Deﬁnition 3 ([29]). A set-valued mapping X. / is called a solution of Eq. (4.1) if it is continuously differentiable in Hukuhara’s sense and satisﬁes system (4.1) everywhere on Œ0; T . The differential Eq. (4.1) is equivalent to the integral equation [25] Z t F .s; X.s// ds; X.t / D X0 C 0

the integral in which is understood in Hukuhara’s sense [58]. The following theorem on existence and uniqueness is true: Theorem 1 ([29]). Assume that F . ; / satisﬁes the conditions: (1) F . ; / is continuous in .t; X / on Œ0; T conv.Rn /; (2) F .t; / has the Lipschitz property with respect to X on conv.Rn /, i.e., there exists a constant L > 0 such that h.F .t; X /; F .t; Y // Lh.X; Y /: Then system (4.1) is uniquely solvable. Example 1. Consider a linear differential equation of the form Dh X D .t /X C F .t /;

X.0/ D X0 ;

(4.2)

126

Chapter 4 Linear Systems with Multivalued Trajectories

where W R ! RC is summable, F W R ! conv.Rn / is measurable, h.F .t /; 0/ k.t /, k W R ! RC is summable, and X0 2 conv.Rn /. By using the properties of the Hukuhara derivative, one can easily show that the set-valued mapping X. / deﬁned, for any t 0, by the formula Z t Rs Rt X.t / D e 0 .s/ ds X0 C F .s/e 0 ./ d ds 0

is a solution of Eq. (4.2). The other interesting result established for differential equations with Hukuhara derivative is the construction of an Euler broken line and estimation of the error. We split the segment Œ0; T into N parts as follows: 0 D t0 < t2 < < tN D T; Ik D Œtk ; tkC1 ;

tkC1 tk D ı;

k D 0; N 1;

and construct the Euler broken line Xk .t / D Xk1 .tk1 / C .t tk1 /F .tk1 ; Xk1 .tk1 //; t 2 Ik1 ;

X0 .t0 / D X0 ;

k D 1; N :

Denote R D sup D.X; Xk /; k

D.X; Xk / D max h.X.t /; Xk .t //; Ik1

k D 1; N :

Theorem 2 ([26]). Assume that F . ; / satisﬁes the conditions: (1) F . ; / is continuous in .t; X / on Œ0; T conv.Rn /; (2) F .t; / satisﬁes the Lipschitz condition in X with constant L; (3) the solution X. / of system (4.1) has the second continuous derivative on Œ0; T such that h.Dh .Dh X.t //; 0/ < K; t 2 Œ0; T : Then the error R satisﬁes the inequality R<

ıK Œ.1=L C ı/.e T L 1/ C ı: 2

In [145], the differential equations with Hukuhara derivative were essentially used for the analysis of some properties of the integral funnel of a differential inclusion. Thus, it was shown that the integral funnel is a subset of the solution of the corresponding equation with Hukuhara derivative. In recent years, the interest in differential equations with Hukuhara derivative increases in connection with their applications to differential equations under the conditions of uncertainty [38, 75, 76].

Section 4.1 Differential Equations with Hukuhara Derivative

127

Consider a system of linear differential equations with Hukuhara derivative Dh Xi .t / D

n X

aij .t /Xj .t / C Fi .t /;

(4.3)

j D1

Xi .0/ D Xi0 ;

i D 1; n;

where t 2 Œ0; T is time and aij W Œ0; T ! R and Fi W Œ0; T ! conv.Rm / are continuous functions. Deﬁnition 4 ([125]). Set-valued mappings Xi W Œ0; T ! conv.Rm /, i D 1; n, are called the solution of problem (4.3) if they are continuously differentiable in Hukuhara’s sense and satisfy system (4.3) everywhere on Œ0; T . We now show that the solution of system (4.3) can be reduced to a solution of a system of ordinary differential equations. Let c.A; / be the support function of a set A. By using the properties of the support functions, we conclude that X X n n c aij .t /Xj .t /CFi .t /; D c.Xj .t /; aij .t / / C c.Fi .t /; / j D1

j D1

D

n X

jaij .t /jc.Xj .t /; sign.aij .t // / C c.Fi .t /; /:

j D1

We now ﬁnd the support function of the left-hand side of Eq. (4.3): ! h Xi .t C t /X i .t / c.Dh Xi .t /; / D c lim ; t t #0 ! h Xi .t C t /X i .t / ; D lim c t t#0

1 h D lim c Xi .t C t /X i .t /; t#0 t 1 d Œc.Xi .t C t /; / c.Xi .t /; / D c.Xi .t C 0/; /: dt t#0 t

D lim Similarly, we get

c.Dh Xi .t /; / D

d c.Xi .t 0/; /; dt

i.e., c.Dh Xi .t /; / D

d c.Xi .t /; /: dt

128

Chapter 4 Linear Systems with Multivalued Trajectories

In view of the convexity of the sets Xi .t / and Fi .t /, system (4.3) is equivalent to the system n X d c.Xi .t /; / D jaij .t /jc.Xj .t /; sign.aij .t // / C c.Fi .t /; /; dt

(4.4)

j D1

c.Xi .0/; / D c.Xi0 ; /;

i D 1; n;

for all 2 S1 .0/ Rm . We now introduce the following functions: fiC .t; / D c.Fi .t /; /;

fi .t; /

D c.Fi .t /; /;

xiC .t;

xi .t;

D c.Xi .t /; /:

/ D c.Xi .t /; /;

/

This enables us to rewrite system (4.4) in the form n X d C 1 C sign.aij .t // C xi .t; / D xj .t; / jaij .t /j dt 2 j D1

C

1 sign.aij .t // xj .t; / C fiC .t; /; 2

n X 1 sign.aij .t // C d jaij .t /j x .t; / D xj .t; / dt i 2 j D1

xiC .t; / D c.Xi0 ; /;

1 C sign.aij .t // C xj .t; / C fi .t; /; 2 xi .t; / D c.Xi0 ; /;

(4.5)

i D 1; n:

Thus, we arrive at a system of 2n ordinary differential equations with a parameter 2 S1C .0/ D ¹ 2 S1 .0/ W 1 0º. Since the sets Xi .t /; i D 1; n, are convex, they are uniquely determined by their support function: \ Xi .t / D ¹x 2 Rm W .x; / xiC .t; /; .x; / xi .t; /º: (4.6) 2S1C .0/

However, the efﬁcient reconstruction of a set according to its support function can be realized only for small values of m. Thus, we have proved the following assertion: Theorem 3 ([125]). A solution of system (4.3) is given by n functions speciﬁed by equalities (4.6), where xi˙ .t; /; i D 1; n, is the solution of the system of ordinary differential Eqs. (4.5).

129

Section 4.1 Differential Equations with Hukuhara Derivative

Example 2. Let m D 2. Consider a system of differential equations with Hukuhara derivative of the form Dh X1 .t / D aX1 .t / C bX2 .t / C Sr1 .0/; Dh X2 .t / D cX1 .t / C dX2 .t / C Sr2 .0/; X1 .0/ D Sı1 .0/;

(4.7)

X2 .0/ D Sı2 .0/:

For this system, we write a system of equations of the form (4.5): d C 1 C sign a C 1 sign a x1 .t; / D jaj x1 .t; / C jaj x1 .t; / dt 2 2 1 sign b 1 C sign b C x2 .t; / C jbj x2 .t; C jbj 2 2 d 1 sign a C 1 C sign a x .t; / D jaj x1 .t; / C jaj x1 .t; / dt 1 2 2 1 C sign b 1 sign b C x2 .t; / C jbj x2 .t; C jbj 2 2 d C 1 C sign c C 1 sign c x .t; / D jcj x1 .t; / C jcj x1 .t; / dt 2 2 2 1 sign d 1 C sign d C x2 .t; / C jd j x2 .t; C jd j 2 2 d 1 sign c C 1 C sign c x .t; / D jcj x1 .t; / C jcj x1 .t; / dt 2 2 2 1 C sign d 1 sign d C x2 .t; / C jd j x2 .t; C jd j 2 2 x1C .0; / D x1 .0; / D ı1 ; x2C .0; / D x2 .0; / D ı2 :

/ C r1 ;

/ C r1 ;

/ C r2 ;

/ C r2 ;

As a result of the term-by-term summation and subtraction of equations of this system, we ﬁnd d C .x .t; / C x1 .t; // D jaj.x1C .t; / C x1 .t; // dt 1 C jbj.x2C .t; / C x2 .t; // C 2r1 ; d C .x .t; / x1 .t; // D a.x1C .t; / x1 .t; // dt 1 C b.x2C .t; / x2 .t; //; d C .x .t; / C x2 .t; // D jcj.x1C .t; / C x1 .t; // dt 2 C jd j.x2C .t; / C x2 .t; // C 2r2 ;

130

Chapter 4 Linear Systems with Multivalued Trajectories

d C .x .t; / x2 .t; // D c.x1C .t; / x1 .t; // dt 2 C d.x2C .t; / x2 .t; //; x1C .0; / C x1 .0; / D 2ı1 ;

x1C .0; / x1 .0; / D 0;

x2C .0; / C x2 .0; / D 2ı2 ;

x2C .0; / x2 .0; / D 0:

Further, we solve the second and fourth equations separately and obtain x1C .t; / x1 .t; /

and x2C .t; / x2 .t; /:

Substituting these relations in the ﬁrst and third equations, we get d C x .t; / D jajx1C .t; / C jbjx2C .t; / C r1 ; dt 1 d C x .t; / D jcjx1C .t; / C jd jx2C .t; / C r2 ; dt 2 x1C .0; / D ı1 ; x2C .0; / D ı2 : Since the right-hand sides of the system are independent of

, we ﬁnd

x1C .t; / D x1 .t / and x2C .t; / D x2 .t /; and the required sets are X1 .t / D Sx1 .t/ .0/

and X2 .t / D Sx2 .t/ .0/:

By the direct substitution in Eq. (4.7), we show that the required solution is indeed given by the obtained set-valued mappings.

4.2

Approximation of the Integral Funnel of a Linear Differential Inclusion with the Help of Systems of Differential Equations with Hukuhara Derivative

Consider a linear differential inclusion xP 2 A.t /x C F .t /;

x.0/ 2 X0 2 conv.Rn /;

(4.8)

where t 2 Œ0; T , x 2 Rn is the phase vector, A.t / is a continuous n n matrix, and F W Œ0; T ! conv.Rn / is a continuous set-valued mapping. Inclusion (4.8) is equivalent, e.g., to a linear control system xP D A.t /x C D.t /u;

x.0/ 2 X0 ;

(4.9)

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion

131

where u.t / 2 U.t / 2 conv.Rp / is a control vector, U.t / is a continuous set-valued mapping, and D.t / is a continuous n p matrix. In this case, F .t / D ¹y 2 Rn W y D D.t /u.t /; u.t / 2 U.t /º. The investigation of the properties of the integral funnel of inclusion (4.8) [the set of attainability of system (4.9)] is of high signiﬁcance for the qualitative theory and the problems of control. In this connection, numerous authors studied the properties of the set of attainability [6, 91, 101, 145] and various approximate methods used for its construction: the method of ellipsoids for linear systems [30, 73, 98], asymptotic methods [41, 115], and numerical methods [91, 96, 149]. We now write the differential equation with Hukuhara derivative corresponding to the differential inclusion (4.8): Dh X.t / D A.t /X.t / C F .t /;

X.0/ D X0 :

(4.10)

Here, the solution X W Œ0; T ! conv.Rn / is a set-valued mapping continuously differentiable in Hukuhara’s sense. In [145], it is proved that R.t / X.t /; (4.11) where R.t / is the set of attainability of system (4.8). The practical construction of a given approximation for the bundle of solutions of inclusion (4.8) is connected with signiﬁcant computational difﬁculties encountered in ﬁnding the solutions of Eq. (4.10) for n > 2. The matrix A.t / is represented in the form 0 1 A11 .t / A12 .t / A1m .t / B A21 .t / A22 .t / A2m .t / C B C A.t / D B (4.12) C; :: :: :: :: @ A : : : : Am1 .t / Am2 .t / Amm .t / P where Aij .t / 2 Rni nj , m iD1 ni D n. Equation (4.10) is associated with the following system of linear equations with Hukuhara derivative: Dh Xi .t / D

m X

Aij .t /Xj .t / C Fi .t /;

Xi .0/ D Xi0 2 conv.Rni /; i

D 1; m;

j D1

(4.13) where F .t / F .t / D F1 .t / Fm .t /, Fi W Œ0; T ! conv.Rni / are continuous 0 , and X W Œ0; T ! conv.Rni / are functions set-valued mappings, X0 X10 Xm i continuously differentiable in Hukuhara’s sense. Consider a set X .t / D X1 .t / Xm .t /.

132

Chapter 4 Linear Systems with Multivalued Trajectories

Theorem 4 ([124]). The following inclusion is true for Eqs. (4.10) and (4.13) and any t 2 Œ0; T : X.t / X .t /: Proof. The sets X.t / and X .t / are convex compact sets in Rn . Hence, it sufﬁces to show that the inequality c.X.t /; / c.X .t /; / holds for all 2 Rn , We split the segment Œ0; T by the points tk D of Euler broken lines for Eqs. (4.10) and (4.13):

kT N

; k D 0; N . Consider a family

X N .tkC1 / D X N .tk / C hŒA.tk /X N .tk / C F .tk /; X N .0/ D X0 ; X m XiN .tkC1 / D XiN .tk / C h Aij .tk /XjN .tk / C Fi .tk / ; j D1

XiN .0/

Xi0 ;

D

Let

i D 1; m;

k D 0; N 1:

N

N X .t / D X1N .t / Xm .t /: N

Since X N .0/ X .0/, we have N

c.X N .0/; / c.X .0/; / for all

2 Rn . Assume that the inequality N

c.X N .tk /; / c.X .tk /; / holds for all

2 Rn and let 1

0 1

C B D @ ::: A;

i

2 Rni :

m

Then c.X N .tkC1 /; / D c.X N .tk / C hA.tk /X N .tk / C hF .tk /; / D c.X N .tk /; / C hc.X N .tk /; AT .tk / / C hc.F .tk /; / N

N

c.X .tk /; / C hc.X .tk /; AT .tk / / C hc.F .tk /; /

(4.14) (4.15)

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion m X D Œc.XiN .tk /;

C hc.XiN .tk /; .AT .tk / /i / C hc.Fi .tk /;

i/

133

i /

i D1

8 10 00 T 11 A11 .tk / AT21 .tk / ATm1 .tk / ˆ ˆ 1 ˆ BB AT .t / AT .t / AT .t / C B ˆ CC < BB 12 k 22 k m2 k C B 2 CC T C B B D .A .tk / /i D BB : CC :: :: :: :: CB ˆ @ :: AC : ˆ : : : A @ A @ ˆ ˆ : T .t / AT .t / AT .t / m A1m k mm k 2m k i 9 1 0 Pm T > j D1 Aj1 .tk / j > > C B Pm > m = X B j D1 AjT2 .tk / j C T C D DB A .t / ji k j > C B :: > A @ : j D1 > > Pm ; T j D1 Aj m .tk / j i m m X X N N T D Aj i .tk / j C hc.Fi .tk /; i / c.Xi .tk /; i / C hc Xi .tk /; i D1

m X

j D1

c.XiN .tk /;

Ch

i/

D

c.XiN .tk /; AjTi .tk / j /

C hc.Fi .tk /;

i/

j D1

iD1 m X

m X

c.XiN .tk /;

i/

Ch

i D1

m X

c.Fi .tk /;

i/

Ch

m X m X

c.XiN .tk /; AjTi .tk /

j /:

iD1 j D1

iD1

(4.16) N

We now ﬁnd the support function of the set X .tkC1 /: N

c.X .tkC1 /; / D

m X

c.XiN .tkC1 /;

i/

iD1

m m X X D c XiN .tk / C h Aij .tk /XjN .tk / C hFi .tk /; j D1

iD1

D

m X

c.XiN .tk /;

i/

Ch

D

iD1

c.XjN .tk /; ATij .tk / i /

i

C hc.Fi .tk /; i /

j D1

iD1 m X

m X

c.XiN .tk /;

i/

Ch

m X iD1

c.Fi .tk /;

i/ C

h

m X m X

c.XjN .tk /; ATij .tk /

i /:

iD1 j D1

(4.17)

134

Chapter 4 Linear Systems with Multivalued Trajectories

In view of relations (4.16) and (4.17), we obtain N

c.X N .tkC1 /; / c.X .tkC1 /; / for all 2 Rn . As N ! 1, the Euler broken lines (4.14) and (4.15) converge to solutions of (4.10) and (4.13), respectively [26]. Hence, passing to the limit and using the property of continuity of the support functions, we conclude that c.X.t /; / c.X .t /; / for all 2 Rn . Thus, X.t / X .t / for all t 2 Œ0; T . The theorem is proved. We now study the problem of variation of the set X .t / in the case of subsequent decomposition of the matrix. Assume that the th row and the th column of the matrix A.t / 2 1; m are split into the following matrices: 0 11 1 1s A .t / A12 .t / A .t / B 21 C 2s B A .t / A22 .t / A .t / C; A .t / D B : : : : :: :: :: C @ :: A s2 .t / Ass .t / As1 .t / A P s lp lq and where Apq .t / 2 R pD1 lp D n ; Ai .t / D . A1i .t / A2i .t / Asi .t / /;

(4.18)

ni lp and i D 1; m, i ¤ ; where Ap i .t / 2 R 0 1 1 Ai .t / B 2 C B A .t / C lp ni and i D 1; m; i ¤ : Ai .t / D B i:: C; where Ap i .t / 2 R @ : A Asi .t /

Together with system (4.13), we consider a system Dh XQ i .t / D

m X j D1 j ¤

s X

Aij .t /XQj .t / C

Ai .t /Xp .t / C FQi .t /; p

pD1

XQ i .0/ D Xi0 ; i D 1; m; i ¤ ; Dh Xq .t / D

m X j D1 j ¤

Aqj .t /XQj .t / C

s X

Aqp .t /Xp .t / C Fq .t /;

(4.19)

pD1 0 ; q D 1; s; Xq .0/ D Xq

where Fi .t / D FQi .t /; i ¤ , F .t / FQ .t / D F1 .t / Fs .t /, Fq W Œ0; T ! 0 X0 , X Qi W conv.Rlq / are continuous set-valued mappings, X0 2 XQ0 D X1 s

135

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion

Œ0; T ! conv.Rni / .i ¤ /, and Xq W Œ0; T ! conv.Rlq / .q D 1; s/ are setvalued mappings continuously differentiable in Hukuhara’s sense. Consider a set XQ .t / D XQ 1 .t / XQ 1 .t / X1 .t / Xs .t / XQ C1 .t / XQ m .t /: Theorem 5 ([124]). The following inclusion holds for systems (4.13) and (4.19) and any t 2 Œ0; T : X .t / XQ .t /. Proof. The sets X .t / and XQ .t / are convex compact sets in Rn . Hence, it is sufﬁcient to show that the inequality c.X .t /; / c.XQ .t /; / is true for all vectors 2 Rn . We split the segment Œ0; T by the points tk D kT N ; k D 0; N and consider the family of Euler broken lines for Eqs. (4.13) and (4.19), i.e., equalities (4.15) and XQiN .tkC1 / D XQ iN .tk / C h

X m

Aij .tk /XQjN .tk / C

j D1 j ¤

s X

N Ap i .tk /Xp .tk /

Q C Fi .tk / ;

pD1

XQ iN .0/ D Xi0 ; i D 1; m; i ¤ ;

N Xq .tkC1 /

D

N Xq .tk /

Ch

X m

Aqj .tk /XQjN .t /

s X

N Aqp .tk /Xp .tk /

C Fq .tk / ;

N 0 Xq .0/ D Xq ; q D 1; s; k D 0; N 1:

(4.20)

C

j D1 j ¤

pD1

Let N N N N N .t / X1 .t / Xs .t / XQ C1 .t / XQ m .t /: XQ N .t / D XQ 1N .t / XQ 1 N

N

Since X .0/ XQ N .0/, we have c.X .0/; / c.XQ N .0/; / for all vectors 2 Rn . Assume that the inequality N c.X .tk /; / c.XQ N .tk /; /

holds for all

2 Rn . Then, by virtue of (4.17), we get

N

c.X .tkC1 /; / D

m X

c.XiN .tk /;

i/

Ch

m X iD1

c.Fi .tk /;

i/

Ch

i D1

iD1

m X

c.XQ iN .tk /;

i/ C h

m X iD1

m X m X

c.XjN .tk /; ATij .tk /

i/

c.XQjN .tk /; ATij .tk /

i/

i D1 j D1

c.FQi .tk /;

i/ C h

m m X X iD1 j D1

136 D

Chapter 4 Linear Systems with Multivalued Trajectories

m X i D1 i ¤

Ch

c.XQ iN .tk /; m X m X i D1

D

D

8 ˆ <

N i / C c.XQ .tk /;

c.XQjN .tk /; ATij .tk /

j D1 j ¤

1

1

C B D @ ::: A;

q

2 Rlq

s

m X

c.XQ iN .tk /;

i /C

i D1 i ¤

s X

s X

m X

c.Fp .tk /;

p /

p /Ch

m X m X

i/

Ch

m X

p /

Ch

m X m X

i/

i/

c.XQjN .tk /; ATj .tk /

/

p /Ch

m X

c.Fi .tk /;

i/

iD1 i ¤

c.XQjN .tk /; ATij .tk /

i/

iD1 j D1 i¤ j ¤

N c.Xp .tk /; .ATi .tk /

i /p /

Ch

pD1

j D1 j ¤

8 ˆ ˆ < D .ATi .tk / ˆ ˆ :

s X

N c.Xp .tk /; .AT .tk /

/p /

pD1

m s X X T Ch c XQjN .tk /; .Ap j .tk //

D

c.Fi .tk /;

c.XQjN .tk /; ATij .tk /

s X N c.Xp .tk /; i /C pD1

c.Fp .tk /;

m X s X iD1 i¤

m X

j D1 j ¤

pD1

Ch

i/

iD1 i ¤

Ch

c.XQ N .tk /; ATi .tk /

i D1 i ¤

s X

iD1 j D1 i¤ j ¤

c.XQ iN .tk /;

Ch

C hc.FQ .tk /;

> ;

i D1

D

i/

9 > =

N c.Xp .tk /;

pD1 m X

c.Fi .tk /;

Q N .tk /; ATi .tk / i / C c.X

pD1

Ch

Ch

h

iD1 i¤

0

ˆ :

/ C

m X

p

pD1

00

.A1i .tk //T BB .A2 .tk //T BB i i /p D BB :: @@ :

p .Ai .tk //T

1 C C C A

.Asi .tk //T iI

.AT .tk / /p

1 C C iC A p

11

00

.A1i .tk //T BB .A2 .tk //T BB i D BB :: @@ : .Asi .tk //T

i i

i

CC CC CC AA p

/

137

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion

10 11 T .As1 .t //T .A11 1 .tk // k C B :: CC BB :: :: :: D @@ A @ : AA : : : T .Ass .t //T .A1s .t // s k k p 1 0 Ps q1 T q s qD1 .A .tk // X C B :: T D@ D .Aqp A .tk // : Ps qs T qD1 q p qD1 .A .tk // 00

D

m X

c.XQ iN .tk /;

i/

C

i D1 i ¤

Ch

s X

N c.Xp .tk /;

s X

c.Fp .tk /;

p /

Ch

m X m X

pD1

c.XQ iN .tk /;

i/

C

s X

c.Fp .tk /;

Ch

N c.Xp .tk /;

p / C

p /

Ch

h

m X

c.Fi .tk /;

i/

iD1 i¤

m X m X

c.XQjN .tk /; ATij .tk /

N T c.Xp .tk /; .Ap i .tk //

i/ C h

s s X X

i/

N T c.Xp .tk /; .Aqp .tk //

T c.XQjN .tk /; .Ap j .tk //

p /:

(4.21)

pD1

c.XQ N .tkC1 /; / m X i D1 i¤

q /

pD1 qD1

We now ﬁnd the support function of the set XQ N .tkC1 /:

D

q

qD1

p

pD1

m X s X j D1 j ¤

iD1 j D1 i¤ j ¤

m X s X iD1 i¤

i/

pD1

pD1

Ch

i/

pD1

iD1 i¤

Ch

c.XQjN .tk /; ATij .tk /

pD1

j D1 j ¤

s X

c.Fi .tk /;

s s X X N T c Xp .tk /; .Aqp i/ C h .tk //

N T c.Xp .tk /; .Ap i .tk //

m s X X N T Q Ch c Xj .tk /; .Ap j .tk //

> > ;

iD1 j D1 i¤ j ¤

m X s X iD1 i¤

m X

Ch

q

iD1 i¤

pD1

Ch

p /

pD1

m X

9 > > =

c.XQ iN .tkC1 /;

i/ C

s X qD1

N c.Xq .tkC1 /;

q /

138

Chapter 4 Linear Systems with Multivalued Trajectories

D

m X

c.XQ iN .tk /;

i/

Ch

i D1 i¤

m X

c.XQjN .tk /; ATij .tk /

i/

j D1 j ¤

Ch

s X

p N c.Xp .tk /; .Ai .tk //T

i/

C hc.Fi .tk /;

i/

pD1

C

s X

N .tk /; c.Xq

q /

Ch

m X

c.XQjN .t /; .Aqj .tk //T

q /

j D1 j ¤

qD1

Ch

s X

N T c.Xp .tk /; .Aqp .tk //

q /

C hc.Fq .tk /;

q /

pD1

D

m X

c.XQ iN .tk /;

i/

Ch

i D1 i¤

Ch

m X s X

s X

p N c.Xp .tk /; .Ai .tk //T

i/

Ch

pD1

i/

N c.Xp .tk /;

s X s X

m X

c.Fi .tk /;

i/

iD1 i ¤

p /Ch

m X s X j D1 j ¤

pD1

Ch

c.XQjN .tk /; ATij .tk /

iD1 j D1 i¤ j ¤

iD1 i ¤

C

m X m X

p /

pD1

N T c.Xp .tk /; .Aqp .tk //

pD1 qD1

T c.XQjN .tk /; .Ap j .tk // s X

c.Fp .tk /; q /Ch pD1

p /:

(4.22)

In view of relations (4.21) and (4.22), we obtain N

c.X .tkC1 /; / c.XQ N .tkC1 /; / for all 2 Rn . As N ! 1, the Euler broken lines (4.15) and (4.20) converge to the solutions of (4.13) and (4.19), respectively [26]. Hence, passing to the limit and using the property of continuity of the support functions, we conclude that c.X .t /; / c.XQ .t /; / for all 2 Rn . Thus, X .t / XQ .t / for all t 2 Œ0; T . The theorem is proved. Corollary 1 ([124]). Let 1 and 2 be decompositions of the matrix A.t / for Eq. (4.10). Assume that 2 can be obtained from 1 by additional decomposition. Let X 1 .t / and X 2 .t / be solutions of systems of the form (4.13) corresponding to the indicated decompositions. Then X 1 .t / X 2 .t / for all t 2 Œ0; T .

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion

139

Corollary 2 ([124]). Denote by Ym .t / the intersection of all sets X .t / of solutions of systems of the form (4.13) for all possible decompositions of the matrix A.t / into the matrices Aij .t /, i; j D 1; m. Then, by Theorems 4 and 5, X.t / D Y1 .t / Y2 .t / Yn .t / for all t 2 Œ0; T : The inclusion X.t / Yn .t / is proved and illustrated by model examples in [116, 120]. Remark 2 ([116]). For m D n, system (4.13) can be simpliﬁed. Let Xi .t / D xi .t / C yi .t /Œ1; 1 and Fi .t / D fi .t / C gi .t /Œ1; 1. Then system (4.13) can be represented in the form Dh ¹xi .t / C yi .t /Œ1; 1º D

n X

aij .t /¹xj .t /Cyj .t /Œ1; 1ºCfi .t /Cgi .t /Œ1; 1;

j D1

xi .0/ C yi .0/Œ1; 1 D xi0 C yi0 Œ1; 1;

i D 1; n:

By deﬁnition, the Hukuhara derivative Dh ¹xi .t / C yi .t /Œ1; 1º 1 ¹Œxi .t C / yi .t C /; xi .t C / C yi .t C / !0

D lim

Œxi .t / yi .t /; xi .t / C yi .t /º D lim

!0

1 Œxi .t C / yi .t C / .xi .t / yi .t //; C xi .t C / C yi .t C / .xi .t / C yi .t //

D

xi .t C / xi .t / .yi .t C / yi .t // ; xi .t C / xi .t / C .yi .t C / yi .t // lim !0 lim

!0

D Œ.xi .t / yi .t //0 ; .xi .t / C yi .t //0 D xP i .t / C yPi .t /Œ1; 1: Thus, system (4.13) is decomposed into two linear inhomogeneous systems of ordinary differential equations ´ P xP i .t / D jnD1 aij .t /xj .t / C fi .t /; xi .0/ D xi0 ; i D 1; n; ´ P yPi .t / D jnD1 jaij .t /jyj .t / C gi .t /; yi .0/ D yi0 ; i D 1; n; whose solutions are obtained in the explicit form.

140

Chapter 4 Linear Systems with Multivalued Trajectories

Remark 3. As mentioned above, the construction of solutions of the Hukuhara equation in spaces with dimensionality n > 2 encounters serious computational difﬁculties. Therefore, it is reasonable to decompose the matrix A.t / into blocks such that ni 2; i D 1; m. Example 3. Consider a controlled system xP 1 D x2 ;

xP 2 D u;

u 2 Œ1; 1;

x1 .0/ D x2 .0/ D 0:

(4.23)

The set of attainability for this system takes the form ² ³ x2 x2 x2 t t 2 x2 t t2 R.t / D .x1 ; x2 / W 2 C x1 2 C C : 4 2 4 4 2 4 System (4.23) corresponds to the following equation with Hukuhara derivative: 0 1 X.t / C F .t /; X.0/ D 0; Dh X.t / D 0 0 where F .t / D 0 Œ1; 1. We decompose the matrix A into blocks of dimensionality 1 1 and consider a system of differential P equations with Hukuhara derivative of the form (4.13). For any i D 1; 2, the sum j2D1 AjTi j contains at most one nonzero element and, moreover, X0 D X 0 and F .t / F .t /. Hence, by virtue of (4.16) and (4.17), the identity X.t / X .t / is true for all t 2 Œ0; 1. According to Remark 2, system (4.13) is reduced to the following two systems of ordinary differential equations: ´ ´ yP1 D y2 ; yP2 D 1; xP 1 D x2 ; xP 2 D 0; x1 .0/ D x2 .0/ D 0; y1 .0/ D y2 .0/ D 0: As a result of the solution of these systems, we ﬁnd (Figure 1): x1 .t / D x2 .t / D 0;

y1 .t / D

t2 ; 2

and y2 .t / D t:

Thus, we get an approximation of the set of attainability for problem (4.23) in the following form: 2 2 t t R.t / X.t / D X .t / D ; Œt; t : 2 2 Example 4. Consider a linear control problem on the segment Œ0; 1 xP D Ax C u;

x.0/ 2 X0 ;

u.t / 2 U;

(4.24)

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion

141

Figure 1. Approximation of the set of attainability for system (4.23).

where AD

1 0 ; 0 1

(4.25)

X0 is a unit ball centered at the point .1; 2/, and U is a unit ball centered at the origin. For t D 1, we get the approximation depicted in Figure 2.

Figure 2. Approximation of the set of attainability for system (4.24), (4.25).

142

Chapter 4 Linear Systems with Multivalued Trajectories

Example 5. Assume that, in Eq. (4.24), 1 0 ; AD 0 0:1

(4.26)

X0 is a unit ball centered at the origin, and U is a unit square centered at the point .1; 0/. In this case, we arrive at the approximation shown in Figure 3.

Figure 3. Approximation of the set of attainability for system (4.24), (4.26).

It is natural to study the problem of construction of approximations to the bundles of solutions for linear impulsive differential inclusions. Assume that the system described by inclusion (4.8) is subjected to pulse actions at ﬁxed times. In other words, on the segment Œ0; T , we consider a linear impulsive differential inclusion xP 2 A.t /x C F .t /;

t ¤ k ;

x.k C 0/ 2 Bk x.k / C Pk ; x.0/ 2 X0 ;

(4.27)

k D 1; K;

where Bk are n n matrices, Pk 2 conv.Rn /, and the times of pulses are such that 0 1 < < K < T . Inclusion (4.27) is equivalent, e.g., to the following linear impulsive control system xP D A.t /x C D.t /u;

t ¤ k ;

x.k C 0/ D Bk x.k / C Ck vk ; x.0/ 2 X0 ;

k D 1; K;

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion

143

where u.t / 2 U.t / and vk 2 Vk 2 conv.Rr / are control vectors, Ck are .n r/matrices, and in addition, Pk D ¹y 2 Rn W y D Ck vk ; vk 2 Vk º. The equation with Hukuhara derivative corresponding to the impulsive differential inclusion (4.27) takes the form Dh X.t / D A.t /X.t / C F .t /;

t ¤ k ;

X.k C 0/ D Bk X.k / C Pk ; X.0/ D X0 ;

(4.28)

k D 1; K;

where the solution X W Œ0; T ! conv.Rn / is a set-valued mapping piecewise continuously differentiable in Hukuhara’s sense. Assume that the matrix A.t / can be represented in the form (4.12) and that the matrices Bk .k D 1; K/ admit the following representations: 1 0 k k Bk B11 B12 1m B k k C B k B2m C BB k n n Bk D B :21 :22 : :: C; where Bij 2 R i j : : : : @ : : : A : k Bk k Bm1 m2 Bmm Equation (4.28) is associated the following system of linear impulsive differential equations with Hukuhara derivative: Dh Xi .t / D

m X

Aij .t /Xj .t / C Fi .t /;

t ¤ k ;

j D1

Xi .k C 0/ D

m X

Bijk Xj .k / C Pik ;

(4.29)

j D1

Xi .0/ D Xi0 2 conv.Rni /;

i D 1; m; k D 1; K;

k , P k 2 conv.Rni /, and X W Œ0; T ! conv.Rni / where Pk P k D P1k Pm i i are functions piecewise continuously differentiable in Hukuhara’s sense. Consider a set X .t / D X1 .t / Xm .t /.

Theorem 6 ([124]). The inclusion R.t / X.t / X .t /, where R.t / is the set of attainability of (4.27) holds for Eqs. (4.27), (4.28), and (4.29) for any t 2 Œ0; T . Proof. Denote 0 D 0 and KC1 D T . Suppose that the inclusion R.k1 C 0/ X.k1 C 0/ X.k1 C 0/ holds for some k 2 1; K C 1. By virtue of Theorem 4 and inclusion (4.11), we get R.t / X.t / X .t / for all k1 < t k .

144

Chapter 4 Linear Systems with Multivalued Trajectories

We now show that R.k C 0/ X.k C 0/ X .k C 0/; k 2 1; K. The ﬁrst part of the inclusion directly follows from (4.27) and (4.28). We now prove the second inclusion. In view of the convexity of the sets X.k C 0/ and X .k C 0/, it sufﬁces to show that the inequality c.X.k C 0/; / c.X .k C 0/; / holds for all 2 Rn . Equations (4.28) and (4.29) now imply that c.X.k C 0/; / D c.Bk X.k / C Pk ; / D c.X.k /; BkT / C c.Pk ; / c.X .k /; BkT / C c.P k ; / D

m X i D1

D

c.Xi .k /; .BkT /i / C

8 ˆ ˆ ˆ ˆ <

00 BB BB B /i D B BB @@

m X

k /T .B k /T .B21 m1

jm

i

j

C

j D1

m X m X

m X

c.Xi .k /; .Bjki /T

j/ C

c.Xi .k C 0/;

m

i

i/

m X

c.Pik ;

i /I

(4.30)

c.Pik ;

i /:

(4.31)

i/

m X m X c Bijk Xj .k / C Pik ; D

D

c.Pik ;

11 CB C CB 2 C C CB : C C C@ : C C : AA A 1

iD1

i D1

i D1

m X

10

i D1

iD1 j D1

c.X .k C 0/; / D

i/

iD1 k /T .B11 k /T .B12

m X c Xi .k /; .Bjki /T

i D1

c.Pik ;

k /T .B k /T .B22 m2 :: :: :: :: ˆ : : : : ˆ ˆ : k /T .B k /T .B k /T .B1m mm 2m 9 1 0 Pm k T > j > j D1 .Bj1 / > > C > B Pm m = k T X B j D1 .B / j C k T j 2 C B DB D .B / j ji C :: > > A @ : j D1 > > Pm > k T ; .B / j

.B T ˆ k

j D1

D

m X

i

j D1

m X m X i D1 j D1

c.Xj .k /; .Bijk /T

i/

C

n X iD1

In view of (4.30) and (4.31), we conclude that the inequality c.X.k C 0/; / c.X .k C 0/; /

145

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion

is true for all 2 Rn and, hence, X.k C 0/ X .k C 0/. The theorem is thus proved. We now study the problem of variation of the set X .t / in the case of subsequent decomposition of the matrix. Assume that the th row and th column of the matrix A.t / 2 1; m are decomposed as in (4.18) and the th rows and th columns of the matrices Bk are decomposed into the following matrices: 1 0 k11 k12 k1s B B B B k21 k22 s k2s C X C B B B B kpq k lp lq C; where B B 2 R ; lp D n I B D B : :: : : : C : :: A @ :: : pD1 ks1 ks2 kss B B B k1 k2 kp k ks ; Bi D Bi 2 Rni lp ; i D 1; m; i ¤ I (4.32) where Bi Bi Bi 1 0 k1 Bi B k2 C B Bi C kp k lp ni C DB ; i D 1; m; i ¤ : Bi B :: C; where Bi 2 R @ : A ks Bi Parallel with system (4.13), we consider a system Dh XQ i .t / D

m X j D1 j ¤

XQ i .k C 0/ D

m X

Bijk XQj .k / C

s X

m X

Bi Xp .k / C PQik ; kp

i D 1; m; i ¤ ; Aqj .t /XQj .t / C

j D1 j ¤

Xq .k C 0/ D

t ¤ k ;

pD1

XQ i .0/ D Xi0 ; m X

Q Ap i .t /Xp .t / C Fi .t /;

pD1

j D1 j ¤

Dh Xq .t / D

s X

Aij .t /XQj .t / C

s X

Aqp .t /Xp .t / C Fq .t /;

(4.33)

pD1 kq Q Bj Xj .k / C

j D1 j ¤

0 ; Xq .0/ D Xq

s X

k

k Bqp Xp .k / C Pq ;

pD1

q D 1; s; k D 1; K;

k P k , P k 2 conv.Rlq /, X 0 2 where Pik D PQik , i ¤ , Pk PQk D P1 s q 0 X0 , X Q i W Œ0; T ! conv.Rni / (i ¤ ), and Xq W Œ0; T ! XQ0 D X1 s conv.Rlq / (q D 1; s) are set-valued mappings piecewise continuously differentiable in Hukuhara’s sense.

146

Chapter 4 Linear Systems with Multivalued Trajectories

Consider a set XQ .t / D XQ 1 .t / XQ 1 .t / X1 .t / Xs .t / XQ C1 .t / XQ m .t /: Theorem 7 ([124]). The following inclusion is true for systems (4.29) and (4.33) for any t 2 Œ0; T : X .t / XQ .t /: Proof. Denote 0 D 0 and KC1 D T . Assume that the inclusion X.k1 C 0/ XQ .k1 C 0/ holds for some k 2 1; K C 1. By Theorem 5, for all k1 < t k , we have X .t / XQ .t /. Q k C 0/; k 2 1; K. In view of the convexity of We now show that X .k C 0/ X. Q k C 0/, it sufﬁces to show that the inequality the sets X .k C 0/ and X. c.X .k C 0/; / c.XQ .k C 0/; / is true for all

2 Rn . By virtue of (4.31), Eqs. (4.29) and (4.33) imply that

c.X .k C 0/; / D

m X m X

c.Xj .k /; .Bijk /T

i/

C

iD1 j D1

m X m X

c.XQj .k /; .Bijk /T

m X m X iD1

C

m X

c.Pik ;

i/

c.PQik ;

i/

iD1 i/ C

iD1 j D1

D

m X

m X iD1

c.XQj .k /; .Bijk /T

k T i / C c.XQ .k /; .Bi /

i/

j D1 j ¤

c.Pik ;

i/

C c.PQk ;

/

iD1 i ¤

D

m X m X

c.XQj .k /; .Bijk /T

i/ C

m X j D1 j ¤

k T c.XQj .k /; .Bj /

k T c.XQ .k /; .Bi /

i/

i D1

iD1 j D1 i¤ j ¤

C

m X

/C

m X i D1 i¤

c.Pik ;

i /C

s X

k c.Pp ;

pD1

p /

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion

D

m X m X

c.XQj .k /; .Bijk /T

m X s X k T c.Xp .k /; ..Bi / i /C

i D1 j D1 i¤ j ¤

C

s X

m X

i /p /

iD1 pD1 i ¤

k T c.Xp .k /; ..B /

C

/p /

m X

k T c.XQj .k /; .Bj /

/

j D1 j ¤

pD1

C

147

c.Pik ;

i/ C

s X

k c.Pp ;

p /

pD1

iD1 i ¤

8 ˆ ˆ < k T / D ..Bi ˆ ˆ : k T .Bj /

1 k1 /T .Bi C BB :: A i /p D @@ : ks T .Bi /

1

00

D

s X

kp T .Bj /

C

kp T D .Bi /

iA

iI

p

p I

pD1

11 k11 T ks1 T 1 0 / .B / .B 1 C B :: CC BB :: :: k T :: / /p D @@ ..B A @ : AA : : : k1s T kss T s .B / .B / p 9 1 0P kq1 T s > .B / q > s = X C B qD1 : k qp T C D :: DB .B / q A @ > > Ps kqs T qD1 ; .B / q qD1 p 00

D

m X m X

c.XQj .k /; .Bijk /T

i /C

iD1 j D1 i¤ j ¤

iD1 i¤

s s X X k C c Xp .k /; .Bqp /T pD1

j D1 j ¤

m X iD1 i ¤

q

p

pD1

c.Pik ;

i/ C

s X pD1

k c.Pp ;

p /

kp

c.Xp .k /; .Bi /T

pD1

qD1

m s X X kp T C c XQj .k /; .Bj /

C

m X s X

i/

148

Chapter 4 Linear Systems with Multivalued Trajectories m X m X

c.XQj .k /; .Bijk /T

i/

iD1 j D1 i ¤ j ¤

m X s X

C

iD1 i ¤

kp T c.Xp .k /; .Bi /

s s X X

C

i/

pD1 k

c.Xp .k /; .Bqp /T

q /

pD1 qD1

C

m X s X j D1 j ¤

C

m X

kp T c.XQj .k /; .Bj /

p /

pD1

c.Pik ;

s X

i/ C

k c.Pp ;

p /:

(4.34)

pD1

iD1 i ¤

Q k C 0/: We now ﬁnd the support function of the set X. c.XQ .k C 0/; / D

m X

c.XQ i .k C 0/;

i/ C

s X

c.Xq .k C 0/;

q /

qD1

iD1 i¤

m X m s X X kp D c Bijk XQj .k / C Bi Xp .k / C PQik ; iD1 i¤

C

j D1 j ¤

j D1 j ¤

m X m X

m X

c.XQj .k /; .Bijk /T

i /C

iD1 i¤

C

s X s X qD1 pD1

m X s X iD1 i ¤

c.Pik ;

q

pD1

iD1 j D1 i¤ j ¤

C

i

pD1

X s m s X X k kq Q k c Bj Bqp Xp .k / C Pq ; Xj .k / C qD1

D

i/ C

m s X X qD1

kp T c.Xp .k /; .Bi /

i/

pD1

kq T c.XQj .k /; .Bj /

q /

j D1 j ¤

k c.Xp .k /; .Bqp /T

q /

C

s X

k c.Pq ;

q /:

qD1

(4.35)

Section 4.2 Approximation of the Integral Funnel of a Linear Differential Inclusion

149

By using (4.34) and (4.35), we conclude that the inequality c.X .k C 0/; / c.XQ .k C 0/; / holds for all 2 Rn . Hence, X.k C 0/ XQ .k C 0/. The theorem is thus proved. Corollary 3 ([124]). Let 1 and 2 be decompositions of the matrices A.t / and Bk for Eq. (4.28). Assume that 2 can be obtained from 1 by additional decomposition. Let X 1 .t / and X 2 .t / be solutions of systems of the form (4.29) corresponding to the given partitions. Then X 1 .t / X 2 .t / for all t 2 Œ0; T . Corollary 4 ([124]). Let Ym .t /, be the intersection of all sets X .t / of solutions of systems of the form (4.29) for all possible decompositions of the matrices A.t / and Bk into the matrices Aij .t / and Bijk i; j D 1; m. Hence, by virtue of Theorems 6 and 7, X.t / D Y1 .t / Y2 .t / Yn .t / for all t 2 Œ0; T : The estimate X.t / Yn .t / is proved in [116]. Remark 4 ([116]). For m D n, system (4.29) decomposes into two systems of linear impulsive differential equations. Let Xi .t / D xi .t / C yi .t /Œ1; 1, Fi .t / D fi .t / C gi .t /Œ1; 1, and Pik D pik C qik Œ1; 1. Thus, by analogy with Remark 2, we ﬁnd 8 Pn .t /xj .t / C fi .t /; t ¤ k ; k D 1; N ; ˆ <xP i D j D1 aijP k x . / C p k ; xi .k C 0/ D jnD1 bij j k i ˆ : xi .0/ D xi0 : 8 Pn t ¤ k ; k D 1; N ; ˆ ij .t /jyj .t / C gi .t /;
(4.36)

(4.37)

Example 6. Consider the following linear impulsive equation with Hukuhara derivative: 2 1 X.t /; t ¤ i; i 2 N; Dh X.t / D 1 2 0 2 1 3e14 3e 4 A X.i /; (4.38) X.i C 0/ D @ 1 3e24 3e 4 X.0/ D X0 2 conv.R2 /:

150

Chapter 4 Linear Systems with Multivalued Trajectories

Let X0 XQ 0 D X10 X20 , where Xk0 D xk0 C yk0 Œ1; 1; k D 1; 2. In this case, systems (4.36) and (4.37) take the form 8 ˆ xP D 2x1 C x2 ; ˆ ˆ 1 ˆ ˆ ˆ xP 2 D x1 2x2 ; t ¤ i; ˆ ˆ ˆ ˆ ˆ ˆ <x1 .i C 0/ D 24 x1 .i/ 14 x2 .i/; 3e 3e (4.39) 1 2 ˆ .i C 0/ D x .i/ x .i/; i 2 N; x ˆ 2 1 2 4 4 3e 3e ˆ ˆ ˆ ˆ 0 ˆ ˆx1 .0/ D x1 ; ˆ ˆ ˆ ˆ : x2 .0/ D x20 ; 8 ˆ yP D 2y1 C y2 ; ˆ ˆ 1 ˆ ˆ ˆ ˆ ˆyP2 D y1 C 2y2 ; t ¤ i; ˆ ˆ ˆ ˆ
Section 4.3 Linear Differential Equations with -Derivative

4.3

151

Linear Differential Equations with -Derivative

As already indicated, the spaces comp.Rn / and conv.Rn / are not linear. In [146] and [15], the operation of subtraction in the space conv.Rn / is introduced with the help of imbedding of this space in a linear space. By the theorem of imbedding [129], there exist a normalized linear space B and an isometric mapping W conv.Rn / ! B such that .conv.Rn // is a convex cone in B. In addition, the operations of summation and multiplication by a nonnegative scalar in B induce the corresponding operations in conv.Rn /. In order to deﬁne the space B, we introduce the relation of equivalence in the space conv.Rn / conv.Rn / as follows: .A; B/ .C; D/

for A C D D B C C:

By hA; Bi we denote the class of equivalence containing .A; B/. The space B is deﬁned as the quotient space conv.Rn / conv.Rn /= : In the space B, we introduce the operations of summation and multiplication by a scalar as follows: hA; Bi C hC; Di D hA C C; B C Di; ´ h˛A; ˛Bi; ˛ 0; ˛hA; Bi D hj˛jB; j˛jAi; ˛ < 0: The space B thus deﬁned is linear. The imbedding W conv.Rn / ! B is deﬁned as follows: .A/ D hA; 0i;

A 2 conv.Rn /;

i.e., hA; 0i is the class of equivalence ¹.A C D; D/ W D 2 conv.Rn /º. The metric and norm in the space B are given by the formulas .hA; Bi; hC; Di/ D h.A C D; C C B/; khA; Bik D .hA; Bi; h0; 0i/: The space B is not a complete metric space [127]. At the same time, the sharp convex closed cone corresponding to the complete metric space conv.Rn / [35] is complete [127]. Let A; B 2 conv.Rn /. The difference of the spaces A and B is deﬁned as the element of the space B equal to hA; Bi. Since the space B is linear, the indicated difference possesses all properties of difference in linear spaces.

152

Chapter 4 Linear Systems with Multivalued Trajectories

In the general case, the difference of two sets from conv.Rn / is not an element of the space conv.Rn /. At the same time, if the Hukuhara difference of two sets from the space conv.Rn / exists, then h B; 0i: hA; Bi D hA

Deﬁnition 5 ([15, 146]). A set-valued mapping F W R ! conv.Rn / is called differentiable at a point t0 2 R if there exists a continuous linear mapping G.t0 / W R ! B such that hF .t /; F .t0 /i D G.t0 /.t t0 / C o.kt t0 k/:

(4.41)

If G.t0 /.t t0 / D hA.t0 /.t t0 /; B.t0 /.t t0 /i; A.t0 /; B.t0 / 2 conv.Rn /, then, in terms of the Hausdorff metric, relation (4.41) means that h.F .t / C B.t0 /.t t0 /; F .t0 / C A.t0 /.t t0 // D o.kt t0 k/: Moreover, if G.t0 /.t t0 / D hA.t0 /.t t0 /; 0i, then we say that F . / is conically differentiable for t0 2 R. We now consider the relationship between the differentiability in Hukuhara’s sense and -conic differentiability. Theorem 8 ([15]). If a set-valued mapping F W R ! conv.Rn / is differentiable in Hukuhara’s sense for t0 2 R with the derivative Dh F .t0 /, then F . / is conically differentiable and G.t0 /.t t0 / D .t t0 /hDh F .t0 /; 0i: It is also possible to show that if F W R ! conv.Rn / is conically differentiable for t0 2 R and the Hukuhara differences exist for sufﬁciently small t, then F . / is differentiable in Hukuhara’s set at the point t0 2 R. Moreover, if G.t /t D b .t /. b .t /; 0i; t 2 R, then Dh F .t / D F t hF However, generally speaking, the -differentiability of F W R ! conv.Rn / at a point t0 2 R does not imply the differentiability of F . / in Hukuhara’s sense. Example 7. Consider a set-valued mapping F .t / D .2 C sin.t //S1 .0/;

t 2 Œ0; 2:

This mapping is -differentiable on Œ0; 2 and 1 C sign.cos.t // 1 sign.cos.t // j cos.t /jS1 .0/; j cos.t /jS1 .0/ : G.t / D 2 2 At the same time, it is not differentiable in Hukuhara’s sense because the function diam.F . // is not nondecreasing on Œ0; 2.

153

Section 4.3 Linear Differential Equations with -Derivative

The differential equations with -derivative can be regarded as a natural generalization of the notion of differential equations with Hukuhara derivative. Consider a system of linear differential equations with -derivative D hXi .t /; Yi .t /i D

n X

aij .t /hXj .t /; Yj .t /i C hFi .t /; Gi .t /i;

(4.42)

j D1

hXi .0/; Yi .0/i D hXi0 ; Yi0 i;

i D 1; n;

where t 2 Œ0; T is time and aij W Œ0; T ! R and Fi ; Gi W Œ0; T ! conv.Rm / are continuous functions. Deﬁnition 6 ([121]). The couples hXi .t /; Yi .t /i, Xi ; Yi W Œ0; T ! conv.Rm /, i D 1; n, are called the solution of problem (4.42) if they are continuously -differentiable and satisfy (4.42) everywhere on Œ0; T . We now show that the solution of system (4.42) can be reduced to the solution of a system of ordinary differential equations. Deﬁnition 7 ([121]). A function c.hA; Bi; / D c.A; / c.B; / is called the support function of the couple hA; Bi in the direction of a vector

2 Rm .

This deﬁnition is correct, i.e., independent of the choice of a representative .A; B/ of the class of equivalence hA; Bi: Let .A; B/; .C; D/ 2 hA; Bi. Then c.A; / C c.D; / D c.B; / C c.C; / ) c.A; / c.B; / D c.C; / c.D; /: Moreover, the class hA; Bi is uniquely deﬁned by its support function. Assume the contrary, i.e., that there exist two classes hA; Bi and hC; Di with the same support function. According to the deﬁnition of the support function of the couple, we have c.A; / c.B; / D c.C; / c.D; / for all

2 Rm . Hence, c.A; / C c.D; / D c.B; / C c.C; / ) c.A C D; / D c.B C C; /

for all

for all

2 Rm

2 Rm :

In view of the convexity of the sets A; B; C , and D, we ﬁnd A C D D D B C C . Hence, the classes hA; Bi and hC; Di coincide.

154

Chapter 4 Linear Systems with Multivalued Trajectories

Lemma 1 ([121]). The following assertions are true: (a) c.hA; Bi C hC; Di; / D c.hA; Bi; / C c.hC; Di; /; (b) c. hA; Bi; / D c.hA; Bi; /; (c) c.hA; Bi; / is a continuous function of its arguments; (d) c.D hX.t /; Y.t /i; / D

d c.hX.t /; Y.t /i; dt

/.

Proof. (a) c.hA; Bi C hC; Di; / D c.hA C C; B C Di; / D c.A C C; / c.B C D; / D c.A; / C c.C; / .c.B; / C c.D; // D c.A; / c.B; / C c.C; / c.D; / D c.hA; Bi; / C c.hC; Di; /: (b) Let 0. Then c. hA; Bi; / D c.h A; Bi; / D c. A; / c. B; / D c.A; / c.B; / D c.hA; Bi; /: At the same time, if < 0, then c. hA; Bi; / D c.hj jB; j jAi; / D c.j jB; / c.j jA; / D j j.c.B; / c.A; // D c.hA; Bi; /: (c) This assertion follows from the deﬁnition of the support function of a couple and from the continuity of the support function c. ; / in its arguments. (d) c.D hX.t /; Y.t /i; / hX.t C t /; Y .t C t /i hX.t /; Y.t /i

; D c lim t !0 t

1 D c lim hX.t C t / C Y .t /; Y .t C t / C X.t /i; t !0 t 1 D lim c.hX.t C t / C Y .t /; Y .t C t / C X.t /i; / t !0 t 1 Œc.X.t C t /; / C c.Y .t /; / D lim t !0 t .c.Y .t C t /; / C c.X.t /; //

Section 4.4 Extension of the Space conv.Rn / for n D 1

155

1 Œc.X.t C t /; / c.Y .t C t /; / t !0 t

D lim

.c.X.t /; / c.Y .t /; // D

d d .c.X.t /; / c.Y .t /; // D c.hX.t /; Y.t /i; /: dt dt

The lemma is proved. By virtue of the lemma, system (4.42) is equivalent to the following system: n X d c.hXi .t /; Yi .t /i; / D aij .t /c.hXj .t /; Yj .t /i; / C c.hFi .t /; Gi .t /i; /; dt j D1

c.hXi .0/; Yi .0/i; / D c.hXi0 ; Yi0 i; /;

i D 1; n;

for all 2 Rm . Denote zi .t; / D c.hXi .t /; Yi .t /i; /;

fi .t; / D c.hFi .t /; Gi .t /i; /:

As a result, the previous system can be rewritten in the form n X d zi .t; / D aij .t /zj .t; / C fi .t; /; dt

(4.43)

j D1

zi .0; / D c.hXi0 ; Yi0 i; /;

i D 1; n:

Thus, we get a system of n ordinary differential equations with parameter Hence, we have proved the following assertion:

.

Theorem 9 ([121]). The solution of system (4.42) is given by n couples hXi .t /; Yi .t /i, i D 1; n, uniquely determined by their support functions zi .t; / obtained as solutions of the system of ordinary differential Eqs. (4.43).

4.4

Extension of the Space conv.Rn / for n D 1

The elements of the space conv.R/ are segments. Hence, we present each class of equivalence in the form hŒx; x C ı; 0i; ı 0 or hx; Œ0; ıi; ı > 0. This presentation of the class of equivalence is called canonical. Indeed, any element .Œa; b; Œc; d / belongs to a class of the ﬁrst kind for b a d c [in this case, x D a c and ı D .b a/ .d c/ or to a class of the second kind for b a < d a [in this case, x D a c and ı D .d c/ .b a/. We represent a class of equivalence in the canonical form as .x; ı/. In this case, hŒx; x C ı; 0i for ı 0 and hx; Œ0; ıi D h0; Œx; x ıi for ı < 0.

156

Chapter 4 Linear Systems with Multivalued Trajectories

Lemma 2 ([121]). The following assertions are true: (a) .x; ı/ D .y; / , x D y; ı D ; (b) .x; ı/ C .y; / D .x C y; ı C /; (c) a.x; ı/ D .ax; aı/ for any a 2 R. Proof. (a) The sufﬁciency of this assertion is obvious. Thus, we prove its necessity. Let ı 0 and 0. Then .x; ı/ D hŒx; x C ı; 0i D hŒy; y C ; 0i D .y; /: This yields x D y and ı D . Let ı < 0 and < 0. Then .x; ı/ D hx; Œ0; ıi D h0; Œx; x ıi; .y; / D hy; Œ0; i D h0; Œy; y i; whence it follows that x D y and ı D . Let ı 0 and < 0. Then .x; ı/ D hŒx; x C ı; 0i;

.y; / D h0; Œy; y i;

which means that the equality .x; ı/ D .y; / is impossible. The case where ı < 0 and 0 is analyzed similarly. (b) Let ı 0 and 0. Then .x; ı/ C .y; / D hŒx; x C ı; 0i C hŒy; y C ; 0i D hŒx C y; x C y C ı C ; 0i D .x C y; ı C /: Let ı < 0 and < 0. Then .x; ı/ C .y; / D hx; Œ0; ıi C hy; Œ0; i D hx C y; Œ0; ı i D .x C y; ı C /: Let ı 0 and < 0. Then .x; ı/ C .y; / D hŒx; x C ı; 0i C hy; Œ0; i D hŒx C y; x C y C ı; Œ0; i ´ hŒx C y; x C y C ı ./; 0i; ı D D .x C y; ı C /: hx C y; Œ0; ıi; ı < The case where ı < 0 and 0 is analyzed similarly.

Section 4.4 Extension of the Space conv.Rn / for n D 1

157

(c) For a D 0, the assertion is obvious. Consider the case a > 0. If ı 0, then a.x; ı/ D ahŒx; x C ı; 0i D hŒax; ax C aı; 0i D .ax; aı/: Further, if ı < 0, then a.x; ı/ D ahx; Œ0; ıi D hax; Œ0; aıi D .ax; aı/: Now let a < 0. Thus, if ı 0, then a.x; ı/ D ahŒx; x C ı; 0i D h0; jajŒx; x C ıi D h0; Œjajx; jajx C jajıi D .jajx; jajı/ D .ax; aı/: At the same time, if ı < 0, then a.x; ı/ D ahx; Œ0; ıi D hjajŒ0; ı; jajxi D hŒjajx; jajx jajı; 0i D .jajx; jajı/ D .ax; aı/: Lemma 3 ([121]). lim .xk ; ık / D .x; ı/ , lim xk D x; lim ık D ı:

k!1

k!1

k!1

Proof. First, we prove necessity. Assume that all ık 0 beginning with some k0 . Then .xk ; ık / D hŒxk ; xk C ık ; 0i: Further, let ı 0. Then

.x; ı/ D hŒx; x C ı; 0i:

Since lim .xk ; ık / D .x; ı/;

k!1

we get ..xk ; ık /; .x; ı// D .hŒxk ; xk C ık ; 0i; hŒx; x C ı; 0i/ D h.Œxk ; xk C ık ; Œx; x C ı/ D max¹kxk xk; kxk C ık x ıkº ! 0 as k ! 1. Hence,

lim xk D x

k!1

and

lim ık D ı:

k!1

If ı < 0, then .x; ı/ D h0; Œx; x ıi and, therefore, ..xk ; ık /; .x; ı// D .hŒxk ; xk C ık ; 0i; h0; Œx; x ıi/ D h.Œxk x; xk x C ık ı; 0/ D max¹kxk xk; kxk C ık x ıkº ¹ 0; which contradicts the condition.

158

Chapter 4 Linear Systems with Multivalued Trajectories

The case where all ık < 0 beginning with some k0 is analyzed similarly. Assume that there are inﬁnitely many ık 0 and inﬁnitely many ık < 0. In this case, we split the sequence of numbers ¹kº into two subsequences ¹k1 W ık1 0º and ¹k2 W ık2 < 0º. Since any subsequence of a convergent sequence converges, we have limk1 !1 .xk1 ; ık1 / D .x; ı/ and, as shown above, limk1 !1 xk1 D x and limk1 !1 ık1 D ı 0, i.e., for any " > 0, there exists k10 such that the following estimates are true for k1 > k10 : kxk1 xk < " and

kık1 ık < ":

Similarly, we obtain limk2 !1 .xk2 ; ık2 / D .x; ı/. Hence, limk2 !1 xk2 D x, limk2 !1 ık2 D ı 0. In other words, for any " > 0, there exists k20 such that the estimates kxk2 xk < " and kık2 ık < " are true for k2 > k20 . Thus, we get ı D 0. Choosing k0 D max¹k10 ; k20 º, we conclude that kxk xk < " and kık k < " for k > k0 . Hence, necessity is proved. Sufﬁciency is proved similarly. Lemma 4 ([121]).

P //; D .x.t /; ı.t // D .x.t P /; ı.t

where D .x.t /; ı.t // is the -derivative [146], [15] of the couple .x.t /; ı.t //. Proof. By using the deﬁnition of the -derivative and Lemmas 2 and 3, we ﬁnd 1 Œ.x.t C /; ı.t C // .x.t /; ı.t // !0 1 D lim Œ.x.t C /; ı.t C // C .x.t /; ı.t // !0 1 D lim .x.t C / x.t /; ı.t C / ı.t // !0 x.t C / x.t / ı.t C / ı.t / ; D lim !0 x.t C / x.t / ı.t C / ı.t / D lim ; lim !0 !0 P //: D .x.t P /; ı.t

D .x.t /; ı.t // D lim

159

Section 4.5 Approximation of the Integral Funnel of a Linear Differential Inclusion

4.5

Approximation of the Integral Funnel of a Linear Differential Inclusion with the Help of Systems of Differential Equations with -Derivative

Example 8. Consider a linear differential inclusion xP 2 ax C Œm; m;

x.0/ D 0

(4.44)

for m > 0 and various a ¤ 0. Tolstonogov ([145], p. 232) studied the relationship between the R-solution R.t / of inclusion (4.44) and a solution of the corresponding equation with Hukuhara derivative Dh X.t / D aX.t / C Œm; m;

X.0/ D 0:

(4.45)

It was shown that the following equality is true for a > 0 and t 2 Œ0; T : R.t / D X.t / D

e at 1 Œm; m: a

At the same time, if a < 0, then R.t / D

e at 1 Œm; m; a

X.t / D e at R.t /

and, hence, R.t / X.t / for t 2 .0; T . Consider the equation with -derivative corresponding to inclusion (4.44): ´ D .x; ı/ D a.x; ı/ C .m; 2m/; (4.46) .x.0/; ı.0// D .0; 0/: By virtue of Lemmas 2 and 4, this equation decomposes into two linear inhomogeneous equations ´ ´ xP D ax m; ıP D aı C 2m; x.0/ D 0; ı.0/ D 0: As a result of the solution of these equations, we conclude that x.t / D Therefore,

m at .e 1/ a

and ı.t / D

2m at .e 1/: a

2m at m at .e 1/ .x.t /; ı.t // D .e 1/; a a Dh m i E e at 1 m D .e at 1/; .e at 1/ ; 0 D Œm; m; 0 : a a a

160

Chapter 4 Linear Systems with Multivalued Trajectories

Hence, for all a, the R-solution of inclusion (4.44) coincides with the integral funnel of inclusion (4.44) and with the solution of the equation with -derivative (4.46). Thus, it is reasonable to consider the problem of approximation of the R-solution of a linear differential inclusion with the help of the solution of the corresponding equation with -derivative. Consider a linear differential inclusion xP 2 A.t /x C F .t /;

x.0/ 2 X0 ;

(4.47)

where t 2 Œ0; T ; x 2 Rn is the phase vector, A.t / is a continuous .n n/-matrix, F .t / is a continuous set-valued mapping Œ0; T ! comp.Rn /, and X0 2 conv.Rn /. The R-solution R.t / of inclusion (4.47) with the initial condition R.0/ D X0 has the form Z t R.t / D ˆ.t; 0/X0 C ˆ.t; s/F .s/ds; (4.48) 0

where ˆ.t; s/ is the matrizant of the system xP 2 A.t /x. Assume that a mapping FQ .t / D F1 .t / Fn .t /, where Fi .t / D Œfi .t /; fi .t / C ri .t /;

i D 1; n;

and a set XQ 0 D X10 Xn0 , where Xi0 D Œxi0 ; xi0 C ıi0 ; i D 1; n, are such that F .t / FQ .t / .F .t / FQ .t // for all t 2 Œ0; T and X0 XQ 0 .X0 XQ 0 /. Inclusion (4.47) is associated with a system of linear differential equations with -derivative of the form D .xi ; ıi / D

n X

aij .t /.xj ; ıj / C .fi .t /; ri .t //;

(4.49)

j D1

.xi .0/; ıi .0// D .xi0 ; ıi0 /;

i D 1; n:

By virtue of Lemmas 2 and 4, system (4.49) decomposes into two systems of linear differential equations ´ P xP i D jnD1 aij .t /xj C fi .t /; (4.50) xi .0/ D xi0 ; i D 1; n; ´ P ıPi D jnD1 aij .t /ıj C ri .t /; (4.51) ıi .0/ D ıi0 ; i D 1; n; whose solutions can be represented in the form Z

t

T

x.t / D .x1 .t /; : : : ; xn .t // D ˆ.t; 0/x0 C Z T

ı.t / D .ı1 .t /; : : : ; ın .t // D ˆ.t; 0/ı0 C

ˆ.t; s/f .s/ds; 0 t

ˆ.t; s/r.s/ds: 0

Section 4.5 Approximation of the Integral Funnel of a Linear Differential Inclusion

161

´ Œxi .t /; xi .t / C ıi .t /; ıi .t / 0; Œxi .t / C ıi .t /; xi .t /; ıi .t / < 0;

Now let

and let XQ .t / D X1 .t / Xn .t /. We now study the relationship between the sets R.t / and XQ .t /. According to the property of the support functions, we ﬁnd Z c.R.t /; / D c.ˆ.t; 0/X0 ; / C c

ˆ.t; s/F .s/ds; 0 t

Z T

D c.X0 ; ˆ .t; 0/ / C

c.F .s/; ˆT .t; s/ /ds

0

./ c.XQ 0 ; ˆT .t; 0/ / C D

n ² X

xi0

ıi0

C

Z

t

c.FQ .s/; ˆT .t; s/ /ds

0

ı0 .ˆ .t; 0/ /i C i j.ˆT .t; 0/ /i j 2

2

iD1

t

³

T

³ ri .s/ ri .s/ T T .ˆ .t; s/ /i C j.ˆ .t; s/ /i j ds C fi .s/ C 2 2 0 i D1 ˇ³ ˇ n n n ² X ˇ ıi0 X ıi0 ˇˇ X 0 j i .t; 0/ j C ˇ j i .t; 0/ j ˇˇ D xi C 2 2 Z tX n ²

iD1

C

n Z t X

j D1

²

fi .s/ C

i D1 0

ri .s/ 2

X n

n X iD1

D

ˇ n ri .s/ ˇˇ X C j i .t; s/ 2 ˇ

n X ıi c.Xi ; i / D xi C 2

n ² X n X iD1

j i .t; s/

j D1

j D1

c.XQ .t /; / D

j D1

j D1

Z

C C

t

ˇ³ ˇ ˇ j ˇ ds D I1 .t; /;

i

iD1

jıi j j C 2

ij

ıj0 ij .t; 0/ xj0 C 2 n X

0 j D1

j

j

rj .s/ ds ij .t; s/ fj .s/ C 2

ˇX n

i j ˇˇ

2 ˇ

j D1

ij .t; 0/ıj0

C

n Z X j D1 0

t

i

ˇ³ ˇ ij .t; s/rj .s/ds ˇˇ D I2 .t; /:

162

Chapter 4 Linear Systems with Multivalued Trajectories

The relationship between I1 .t; / and I2 .t; / is directly connected with the relationship between n ˇX X ˇ n j i .t; 0/ ıi0 ˇˇ J1 .t; / D j D1

iD1

ˇX ˇ Z t ˇ n ˇ ˇ j i .t; s/ ri .s/ˇˇ jˇ C 0

j D1

ˇ ˇ ˇ j ˇ ds

and J2 .t; / D

n X i D1

ˇX ˇ n Z t X ˇ n ˇ 0 j i j ˇˇ ij .t; 0/ıj C ij .t; s/rj .s/ds ˇˇ: j D1 0

j D1

Thus, we have proved the following assertion: Theorem 10 ([121]). Assume that, for the linear differential inclusion (4.47), there exist a continuous function FQ .t / such that F .t / FQ .t / .F .t / FQ .t // and a set XQ 0 such that X0 XQ 0 .X0 XQ 0 / and J1 .t; / J2 .t; / .J1 .t; / J2 .t; // for all 2 Rn and t 2 Œ0; T . Then R.t / XQ .t / .R.t / XQ .t // for all t 2 Œ0; T . Corollary 5. Assume that n D 1. In this case, F .t / D FQ .t /, X0 D XQ0 , and Z t J1 .t; / D ı0 j.t; 0/ j C r.s/j.t; s/ jds 0

ˇ ˇ Z t ˇ ˇ j jˇˇ.t; 0/ı0 C r.s/.t; s/ds ˇˇ D J2 .t; / 0

for all

2 R and t 2 Œ0; T . Then R.t / XQ .t /.

Example 9. Consider the differential inclusion (4.47) with n D 2 and a diagonal matrix ˆ.t; s/. Assume that the mapping FQ .t / and the set XQ 0 are such that F .t / FQ .t / and X0 XQ 0 . In this case, J1 .t; / D ı10 j11 .t; 0/ 1 j C ı20 j22 .t; 0/ 2 j Z t Z t r1 .s/j11 .t; s/ 1 jds C r2 .s/j22 .t; s/ C 0

0

2 jds;

ˇZ t ˇ ˇ ˇ 0ˇ ˇ J2 .t; / D j 1 jˇ r1 .s/11 .t; s/ds C 11 .t; 0/ı1 ˇ 0

ˇZ t ˇ ˇ ˇ 0ˇ ˇ r2 .s/22 .t; s/ds C 22 .t; 0/ı2 ˇ: C j 2 jˇ 0

Since 11 .t; s/ and 22 .t; s/ are sign-preserving functions (otherwise, the matrix ˆ.t; s/ is nondegenerate), we conclude that J1 .t; / D J2 .t; / for all 2 Rn

163

Section 4.5 Approximation of the Integral Funnel of a Linear Differential Inclusion

and t 2 Œ0; T . Thus, by virtue of the already proved theorem, R.t / XQ .t / for all t 2 Œ0; T . If F .t / D FQ .t / and X0 D XQ 0 , then R.t / D XQ .t /. In the case where the matrix ˆ.t; s/ is inversely diagonal, we get the same estimate. Corollary 6. Assume that a matrix ˆ.t; s/ has the following property: Each row and each column of the matrix contain a single nonzero element. Let a mapping FQ .t / and a set XQ 0 be such that F .t / FQ .t / and X0 XQ0 . Then R.t / XQ .t / for all t 2 Œ0; T . It is clear that, in the general case, it is impossible to say which of the quantities J1 .t; / and J2 .t; / is larger. Theorem 11 ([121]). Assume that, for the linear differential inclusion (4.47) and any k … 1; n, all ki .t; s/ are nonnegative (nonpositive) for all i … 1; n and t; s 2 Œ0; T . Moreover, there exist a function FQ .t / such that F .t / FQ .t / and a set XQ 0 such that X0 XQ0 . Then R.t / XQ .t / for all t 2 Œ0; T . Proof. We use Theorem 10. It sufﬁces to check the validity of the inclusion J1 .t; / J2 .t; / for the vectors D ˙ek , k D 1; n, where ek is a unit vector. In this case, Z t n X 0 J1 .t; ˙ek / D ri .s/jki .t; s/jds ; ıi jki .t; 0/j C 0

iD1

ˇX ˇ n Z t X ˇ n ˇ 0 ˇ ki .t; 0/ıi C ki .t; s/ri .s/ds ˇˇ: J2 .t; ˙ek / D ˇ iD1

iD1 0

Since, for any k D 1; n, all ki .t; s/ are nonnegative (nonpositive) for all i D 1; n and 0 s t T , we conclude that J1 .t; ˙ek / D J2 .t; ˙ek / for all k D 1; n and t 2 Œ0; T . Remark 7. If, in Theorem 10, F .t / D FQ .t / and X0 D XQ 0 , then J1 .t; ˙ek / D J2 .t; ˙ek / for all k D 1; n and t 2 Œ0; T . Thus, the set R.t / is inscribed in the set XQ .t /. Example 10. In [84], the differential inclusion (4.47) was considered for n D 1: xP 2 a.t /x C F .t /;

x.0/ 2 X0 :

(4.52)

In the case where a.t / 0 and the mapping F W Œ0; T ! conv.R/ is measurable and integrally bounded, it was shown in [145] that the R-solution R.t / of inclusion (4.52) coincides with a solution of the equation with Hukuhara derivative Dh X D a.t /X C F .t /;

X.0/ D X0 :

164

Chapter 4 Linear Systems with Multivalued Trajectories

Consider the equation with -derivative corresponding to inclusion (4.52): D .x; ı/ D a.t /.x; ı/ C .f .t /; r.t //;

(4.53)

.x.0/; ı.0// D .x0 ; ı0 /: In this case, F .t / D FQ .t /, X0 D XQ 0 , and Z t a. /d > 0 .t; s/ D exp s

for all t; s 2 Œ0; T . Thus, by virtue of Theorem 11 and Corollary 5, we get R.t / D XQ .t / Z t Rt Rt Rt a.s/ds 0 C f .s/e s a./d ds; .x0 C ı0 /e 0 a.s/ds D x0 e 0

Z C

t 0

.f .s/ C r.s//e

Rt s

a./d

ds :

Example 11. Consider a linear inhomogeneous inclusion 0 2 1 x1 xP 1 ; 2 C 1 2 xP 2 x2 Œ0; 2 sin2 t

(4.54)

x1 .0/ D x2 .0/ D 0: The matrizant of the corresponding homogeneous system takes the form 1 e .ts/ C e 3.ts/ e .ts/ e 3.ts/ ˆ.t; s/ D : 2 e .ts/ e 3.ts/ e .ts/ C e 3.ts/ Since all ij .t; s/ > 0 for i; j 2 1; 2 and all 0 s t T , by virtue of Theorem 11, the R-solution R.t / of inclusion (4.54) is a subset of the set XQ .t / speciﬁed by the system of equations with -derivative 8 ˆ D .x1 ; ı1 / D 2.x1 ; ı1 / C .x2 ; ı2 /; ˆ ˆ ˆ
Section 4.5 Approximation of the Integral Funnel of a Linear Differential Inclusion

165

whose solutions can be represented in the form x1 .t / 0; 2 ı1 .t / D e t C 5 2 ı2 .t / D e t 5

x2 .t / 0;

1 1 10 3t 8 e sin 2t C cos 2t C ; 195 65 65 3 14 2 10 3t 18 e sin 2t cos 2t C : 195 65 65 3

Thus, XQ .t / approaches a periodic solution as t ! 1, whereas the solution of the equation with Hukuhara derivative corresponding to inclusion (4.54) is an extending set as t ! 1. Consider a linear impulsive differential inclusion xP 2 A.t /x C F .t /;

t ¤ k ; x.0/ 2 X0 ;

(4.55)

xj tDk 2 Bk x C Pk ; where the times of pulses k are such that 0 1 < < N T , Bk are n nmatrices, and Pk 2 conv.Rn /. Assume that the sets PQk D P1k Pnk , where Pik D Œpik ; pik C qik , i D 1; n, are such that Pk PQk Pk PQk . Inclusion (4.55) is associated with a system of linear differential equations with -derivative of the form D .xi ; ıi / D

n X

aij .t /.xj ; ıj / C .fi .t /; ri .t //;

t ¤ k ;

j D1

.xi ; ıi /j tDk D

n X

k bij .xj ; ıj / C .pik ; qik /;

(4.56)

j D1

.xi .0/; ıi .0// D .xi0 ; ıi0 /;

i D 1; n;

where .xi ; ıi /j tDk D .xi j tDk ; ıi j t Dk /. This system decomposes into two systems of linear inhomogeneous impulsive differential equations 8 Pn ˆ t ¤ k ; ij .t /xj C fi .t /; <xP i D j D1 aP n k k xi j t Dk D j D1 bij xj C pi ; ˆ : xi .0/ D xi0 ; i D 1; n; 8 Pn P ˆ t ¤ k ; ij .t /ıj C ri .t /; <ıi D j D1 aP n k k ıi j tDk D j D1 bij ıj C gi ; ˆ : ıi .0/ D ıi0 ; i D 1; n:

(4.57)

(4.58)

166

Chapter 4 Linear Systems with Multivalued Trajectories

Let XQ .k C 0/ D X1 .k C 0/ Xn .k C 0/, where ´ Œxi .k C 0/; xi .k C 0/ C ıi .k C 0/; Xi .k C 0/ D Œxi .k C 0/ C ıi .k C 0/; xi .k C 0/; xi .k C 0/ D

n X

ıi .k C 0/ 0; ıi .k C 0/ < 0;

k .eij C bij /xj .k / C pik ;

j D1

ıi .k C 0/ D

n X

k .eij C bij /ıj .k / C qik ;

j D1

and eij is the Kronecker symbol. Q We now study the relationship between the sets R. k C 0/ and X.k C 0/ under Q Q the assumption that R.k / X .k / R.k / X .k / . According to the property of the support functions, we ﬁnd c.R.k C 0/; / D c..E C Bk /R.k / C Pk ; / D c.R.k /; .E C Bk /T / C c.Pk ; / ./ c.XQ .k /; .E C Bk /T / C c.PQk ; / D

n X ¹c.Xi .k /; ..E C Bk /T /i / C c.Pik ;

i /º

i D1

n n ² X ıi .k / X D .ej i C bjki / xi .k / C 2 i D1

j D1

ˇ n jıi .k /j ˇˇ X C .ej i C bjki / 2 ˇ j D1

c.XQ .k C 0/; / D

n X

c.Xi .k C 0/;

j

ˇ ˇ qik ˇ j jˇ C 2

qk

C pik C i 2 ³ i j D Ik1 . /;

i/

i D1

n ² X ıi .k C 0/ D xi .k C 0/ C 2 i D1

D

n ²X n X i D1

i

jıi .k C 0/j j C 2

ıj .k / k .eij C bij / xj .k / C 2

j D1

ˇ n 1 ˇˇ X k k C ˇ .eij C bij /ıj .k / C qi 2 j D1

i

i

ij

qk C pik C i 2

³ ij

³

D Ik2 . /:

i

Section 4.5 Approximation of the Integral Funnel of a Linear Differential Inclusion

167

The relationship between Ik1 . / and Ik2 . / is directly connected with the relationship between Jk1 . / D

n ² X iD1

and

ˇX ˇ n .ej i C bjki / jıi .k /jˇˇ j D1

ˇ ˇ k ˇ j ˇ C qi j

ˇ n ˇX X ˇ n ˇ k kˇ ˇ Jk2 . / D ˇ .eij C bij /ıj .k / C qi ˇj

³ ij

i j:

iD1 j D1

Thus, we have proved the following assertion: Theorem 12 ([121]). Let Pk PQk .Pk PQk / and let X.k / XQ .k / .X.k / 2 Rn . Then the following XQ .k //. Assume that Jk1 . / ./Jk2 . / for all Q k C 0/ .R.k C 0/ XQ .k C 0//. inclusion is true: R.k C 0/ X. Corollary 7. For n D 1, we have Pk D PQk and Jk1 . / D jıi .k /.1 C bk / j C qk j j j.1 C bk /ıi .k / C qk j j j D Jk2 . / for all

Q k C 0/. 2 R. Thus, R.k C 0/ X.

Theorem 13 ([121]). Let Pk PQk and let R.k / XQ .k /. If, for any m D 1; n, all k /ı . / are nonnegative for all i D 1; n, then the following inclusion is .emi C bmi i k true: R.k C 0/ XQ .k C 0/. Proof. It is sufﬁcient to check the validity of the inequality Jk1 . / Jk2 . / for the vectors D ˙em ; m D 1; n. In this case, Jk1 .˙em / D

n X

k k jıi .k /.emi C bmi /j C qm ;

iD1

ˇX ˇ ˇ n ˇ k k Jk2 .˙em / D ˇˇ .emi C bmi /ıi .k / C qm ˇˇ: iD1

k /ı . / are nonnegative for all i D 1; n, we Since, for any m D 1; n, all .emi C bmi i k have Jk1 .˙em / D Jk2 .˙em / for all m D 1; n.

Remark 8. If the equalities Pk D PQk and R.k / D XQ .k / in Theorem 13 are true, then Jk1 .˙em / D Jk2 .˙em / for all m D 1; n. Thus, the set R.k C 0/ is inscribed Q k C 0/. in the set X. For n D 1, we conclude that R.k C 0/ D XQ .k C 0/.

168

Chapter 4 Linear Systems with Multivalued Trajectories

Example 12. Consider an impulsive differential inclusion (4.55) for n D 1: x 2 a.t /x C F .t /; xj tDk 2 bk x C Pk ;

t ¤ k ;

x.0/ 2 X0 :

It is associated with the following linear equation with -derivative: D .x; ı/ D a.t /.x; ı/ C .f .t /; r.t //;

t ¤ k ;

.x; ı/j tDk D bk .x; ı/ C .pk ; qk /; .x.0/; ı.0// D .x0 ; ı0 /; which decomposes into two linear impulsive differential equations 8 8 P ˆ ˆ <ı D a.t /ı C r.t /; t ¤ k ; <xP D a.t /x C f .t /; t ¤ k ; xj tDk D bk x C pk ; ıj tDk D bk ı C qk ; ˆ ˆ : : x.0/ D x0 ; ı.0/ D ı0 : In this case, F .t / D FQ .t / and X0 D XQ 0 . Assume that bk 1 for all k 2 N. We now check the validity of the conditions of Theorem 13, R Z 1 R 1 1 .1 C b1 /ı.1 / D .1 C b1 / e 0 a.s/ds ı0 C r.s/e s a.s/ds ds 0: 0

Assume that .1 C bk1 /ı.k1 / 0. Then R Z k k1 a.s/ds ı.k1 C 0/ C .1 C bk /ı.k / D .1 C bk / e

k

r.s/e

k1 R

D .1 C bk / ..1 C bk1 /ı.k1 / C qk1 /e Z C

k

r.s/e k1

R s

k

k k1

R s

k

a.s/ds

ds

a.s/ds

a.s/ds

ds 0:

Thus, by virtue of Example 10 and Remark 7, we get R.t / XQ .t /. Hence, R.t / D XQ .t /.

Chapter 5

Method of Averaging in Systems with Pulse Action

In their monograph “Introduction to Nonlinear Mechanics” [72], Krylov and Bogolyubov used on an elegant example to illustrate the efﬁciency of application of the method of averaging to the investigation of oscillations of a pendulum subjected to a pulse action. Later, their ideas were developed by Mitropol’skii [88] and Samoilenko [133,136,137] for a broader class of systems subjected to the action of pulse perturbations. The cited works initiated mathematical investigations of oscillating processes in systems with pulses and stimulated the investigations in the entire theory of differential equations with pulse action. At present, these investigations are extensively carried out both in the Ukraine and abroad. In what follows, we consider the main results obtained in the development of the method of averaging for systems of differential equations with pulse action.

5.1

Oscillating System with One Degree of Freedom

Consider an oscillating system subjected to a pulse action and described by the equation P xR C ! 2 x D "f .t; x; x/; where " is a small parameter. Impulsive forces may act on the system at ﬁxed times or when the image point hits certain sets of the phase (extended phase) space. It is customary to assume that the action of impulsive forces leads to changes in the velocity of motion of the phase point of the system by a certain value, generally speaking, depending on the position and velocity of the phase point at the time of pulse action. We now consider an autonomous oscillating system subjected to the action of a periodic impulsive force xR C ! 2 x D "f .x; x/; P

t ¤ i ;

xj P tDi D "Ii .x; x/: P

(5.1)

It is supposed that the times of pulse action i and the functions Ii .x; x/ P are such that iCp D i C 2; IiCp .x; x/ P D Ii .x; x/ P (5.2) for all integers i and a natural number p. In analyzing Eqs. (5.1), the resonance and nonresonance cases should be considered separately. In the nonresonance case, i.e., in the case where ! is an irrational number poorly approximated by rational numbers, system (5.1) was studied in [90].

170

Chapter 5 Method of Averaging in Systems with Pulse Action

Nonresonance Case. Assume that Ii .x; x/ P are ﬁnite polynomials, the function f .x; x/ P is continuously differentiable, and its derivatives satisfy the Lipschitz condition with respect to both variables in the domain x2 C

xP 2 R2 : !2

(5.3)

By the change of variables x D a sin ;

xP D a! cos ;

(5.4)

in the system of equations (5.1), we arrive at the equations da " D f .a sin ; a! cos / cos ; dt ! d " D! f .a sin ; a! cos / sin ; t ¤ i ; dt !a " aj tDi D Ii .a sin ; a! cos / cos C "2 ; ! " Ii .a sin ; a! cos / sin C "2 : j tDi D !a

(5.5)

These equations are analyzed by the method of averaging. As shown in [72], the application of the method of averaging is equivalent to a certain sequence of changes aimed at the exclusion of the dependent variable (time) from the system up to quantities of arbitrarily high smallness in ". It turns out that the process of successive changes can also be used for systems subjected to pulse actions. We introduce the following notation: f .a sin ; a! cos / cos

D f .1/ .a; /;

f .a sin ; a! cos / sin

D f .2/ .a; /;

Ii .a sin ; a! cos / cos

D Ii.1/ .a; /;

Ii .a sin ; a! cos / sin

D Ii.2/ .a; /

.1/ .2/ and assume that the functions Ii .a; / and Ii .a; / are ﬁnite trigonometric polynomials. Moreover, let N X @Ii.j / .a; / .j / D .Aik .a/ sin k @ kD1

i D 1; p;

j D 1; 2:

.j /

C Bik .a/ cos k /;

171

Section 5.1 Oscillating System with One Degree of Freedom

By zj .a; ; t / we denote the following functions: zj .a; ; t / p N ² 1 XX .j / D ŒAik .a/ sin k

.j /

C Bik .a/ cos k

1 X cos n.t i / .k!/2 n2

nD1

iD1 kD1

.j / C k!ŒBik .a/ sin k

/ A.j .a/ cos k ik

³ 1 X sin n.t i / : nŒ.k!/2 n2

nD1

By direct veriﬁcation, we can show that, for t ¤ i , these functions satisfy the relation p X @Ii.j / .a; / 1 t i @zj @zj !C D ; @ @t @ 2 2 iD1

where . s/ is a 2-periodic function deﬁned, on a period, by the expression s; 0 < s < 2. In the system of equations (5.5), we perform the following change of variables: aDbC

" .1/ u .b; ; t /; !

D

" .2/ u .b; ; t /; !b

(5.6)

where 1 !

u.j / .b; ; t / D

Z f .j / .b; / C

p 1 X .j / Ii .b; / 2 i D1

p 1 X .j / .j / I i .b/ C zj .b; ; t / f0 .b/ 2 C

p X

.j /

Ii

.b; /

iD1

iD1

1 t i ; 2 2

j D 1; 2:

.j /

Here, f0.j / .b/ and I i .b/ are the mean values of the functions f .j / .b; / and .j / Ii .b; /, respectively, and the integrals stand for the primitives whose mean value is equal to zero. Substituting (5.6) in (5.5), we get db " D F .b/ C "2 R.1/ .b; ; t; "/; dt ! d " D! ˆ.b/ C "2 R.2/ .b; ; t; "/; dt !b .1/

bj t Di D "2 Ji .b; ; "/;

t ¤ i ; .2/

j tDi D "2 Ji .b; ; "/;

(5.7)

172

Chapter 5 Method of Averaging in Systems with Pulse Action

where .1/

F .b/ D f0 .b/ C

p 1 X .1/ I i .b/; 2 iD1

ˆ.b/ D

f0.2/ .b/

p 1 X .2/ C I i .b/; 2 iD1

the functions R.1/ .b; ; t; "/ and R.2/ .b; ; t; "/ satisfy the Lipschitz condition with respect to b and uniformly in t and " and are piecewise continuous in t with discontinuities of the ﬁrst kind at points t D i . The functions Ji.1/ .b; ; "/ and Ji.2/ .b; ; "/ satisfy the Lipschitz condition with respect to b and uniformly in i and ". Neglecting the quantities of the order of smallness "2 in Eq. (5.7), we arrive at the equations of the ﬁrst approximation db " D F .b/; dt !

d " D! ˆ.b/: dt !b

(5.8)

Assume that the equation F .b/ D 0 has a simple positive root b D b0 such that < 0. We now clarify what corresponds to this root in the system of equations (5.7). To this end, we perform certain transformations in this system. Note that the equation db ! D F .b/; dt F 0 .b0 /

has the general solution b D A.ce t / in a neighborhood of the point b D b0 , where

D F 0 .b0 /, c is an arbitrary constant and, in addition, A.0/ D b0 . Denote Z ˆ.h/ D It is clear that

h 0

1 ˆ.A.h// ˆ.b0 / dh:

!h A.h/ b0

ˆ.A/ @ˆ 1

h D ˆ.b0 / : @h !A !b0

In Eqs. (5.7), we perform the change of variables b D A.h/;

D ˆ.h/:

For t ¤ i , we get dh D " h C "2 P .h; ; t; "/; dt

d D !."/ C "2 Q.h; ; t; "/; dt

(5.9)

173

Section 5.1 Oscillating System with One Degree of Freedom

where P .h; ; t; "/ D

R.1/ .A.h/; ˆ.h/; t; "/ ; A0 .h/

Q.h; ; t; "/ D R.2/ .A.h/; ˆ.h/; t; "/ C

@ˆ .1/ R .A.h/; ˆ.h/; t; "/ @h ; A0 .h/

!."/ D !

" ˆ.b0 /: !b0

For t D i , we ﬁnd A.h C h/ A.h/ D "2 Ji.1/ .A.h/; ˆ.h/; "/; Œˆ.h C h/ ˆ.h/ D

.2/ "2 Ji .A.h/;

(5.10)

ˆ.h/; "/:

It is clear that these equations are solvable with respect to h and and their solutions are quantities of the same order as "2 . Thus, by the change of variables (5.9), we reduce system (5.7) to the equations dh d D " h C "2 P .h; ; t; "/; D !."/ C "2 Q.h; ; t; "/; dt dt hj t Di D "2 Hi .h; ; "/; j tDi D "2 Gi .h; ; "/:

(5.11)

In this case, there exist positive numbers 0 and "0 such that, in the domain jhj 0 ;

1 < < 1;

1 < t < 1;

0 < " "0 ;

(5.12)

the functions P .h; ; t; "/, Q.h; ; t; "/, Hi .h; ; "/, and Gi .h; ; "/ are deﬁned and continuous in h; , and ", piecewise continuous in t with discontinuities of the ﬁrst kind at t D i , periodic in and t with period 2, and satisfy the inequalities jP .h0 ; 0 ; t; "/ P .h00 ; 00 ; t; "/j C jQ.h0 ; 0 ; t; "/ Q.h00 ; 00 ; t; "/j C jHi .h0 ; 0 ; "/ Hi .h00 ; 00 ; "/j C jGi .h0 ; 0 ; "/ Gi .h00 ; 00 ; "/j L.jh0 h00 j C j 0 00 j/;

(5.13)

jP .h; ; t; "/j C jQ.h; ; t; "/j C jHi .h; ; "/j C jGi .h; ; "/j M;

(5.14)

where L and M are positive constants. By virtue of the assumption that the equation F .b/ D 0 possesses a simple positive root, Eq. (5.8) has a stationary solution b D b0 , D !."/t C 0 , where 0 is an arbitrary constant. The approximate stationary solution of Eq. (5.1) corresponding to this solution has the form x D b0 sin.!."/t C 0 / " C .u.1/ .b0 / sin.!."/t C 0 / u.2/ .b0 / cos.!."/t C 0 //: !

(5.15)

174

Chapter 5 Method of Averaging in Systems with Pulse Action

Theorem 1 ([90]). All solutions of the differential equation with pulse action (5.1) whose initial values are sufﬁciently close to the initial values of the approximate stationary solutions (5.15) are uniformly bounded on the entire real semiaxis .t 0/, at least for sufﬁciently small values of the parameter ". The proof of Theorem 1 is based on the following two lemmas: Lemma 1. Assume that, for t t0 , a nonnegative piecewise continuous function y.t / satisﬁes the inequality Z y.t / ˛ C

t t0

Œˇe ı.st0 / C y.s/ds C

X

Œˇe ı.i t0 / C y.i 0/;

t0
where ˛ 0, ˇ 0, ı > 0, > 0, and i are points of discontinuity of the ﬁrst kind of the function y.t /. Then n

y.t / ˛.1 C / C ˇ

n X

e

.ı/.i i 1 /

.1 C /

i1

e .tt0 /

iD1

C

ˇ.e ı.tt0 / e .tt0 / / ; ı

tn < t < tnC1 :

The lemma is proved by induction by using Theorem 1.5 from [46] in each interval of continuity of the function y.t /. Corollary 1. If, in the conditions of the lemma, ı > and the sequence of times i is such that iCp D i C T , then, for all t t0 , the function y.t / admits the following estimate: p p .1 C /p e T .t t0 / .tt0 / y.t / ˛.1 C /p e T .tt0 / C ˇe .ı/T e ˇ ı.tt0 / e C : ı

Moreover, if ı > .1 C

p T /,

then

e ı.t t0 / y.t / ˛.1 C /p C

ˇpe .ı/T ˇ : C p T .ı .1 C T // ı

Lemma 2. There exists a positive number " "0 such that the solutions h t .; "/ of the system of equations (5.10) are uniformly bounded independently of t if the initial values h0 are sufﬁciently small for all 0 < " < " and t t0 .

Section 5.1 Oscillating System with One Degree of Freedom

175

Indeed, we ﬁx a sufﬁciently small positive number ı0 < 0 and choose " from the condition .1 C "2 L/p ı0 C

"Mpe " T "M 0 : C p T Œj j ".1 C T /L j j "L

In view of inequalities (5.13) and (5.14), it follows from Eqs. (5.11) that Z t e " .t t0 / jh t j jh0 j C "2 Œe " .st0 / M C Le " .st0 / jhs jds C "2

X

t0

.e " .i t0 / M C Le " .i t0 / jhi j/;

t0
whence, by virtue of Corollary 1, we get jh t .; "/j .1 C "2 L/p jh0 j C

"Mpe " T "M : C p T Œj j ".1 C T /L j j "L

Therefore, jh t .; "/j 0 for all t t0 and 0 < " " provided that jh0 j < ı0 , which completes the proof of Lemma 2. In view of the changes of variables (5.4), (5.6), and (5.9), Lemma 2 implies the assertion of Theorem 1 because the initial condition h0 in the variables b takes the form b D b0 . Hence, if " is sufﬁciently small and h0 is close to zero, then the initial values x0 and xP 0 of the original variables are close to the initial values of the corresponding approximate stationary solutions (5.15). We now prove the existence of an invariant toroidal set of the system of equations (5.11). We seek the required invariant set ."/ in the form ."/W h D u.; t; "/;

(5.16)

where u.; t; "/ is a function continuous in , piecewise continuous in t with discontinuities of the ﬁrst kind at t D i , and periodic in and t with period 2. Expression (5.16) speciﬁes ."/ whenever du. t .s; /; t; "/ " u. t .s; /; t; "/ dt C "2 P .u. t .s; /; t; "/; t .s; /; t; "/ for all t 2 .1; 1/; t ¤ i , and u.i .s; /; i ; "/ D "2 Hi .u.i .s; /; i ; "/; i .s; /; "/;

(5.17)

176

Chapter 5 Method of Averaging in Systems with Pulse Action

where t .s; / is a solution of the system d D !."/ C "2 Q.u.; t; "/; ; t; "/; t ¤ i ; dt j tDi D "2 Gi .u.; i ; "/; ; "/; s .s; / D :

(5.18)

To ﬁnd the invariant set ."/, we use a simple iterative method according to which ."/ is sought as the limit of a sequence of sets .1/ ."/; : : : ; .n/ ."/; : : : each of which is an invariant set .nC1/ ."/W h D u.nC1/ .; t; "/;

n D 0; 1; : : : ;

of the system of equations dh D " h C "2 P .u.n/ .; t; "/; ; t; "/; dt d D !."/ C "2 Q.u.n/ .; t; "/; ; t; "/; dt

t ¤ i ;

(5.19)

hj tDi D "2 Hi .u.n/ .; i ; "/; ; "/; j tDi D "2 Gi .u.n/ .; i ; "/; ; "/: .s; / D be the general solution of the equation obtained Let t.n/ .s; /, .n/ i from (5.18) as a result of the replacement of u.; t; "/ with u.n/ .; t; "/. Then Z t 2 h t .; "/ D " e " .t s/ P .u.n/ ..n/ .s; /; ; "/; s.n/ .; /; ; "/d 1

2

C"

X

e " .t i / Hi .u.n/ ..n/ .s; /; i ; "/; .n/ .s; /; "/ i i

i
is a family of bounded solutions of the equation dh D " h C "2 P .u.n/ . t.n/ .s; /; t; "/; t.n/ .s; /; t; "/; dt

t ¤ i ;

.s; /; i ; "/; .n/ .s; /; "/ hj t Di D "2 Hi .u.n/ ..n/ i i depending on ; s, and " as on parameters. This family covers the invariant set .nC1/ ."/W h D u.nC1/ .; t; "/ Z t .n/ 2 e " P .u.n/ . t C .t; /; t C ; "/; t C .t; /; t C ; "/d " 1

2

C"

X

i
e " .ti / Hi .u.n/ ..n/ .t; /; i ; "/; i .t; /; "/: i

(5.20)

177

Section 5.1 Oscillating System with One Degree of Freedom

It can be directly shown that the function u.nC1/ .; t; "/ is periodic in and t with period 2. Prior to proving the convergence of the sequence of functions u.n/ .; t; "/, we es.n/ tablish some properties of these functions and the functions t .s; /. To this end, we need the following assertion: Lemma 3. There exist positive numbers "1 .0 < "1 "0 / and N D N."/ .N."/ ! 0 as " ! 0/ such that, for all 0 < "1 "0 , 1 < t < 1, 1 < s < 1, 0 , 00 , t s, and n D 0; 1; : : : , ju.nC1/ . 0 ; t; "/ u.nC1/ . 00 ; t; "/j N."/j 0 00 j; j t.n/ .s; 0 / t.n/ .s; 00 /j Œ1 C N 0 ."/p e

(5.21)

p N 0 ."/.1C T

/.ts/

j 0 00 j;

N 0 ."/ D "2 L.1 C N."//: The lemma is proved by induction. For n D 0, by using the following representa.0/ tion for t .s; /: Z

t.0/ .s; / D C !."/.t s/ C "2 2

C"

X

t s

Q.0; .0/ .s; /; ; "/d

Gi .0; .0/ .s; /; "/; i

s
we ﬁnd j t.0/ .s; 0 / t.0/ .s; 00 /j j 0 00 j C "2 L X

2

C" L

Z

t s

j.0/ .s; 0 / .0/ .s; 00 /jd

j.0/ .s; 0 / i

.0/ .s; 00 /j: i

s
By virtue of Lemma 1 in Chapter 1, this yields j t .s; 0 / t .s; 00 /j e " .0/

2 L.ts/

.0/

.1 C "2 L/Œ

t s T C1p

2 L.1C p /.t s/ T

.1 C "2 L/p e "

j 0 00 j

j 0 00 j;

(5.22)

i.e., inequality (5.20) holds for n D 0. For n D 0, it follows from (5.20) that .1/

ju

0

.1/

. ; t; "/ u

00

Z

0

2

. ; t; "/j " L 2

1

C" L

e " j Ct .t; 0 / Ct .t; 00 /jd

X

i
.0/

.0/

e " .ti / j.0/ .t; 0 / .0/ .t; 00 /j: i i

178

Chapter 5 Method of Averaging in Systems with Pulse Action

In view of inequality (5.22), this yields ju.1/ . 0 ; t; "/ u.1/ . 00 ; t; "/j "LN0 ."/j 0 00 j; where

2

p

N0 ."/ D .1 C " L/

1 j j "L.1 C

Note that N0 ."/ !

p T/

C

p

1 1C j j T

"pe "j jT p

1 e "Œj j"L.1C T /T

:

as " ! 0:

Finally, we can write ju.1/ . 0 ; t; "/ u.1/ . 00 ; t; "/j N1 ."/j 0 00 j; where N1 ."/ D "LN0 ."/. Assume that the following inequalities hold for n D 1; k 1: ju.nC1/ . 0 ; t; "/ u.nC1/ . 00 ; t; "/j NnC1 ."/j 0 00 j; j t.n/ .s; 0 / t.n/ .s; 00 /j 2 L.1CN

Œ1 C "2 L.1 C Nn ."//p e "

p n ."//.1C T /.t s/

j 0 00 j:

Then, for n D k, the representation .k/

t .s; / D C !."/.t s/ Z t 2 Q.u.k/ ..k/ .s; /; ; "/; .k/ .s; /; ; "/d C" s

2

C"

X

Gi .u.k/ ..k/ .s; /; i ; "/; .k/ .s; /; "/ i i

(5.23)

s
implies that j t .s; 0 / t .s; 00 /j j 0 00 j Z t 2 .1 C Nk ."//j.k/ .s; 0 / .k/ .s; 00 /jd C" L .k/

.k/

s

2

C " L.1 C Nk ."//

X

j.k/ .s; 0 / .k/ .s; 00 /j: i i

s
By virtue of Lemma 1 in Chapter 1, this inequality yields the following estimate: .k/ .k/ j t .s; 0 / t .s; 00 /j 2 L.1CN

.1 C "2 L.1 C Nk ."//p e "

p k ."//.1C T

/.ts/

j 0 00 j:

(5.24)

179

Section 5.1 Oscillating System with One Degree of Freedom

Further, it follows from (5.20) that ju.kC1/ . 0 ; t; "/ u.kC1/ . 00 ; t; "/j Z 0 .k/ .k/ 2 e " j tC .t; 0 / tC .t; 00 /jd " L.1 C Nk ."// 1

C

X

e

" .ti /

j.k/ .t; 0 / i

.k/ .t; 00 /j i

:

i
In view of inequality (5.24), we obtain ju.kC1/ . 0 ; t; "/ u.kC1/ . 00 ; t; "/j NkC1 ."/j 0 00 j; where NkC1 ."/ D "L.1 C Nk ."//M.Nk ."/; "/; 1 M.N; "/ D .1 C "2 L/.1 C N / j j "L.1 C N /.1 C C

p T/

"pe "j jT p

1 e "Œj j"L.1CN /.1C T /T

:

Thus, by induction, we conclude that, for all n D 0; 1; : : : , ju.n/ . 0 ; t; "/ u.n/ . 00 ; t; "/j Nn ."/j 0 00 j; where Nn ."/ satisfy the recurrence relation NnC1 ."/ D "L.1 C Nn ."//M.Nn ."/; "/: We choose 0 < "1 "0 sufﬁciently small to guarantee that 1 "1 LM.1; "1 / < : 2 This choice is possible because M.1; "/ !

p

1 1C j j T

as " ! 0:

Thus, in view of the monotonicity of the function M.N; "/, we get NnC1 ."/ 2"LM.1; "/ for all n D 0; 1; : : : and 0 < " "1 . Hence, to complete the proof of Lemma 3, it sufﬁces to set N."/ D 2"LM.1; "/:

180

Chapter 5 Method of Averaging in Systems with Pulse Action .n/

We now prove the uniform convergence of the functions u.n/ .; t; "/ and t .s; /. To do this, we estimate the differences .nC1/

t

.n/

u.nC1/ .; t; "/ u.n/ .; t; "/:

.s; / t .s; / and

By using representation (5.23) for t.n/ .s; / and inequality (5.21), we get Z t .nC1/ .n/ 2 j t .s; / t .s; /j " L Œku.nC1/ .; t; "/ u.n/ .; t; "/k s

C .1 C N."//j.nC1/ .s; / .n/ .s; /jd X Œku.nC1/ .; t; "/ u.n/ .; t; "/k C "2 L s
C .1 C N."//j.nC1/ .s; / .n/ .s; /j; i i where kuk D sup0 2;0t2 ju.; t; "/j. By virtue of Lemma 3 in Chapter 1, this yields .nC1/

j t

.n/

.s; / t .s; /j

(5.25)

p ku.nC1/ u.n/ k 2 Œ.1 C "2 L.1 C N."///p e " L.1CN."//.1C T /.t s/ 1; 1 C N."/

t > s:

Representation (5.20) and estimate (5.21) now imply that ju.nC1/ .; t; "/ u.n/ .; t; "/j Z 0 2 e " Œku.n/ .; t; "/ u.n1/ .; t; "/k " L 1

C "2 L

X

.n1/ C .1 C N."//j t.n/ C .t; / tC .t; /jd

e " .t i / Œku.n/ .; t; "/ u.n1/ .; t; "/k

i
.t; / .n1/ .t; /j: C .1 C N."//j.n/ i i Applying Lemma 3 in Chapter 1 and estimate (5.25) to the last inequality, we ﬁnd ku.nC1/ .; t; "/ u.n/ .; t; "/k ."/ku.n/ .; t; "/ u.n1/ .; t; "/k; where

1 "pe "j 1 ."/jT C ; j 1 ."/j 1 e " 1 ."/T p

:

1 ."/ D C "L.1 C N."// 1 C T ."/ D "L.1 C "2 L.1 C N."///p

181

Section 5.1 Oscillating System with One Degree of Freedom

Hence, for all n D 1; 2; : : : , we conclude that ku.nC1/ .; t; "/ u.n/ .; t; "/k n ."/ ku.1/ .; t; "/k;

(5.26)

j t.n/ .s; /

(5.27)

t.n1/ .s; /j

n1 ."/ 1 C N."/

2 L.1CN."//.1C p /.t s/ T

Œ.1 C "2 L.1 C N."///p e "

1ku.1/ .; t; "/k:

It follows from (5.20) that ku.1/ .; t; "/k "M0 ."/;

M0 ."/ D

1 "pe " T C M: j j 1 e " T

We now choose 0 < "0 "1 sufﬁciently small to guarantee that the inequalities "M0 ."/ < 0 ;

."/ < 1

are simultaneously true for all 0 < " "0 . Then, for all 0 < " "0 and n D 1; 2; : : : , we get ku.n/ .; t; "/k 0 : It follows from inequalities (5.26) and (5.27) that, in any ﬁnite interval of the real line and, hence, for all t 2 R, the sequence of functions t.n/ .s; / is uniformly convergent and the same is true for the sequence of functions u.n/ .; t; "/. Denote lim t.n/ .s; / D t .s; /;

n!1

lim u.n/ .; t; "/ D u.; t; "/:

n!1

(5.28)

We now show that the limiting function u.; t; "/ speciﬁes the invariant set ."/W h D u.; t; "/ of the system of equations (5.11). Indeed, since h D u.nC1/ .; t; "/ is an invariant set of the system of equations (5.19), the trajectories t.n/ and h.n/ t lying in this set satisfy the relations Z t t.n/ .s; / D C Œ!."/ C "2 Q.u.n/ ..n/ ; ; "/; .n/ .s; /; ; "/d s

2

C"

X

Gi .u.n/ ..n/ ; i ; "/; .n/ ; "/; i i

s
u

.nC1/

.n/ . t ; t; "/

.nC1/

Du

.; s; "/ C

Z

t s

Œ" u.nC1/ ..n/ ; ; "/

C "2 P .u.n/ ..n/ ; ; "/; .n/ ; ; "/d X C "2 Hi .u.n/ ..n/ ; i ; "/; .n/ ; "/: i i s
(5.29)

(5.30)

182

Chapter 5 Method of Averaging in Systems with Pulse Action

The assumptions of continuity of the functions P , Q; Hi , and Gi in h and guarantee the validity of all permutations of the limits required to derive the identities Z t Œ!."/ C "2 Q.u. ; ; "/; ; ; "/d t .s; / D C s

C "2

X

Gi .u.i ; i ; "/; i ; "/;

s
Z

u. t ; t; "/ D u.; s; "/ C C "2

X

t s

(5.31)

Œ" u. ; ; "/ C "2 P .u. ; ; "/; ; ; "/d

Hi .u.i ; i ; "/; i ; "/

s
from relations (5.28)–(5.30). It follows from identities (5.31) that the function u.; t; "/ continuous in , piecewise continuous in t , and periodic in and t with period 2 for all t 2 R satisﬁes the system of equations (5.17), (5.18). Hence, h D u.; t; "/ is an invariant set of the system of equations (5.11). Thus, the following theorem is proved: Theorem 2 ([90]). Assume that the system of equations (5.11) satisﬁes inequalities (5.13) and (5.14) in domain (5.12). Then there exists a positive number "0 " such that, for all 0 < " "0 , the system of equations (5.11) possesses an invariant set ."/W h D u.; t; "/, where the function u.; t; "/ is periodic in and t with period 2, satisﬁes the Lipschitz condition with respect to with a constant proportional to ", and is piecewise continuous in t with discontinuities of the ﬁrst kind for t D i . On the invariant set ."/, the system of equations (5.11) reduces to the system d D !."/ C "2 Q.u.; t; "/; ; t; "/; dt j tDi D "2 Gi .u.; i ; "/; ; "/:

t ¤ i ;

Note that the function u.; t; "/ continuously depends on the parameter " and approaches zero as " ! 0. In view of Theorems 1 and 2, with regard for the changes of variables (5.4), (5.6), and (5.9) reducing Eq. (5.1) to the system of equations (5.11), we can now formulate the following assertion for the original system of equations (5.1): Theorem 3 ([90]). Assume that the system of equations with pulse action (5.1) satisﬁes the following conditions: (1) the function f .x; x/ P is continuously differentiable in domain (5.3) with respect to both variables and its derivatives in this domain satisfy the Lipschitz condition;

Section 5.1 Oscillating System with One Degree of Freedom

183

(2) the functions Ii .x; x/ P deﬁned as ﬁnite polynomials of their variables and the times i satisfy equalities (5.2); (3) the equation F .a/

.1/ f0 .a/

p 1 X .1/ C I i .a/ D 0; 2 iD1

where I0.1/ D

1 2

Z

2

f .a sin ; a! cos / cos d ; 0

possesses an isolated positive root a D a0 such that F 0 .a0 / < 0. Then there exists a positive number " such that, for all 0 < " " , (1) the system of equations possesses an invariant set r 1 x 2 C 2 xP 2 D u .; t; "/; ! where the function u .; t; "/ satisﬁes the Lipschitz condition with respect to , is piecewise continuous in t with discontinuities of the ﬁrst kind for t D i , periodic in and t with period 2, and such that u .; t; "/ ! a0 as " ! 0 uniformly in and t ; (2) there exists a ı0 ."/-neighborhood .ı0 ."/ ! 0 as " ! 0/ of the cylinder x2 C

1 2 xP D a02 !2

such that any solution of Eq. (5.1) starting at t D t0 from this neighborhood remains bounded together with its derivative for all t t0 . Resonance Case. In the case where ! is close to one of the numbers of the form rs , where r and s are natural numbers, system (5.1) is investigated in [139]. In this case, the ﬁrst approximations to the solutions of Eqs. (5.1) should be taken in the form x D a sin. rts C /, where a and are solutions of the equations aP D "s

A.a/ C I .1/ .a; / ; r

P D "s

a 2

C B.a/ C I .2/ .a; / ; ra

where, in turn, Z

r f a sin ; a cos cos d; s 0 Z 2

1 r B.a/ D f a sin ; a cos sin d; 2 0 s A.a/ D

1 2

2

184

I

Chapter 5 Method of Averaging in Systems with Pulse Action

.1/

ps r

r

r

r 1 X .a; / D Ii a sin i C ; a cos i C cos i C ; 2s s s s s iD1

I .2/ .a; / D

ps r

r

r

r 1 X Ii a sin i C ; a cos i C sin i C ; 2s s s s s iD1

2

r ! 2: s2 As an example, we study the inﬂuence of pulse action on a linear oscillator under the assumption that this action leads to an increase in the kinetic energy of the oscillator by a constant value "I . In other words, we study the system of equations " D

"

; t ¤ i ; xP 2 D 2"I; x !i D 2 i; i D 0; ˙1; : : : :

xR C ! 2 x D

> 0; I > 0; (5.32)

In these equations, we perform the change of variables x D a sin.!t C /;

x D a cos.!t C /

and average the equations obtained as a result over the explicit time. As a result, in the ﬁrst approximation, we get the following equations: I

a a! tanh ; P D "I : 2 2a2 ! The ﬁrst equation has the stationary solution 12 I a D a0 D ;

!

aP D "

(5.33)

and the second equation has two stationary solutions D 0.mod 2/ and

D .mod 2/:

By using the relations of the improved ﬁrst approximations, we get x.t / D a sin.!t C / " a

1 "I sin !t X sin k!t cos.!t C / C ; 4 a! 2 cos k kD1

where a and are the solutions of the averaged equations (5.33). We now analyze system (5.32) in the ﬁrst approximation. It is easy to see that all solutions of this system approach, as functions of time, one of two asymptotically stable periodic solutions 1 cos !t "I sin !t X sin k!t x.t / D ˙a0 sin !t ˙ " a0 ˙ : 4 a0 ! 2 k kD1

(5.34)

185

Section 5.1 Oscillating System with One Degree of Freedom

Moreover, the solutions corresponding to the stationary solution of the ﬁrst equation in (5.33) cover an integral set determined (in the extended phase space .x; x; P t // by the equation 2 2

2

1 2

.! x C xP / D

! 12

sin 2.!t C / 1 "

4

I

C" !

12 X 1 sin k!t : k kD1

This set is asymptotically stable and the solutions lying in this set are attracted, in the course of time, by one of the two periodic solutions (5.34). The possibility of application of the method of averaging to Eqs. (5.1) in the resonance case is justiﬁed by the following assertion: Theorem 4 ([139]). Assume that the functions f .x; x/ P and Ii .x; x/, P i D 1; p in the system of equations (5.1) are deﬁned and satisfy the Lipschitz condition with respect to their variables in a certain domain 0 < ˛ 2 x 2 C ! 2 xP 2 ˇ 2 : Suppose that the system of equations A.a/ C I .1/ .a; / D 0;

B.a/ C

a C I .2/ .a; / D 0 2

(5.35)

has an isolated solution a D a0 , D 0 from the strip ˛ < a0 < ˇ together with a certain its -neighborhood and is such that the index of the point .a0 ; 0 / under the mapping speciﬁed by the left-hand sides of Eqs. (5.35) differs from zero. Then there exists a positive number "0 such that, for all 0 < " "0 , system (5.1) possesses, in the resonance case, a solution 2-periodic in rts . An oscillating system subjected to a pulse action when the phase point .x; x/ P passes through the position x D x0 , namely,

xj P xDx0

xR C ! 2 x D "f .x; x/; P x ¤ x0 ; ´ "I.xP /; x D x0 ; xP 0; D xP C xP D 0; x D x0 ; xP < 0;

(5.36) (5.37)

where xP and xP C are the velocities of the point prior to and after the action of the instantaneous pulse, is analyzed by the method of averaging in [133, 136, 141]. System (5.36), (5.37) can be interpreted as a motion of oscillator (5.36) subjected to the action of instantaneous pulses increasing the velocity of the oscillator by "I.xP / every time when the oscillator passes through the position x D x0 (with a nonnegative velocity). As a simple example of this oscillator, we can mention an impact model of a clock studied by many authors [16, 72].

186

Chapter 5 Method of Averaging in Systems with Pulse Action

It is clear that the motions of system (5.36), (5.37) may have the following form: (1) Motions without pulse actions. These motions are described by Eq. (5.36) and satisfy the inequality x.t / ¤ x0 for all t 0. (2) Motions subjected to the pulse action for t 0 only ﬁnitely many times. These are motions for which x.t / D x0 for ﬁnitely many values of t 0. (3) Motions subjected to the pulse action for t 0 inﬁnitely many times. Motions of the ﬁrst two types either approach the inﬁnity as t ! 1 or are attracted to the equilibrium position and cycles of Eq. (5.36) lying in the domain x < x0 of the phase plane .x; x/. P In this case, periodic motions may appear only for the ﬁrst type of motions. They are investigated without using conditions (5.37) according to the standard schemes of asymptotic methods. Motions of the third type either approach the inﬁnity as t ! 1 or are attracted to discontinuous cycles of system (5.36), (5.37). Motions of the last type are not cycles of Eq. (5.36) and generate periodic motions with exactly one pulse action per period. They can be studied by using the scheme of the method of averaging proposed in [132]. Suppose that system (5.36), (5.37) possesses a periodic solution. In view of the autonomy of system (5.36), (5.37), the function obtained from this solution with t replaced by t C s is also a periodic solution of system (5.36), (5.37). Therefore, we can always assume that the analyzed periodic solution takes the value x.0/ D x0 at time t D 0 and, in addition, x.C0/ P > 0: By we denote the frequency of this solution. For this solution, the times k of pulse action are given by the formula

k D 2k;

k D 1; 2; : : : ;

and the jumps of velocity xP at these times are given by the following formula: xP D "I.xP / D "I0 : By virtue of periodicity of the solution, the quantity I0 is constant for all k D 2k. Hence, the periodic solution of system (5.36), (5.37) satisﬁes the equation xR C ! 2 x D "f .x; x/; P

t ¤ 2k;

(5.38)

and the condition xP D "I0 ;

t D 2k;

(5.39)

where k can be set equal to 0; ˙1; ˙2; : : : . On the other hand, assume that, for some and t0 , system (5.38), (5.39) possesses periodic solutions with period 2 with respect to t W x D x.t /. It is clear that these solutions are solutions of the original system (5.36), (5.37) whenever x.0/ D x0 ; I0 D I.xP /; 2 0 0; xP C 0: xP D xP

(5.40)

187

Section 5.1 Oscillating System with One Degree of Freedom

Thus, the problem of ﬁnding periodic solutions of system (5.36), (5.37) is reduced to the determination of the periodic solutions of system (5.38), (5.39) with period 2 with respect to t and the choice of the parameters and t0 for these solutions satisfying relations (5.40). In view of the fact that any cycle of system (5.36), (5.37) is formed by a part of the spiral of Eq. (5.36) for which the difference between its two successive crossings of the beam xP 0 by the straight line x D x0 is equal to "I0 , we conclude that the frequency of periodic solutions of system (5.36), (5.37) for small " should be taken close to the frequency of natural oscillations of system (5.36), i.e., close to the value !. Therefore, we seek periodic solutions of system (5.38) under the condition ' !: We represent in the form

2 D ! 2 C "; (5.41) where is an unknown constant. In view of (5.41), system (5.38), (5.39) can be rewritten in the form P xR C 2 x D "Œx C f .x; x/; xP D "I0 ;

t ¤ 2k;

t D 2k:

(5.42) (5.43)

By using the formalism of generalized functions, we pass from system (5.42), (5.43) to the equation 1 X 2k : P C "I0 ı t xR C 2 x D "Œ x C f .x; x/

kD1

Further, in view of the properties of the delta function, we get 1 I0 1 X xR C 2 x D "Œ x C f .x; x/ P C" cos k t : C 2

(5.44)

kD1

In (5.44), we perform the change of variables 1 I0 1 X cos k t xD" C z: 2 k2 1

(5.45)

kD2

This yields the following ordinary differential equation for z: P // C "I0 . t / zR C 2 z D "Œz C f .z C "I0 . t /; zP C "I0 . t C" where

I0 cos t;

(5.46)

1 1 1 X cos k t . t / D : 2 k2 1 kD2

188

Chapter 5 Method of Averaging in Systems with Pulse Action

Relations (5.40) for the variable z take the form "I0 z.0/ D "I0 .0/ D x0 ; I0 D I zP .0/ ; 2

zP .0/

"I0 : 2

(5.47)

Hence, it is necessary to ﬁnd periodic solutions z.t / of Eq. (5.46) with period 2 with respect to t satisfying relations (5.49). To do this, we perform the change of variables z D a cos ; zP D a sin ; D t C : (5.48) As a result, we arrive at the following system of equations: " da P C "I0 sin D Œa cos C f .a cos C "I0 ; a sin C "I0 / dt

"I0 cos t sin ; (5.49) d " P C "I0 cos D Œa cos C f .a cos C "I0 ; a sin C "I0 / dt a "I0 cos t cos : a Then relations (5.47) take the form a0 cos 0 D x0 "I0 .0/; a0 sin 0

"I0 I0 D I a0 sin 0 ; 2

(5.50)

"I0 ; 2

where a0 D a.0/ and 0 D .0/. Assume that the function f .x; x/ P satisﬁes the Lipschitz condition with respect to x and xP in an annular domain

2˛2 2x2 C y 2 ˇ2 2: Then, for small ", (5.49) is a T -system .T D

2 /

[134] in the band

0 < ˛ a ˇ:

(5.51)

Hence, any internal point of band (5.51) has a -constant [134] and this constant is determined, to within the quantities of the second order of smallness, from the system " "I0 1 D A.a0 / sin 0 C "2 ;

2 " " "I0 B.a0 / cos 0 C "2 ; 2 D 2 a0 2a0

Section 5.1 Oscillating System with One Degree of Freedom

where

Z

1 A.a/ D A.a; / D 2

2 0

Z

1 B.a/ D B.a; / D 2

2 0

f .a cos ; a sin / sin d;

189

(5.52)

f .a cos ; a sin / cos d:

According to [134], the solution of system (5.49) is periodic with period 2 with respect to t provided that a0 and 0 satisfy the relations "I0 " sin C "2 D 0; A.a0 /

2 " " "I0 B.a0 / cos 0 C "2 D 0: 2 a0 2a0

(5.53) (5.54)

Moreover, the indicated periodic solution is the limit of the sequence Z " t ŒF1 .t; an1 .t /; n1 .t /; "/ an .t / D a0

0 " n .t / D a0

F1 .t; an1 .t /; n1 .t /; "/dt;

Z

t 0

(5.55)

ŒF2 .t; an1 .t /; n1 .t /; "/ F2 .t; an1 .t /; n1 .t /; "/dt;

where " F1 .t; a; / and " F2 .t; a; / are the right-hand sides of the ﬁrst and second equations of system (5.49), respectively, and Z 1 T 2 ; n D 1; 2; : : : : F .t / D F .t /dt; T D T 0

In particular, for n D 1, to within the quantities of the order "2 , relations (5.55) yield the following formulas: " cos 2 0 cos 2. t C 0 / a0

2 4 Z tC 0 " "I0 cos 0 cos.2 t C 0 / 2 ; f .a cos ; a sin / sin d

0 4

a.t / D a0

" sin 2. t C 0 / sin 2 0

2 4 Z tC 0 " 2 2 f .a cos ; a sin / cos d

a0 0

.t / D 0

"I0 sin 0 sin.2 t C 0 / : a0 4

(5.56)

190

Chapter 5 Method of Averaging in Systems with Pulse Action

We now study the problem of solvability of Eqs. (5.53) and (5.54). Since the constant is a continuous function of the point a0 ; 0 and the parameters I0 ; , and " and satisﬁes the same Lipschitz condition with respect to a0 ; 0 , I0 ; , and " [134], Eq. (5.54) is always solvable in the form D .a0 ; 0 ; I0 ; "/:

(5.57)

Moreover, the function is continuous in the collection of variables a0 , 0 , I0 , and " in the domain of their variation and satisﬁes, in this domain, the Lipschitz condition with respect to a0 , 0 , I0 , and ". Equation (5.54) also implies that, to within the quantities of order ", the function .a0 ; 0 ; I0 ; "/ is given by the formula .a0 ; 0 ; I0 ; "/ D .a0 ; 0 ; I0 ; 0/ D 2

B.a0 / I0 ! cos 0 : a0 a0

(5.58)

Substituting .a0 ; 0 ; I0 ; "/ in the ﬁrst equation in (5.53), we arrive at the relation for the initial values of periodic solutions of system (5.49). To within the quantities of order ", this relation takes the form A.a0 / C

I0 ! sin 0 D 0: 2

(5.59)

It is clear that periodic solutions of system (5.49) are also periodic solutions of the original system (5.36), (5.37) provided that their initial values satisfy relations (5.50). Hence, to determine these values, it sufﬁces to solve the system of relations (5.50), (5.53) by substituting relation (5.57) for the function in (5.53). Since a0 is taken from interval (5.51), relations (5.50) imply that s x0 "I0 .0/ 2 x0 "I0 .0/ cos 0 D ; sin 0 D 1 : (5.60) a0 a0 Both relations (5.60) and the inequality in (5.50) taking, in view of of relation (5.60), the form q "I0

a02 .x0 "I0 .0//2 ; 2 are always true for small " if a0 varies within the range ˇ a0 ˛ > jx0 j:

(5.61)

Thus, we assume that a > jx0 j and substitute (5.60) in the remaining relation in (5.50). As a result, instead of (5.50), we get relations (5.60) and the equation q "I0 2 2 I0 D I a0 .x0 "I0 .0// : (5.62) 2

Section 5.1 Oscillating System with One Degree of Freedom

191

We now assume that the function I.x/ P satisﬁes the Lipschitz condition in the segment ˇ xP 0: Then Eq. (5.62) is solvable for small " in the form I0 D I0 .a0 ; "/:

(5.63)

Moreover, the function I0 .a0 ; "/ satisﬁes the Lipschitz condition with respect to a0 and " in the domain of their variation. Relation (5.62) also implies that, to within the quantities of order ", the function I0 .a0 ; "/ is given by the formula q 2 (5.64) I0 .a0 ; "/ D I0 .a0 ; 0/ D I ! a0 x02 : Thus, the system of relations (5.50), (5.53), (5.57) is reduced to system (5.53), (5.57), (5.60), (5.63). Substituting relations (5.63) in (5.60), we express 0 in terms of a0 . Then we substitute (5.63) and (5.60) in (5.57) and express via a0 . Similarly, we arrive at the expression s I0 .a0 ; "/ x0 "I0 .0/ 2 1 C " : A.a0 / 2 a0 By virtue of relations (5.53), (5.57), (5.60), and (5.63), this expression satisﬁes the original system (5.50), (5.53), (5.54) or, in view of (5.41) and (5.64), the relation q s I.! a02 x02 / x2 A.a0 / ! 1 02 C " D 0: (5.65) 2 a0 Since the left-hand side of Eq. (5.63) is continuous in a0 and ", Eq. (5.65) is solvable for small " provided that the function q I.! a02 x02 / q A.a0 / ! a02 x02 ; (5.66) 2a0 takes values of different signs in the interval ˛ < a0 < ˇ [135]. Moreover, the solutions of Eq. (5.65) are continuous in " and, to within the quantities of a certain order ı."/, where ı."/ ! 0 as " ! 0, are determined by the solutions of the equation q I.! a02 x02 /! q A.a0 / D a02 x02 : (5.67) 2a0 Thus, to satisfy the system of relations guaranteeing the existence of periodic solutions of system (5.36), (5.37), it is sufﬁcient that Eq. (5.67) have a solution lying in segment (5.61) together with its certain neighborhood and that this solution be neither

192

Chapter 5 Method of Averaging in Systems with Pulse Action

a point of maximum nor a point of minimum of function (5.66). By using relations (5.41), (5.58), (5.60), and (5.64), we get the following formula for the frequency of the periodic solution: q I.! a02 x02 /x0 1 " / B.a 0

D!C C "2 ' ! " C : (5.68) 2 ! a0 ! 2a02 According to the changes of variables (5.45) and (5.48), the periodic solutions of system (5.36), (5.37) are determined (to within the quantities of order ı."// by the formula q 1 "I.! a02 x02 / 1 X cos k .t C s/ x.t / D ! 2 k2 1 kD2

C a.t C s/ cos. .t C s/ C .t C s//; where a.t /; .t / is the function given by relation (5.56), is given by relation (5.68), a0 is the root of Eq. (5.67), and s is an arbitrary constant. This proves the following theorem: Theorem 5 ([136]). Assume that the functions f .x; x/ P and I.x/ P characterizing system (5.36), (5.37) are deﬁned and continuous and that they satisfy the Lipschitz condition with respect to x and xP for x and xP from the domain ˛2 x 2 C

xP 2 ˇ2 ; !2

where ˇ and ˛ are positive constants satisfying the inequality ˇ > ˛ > jx0 j: Suppose that the equation q I.! a2 x02 /! q A.a/ a2 x02 D 0; (5.69) 2a where A.a/ is given by relation (5.52), possesses a solution a D a0 lying in the segment ˛ a0 ˇ together with its certain -neighborhood and such that the left-hand side of Eq. (5.69) turns neither into a maximum nor into a minimum for this solution. Then there exists "0 > 0 such that, for all " smaller than "0 , the system of equations (5.36), (5.37) has a one-parameter family of periodic solutions x D x0 .!1 .t C s/; "/ D a.!1 .t C s// cos .!1 .t C s// satisfying the inequality ja.t / a0 j C j!1 ! ".a0 /j ."/;

193

Section 5.1 Oscillating System with One Degree of Freedom

where s is an arbitrary constant, ."/ is a constant approaching zero as " ! 0, q I.! a02 x02 / B.a0 / .a0 / D x0 ; 2 a0 ! 2a0 and B.a/ is given by relation (5.52). The case where the right-hand side of the ﬁrst equation in system (5.36), (5.37) depends on t , i.e., has the form f . t; x; x/, P is studied in [141]. In the cited work, it is shown that, in the resonance case, i.e., for r 2 !2 D C "; s the isolated solution of the system of equations r q I A.a; / C r a2 x02

q a2 x02 s

q

a2 x02 I s D 0; 2sa q q r a2 x02 r a2 x02 I I a s s B.a; / C r x0 D 0; 2 2sa

where

r

r f ; a sin C ; s 0 r r ar cos C cos C d; s s s Z 2s r 1 C ; f ; a sin B.a; / D 2s 0 s r r ar cos C sin C d; s s s A.a; / D

1 2s

Z

2s

generates a periodic solution (with period 2 in rt s / of the original system (5.36), (5.37) for small values of the parameter ". A similar result is obtained in [97] for the integrodifferential equation subjected to the pulse action Z tCs xR C ! 2 x D "f t; x; x; P h. t; s; x.s/; x.s//ds P ; x ¤ x0 ; t

xP D "I.x/; P where f . t; x; x; P v/ and h. t; s; x; x/ P are functions periodic in t and s with period 2; ! ' , and s and x0 are constants.

194

Chapter 5 Method of Averaging in Systems with Pulse Action

In [63], the method of averaging is substantiated for the nonresonance case under an additional assumption concerning the straight line in which the system is subjected to the pulse action. Under fairly general assumptions imposed on the functions f .t; x; x/ P and I.x/, P it is shown that, for small values of the parameter ", the isolated positive root a D a0 of the equation Z 2 Z 2 1 f .t; a sin ; a! cos / cos t cos dt d F .a/ 4 2 0 0 I.a!/ I.a!/ ; C! 2 generates an integral set of the system of equations xR C ! 2 x D "f . t; x; x; P /;

x ¤ "x0 ; xj P xD"x0 D "I.x/: P

(5.70)

This set is speciﬁed by the equation q .x "x0 /2 C ! 2 xP 2 D u.; t; "/; where the function u.; t; "/ satisﬁes the Lipschitz condition with respect to t , is piecewise continuous in with discontinuities of the ﬁrst kind for D k, 2periodic in and t , and such that u.; t; "/ ! 0 as " ! 0. In addition, if F 0 .a0 / < 0, then there exists a ı0 ."/-neighborhood .ı0 ."/ ! 0 as " ! 0/ of the cylinder x 2 C! 2 xP 2 D a02 such that all solutions of Eq. (5.70) starting from this neighborhood remain bounded together with their ﬁrst derivatives for all t 0. The ﬁrst and improved ﬁrst approximations and equations of the ﬁrst approximation for the equations with slowly varying coefﬁcients subjected to the pulse action d .m.s/x/ P C c.s/x D "f .s; x; x/; P dt .m.s/x/j P tDiT D "I.x; x/; P

t ¤ iT;

where m.s/ and c.s/ are positive functions of the “slow” time s D "t , are constructed in [42].

5.2

Systems with Fixed Times of the Pulse Action

We now consider a system of differential equations with pulse action of the form dx D "X.t; x/ dt xj tDi D "Ii .x/:

for t ¤ i ;

(5.71)

The following assumptions are made for the functions X.t; x/ and Ii .x/ and the times i :

195

Section 5.2 Systems with Fixed Times of the Pulse Action

(a) X.t; x/ and Ii .x/ are continuous functions of their arguments with continuous derivatives with respect to x up to the second order inclusively in a certain domain t 2 R, x 2 D Rn ; (b) the times i at which the state of the system instantaneously changes are enumerated by integer numbers so that i ! 1 as i ! 1 and i ! C1 as i ! C1. Moreover, one can ﬁnd numbers L and d such that any time interval of length L contains at most d points of the sequence ¹i º; (c) the ﬁnite limits 1 lim T !1 T

Z

t CT t

X.s; x/ds D X0 .x/;

1 T !1 T lim

X t
Ii .x/ D I 0 .x/; (5.72)

exist uniformly in t 2 R and x 2 D. Parallel with system (5.71), we consider an averaged system dx D "ŒX0 .x/ C I 0 .x/ "F .x/ dt

(5.73)

and assume that it has an isolated equilibrium position x D x0 , F .x0 / D 0 lying in the domain D together with its certain -neighborhood and that the real parts of the eigenvalues of the matrix ˇ @F ˇˇ AD @x ˇ xDx0

are not equal to zero. We now prove the theorem on the behavior and properties of solutions of the system of equations (5.71) in a neighborhood of the equilibrium position x D x0 of the averaged system of equations (an analog of the famous Bogolyubov theorem). For the sake of convenience, in what follows, we transform the system of equations (5.71) to a certain special form. To this end, we need the following lemma: Lemma 4 ([140]). Let the functions X.t; x/ and Ii .x/ and times i satisfy conditions (a)–(c). If the quantities X0 .x/ and I 0 .x/ are deﬁned by relations (5.72), then, for > 0, there exists a continuous function . /, . / ! 0 as ! 0, such that if Z f .t; x/ D

t 1

e .ts/ ŒX.s; x/ X0 .x/ I 0 .x/ds C

X

Ii .x/e .ti / ;

i
then kf .t; x/k 1 . / for all t 2 R and x 2 D .

(5.74)

196

Chapter 5 Method of Averaging in Systems with Pulse Action

Furthermore,

@f @t X.t; x/ C F .x/ . /

(5.75)

for all t 2 R, t ¤ i , and x 2 D . Proof. It follows from conditions (a)–(c) that there exists a continuous decreasing function ".T / such that ".T / ! 0 as T ! 1 and Z tCT X 1 1 0 ŒX.s; x/ X0 .x/ds C ŒIi .x/ I .x/ ".T / T T t t
for all t 2 R and x 2 D . Moreover, the function f .t; x/ satisﬁes the inequality kf .t; x/k 2T .1 e T /1 ".T / M T d C TM 1 C C ".T / C 1 ; L 1 e T where M

sup t 2R; x2D

ŒkX.t; x/ X0 .x/k C kIi .x/ I 0 .x/k:

We choose T , as a function of , so that 1 e T D ".T /. Since ".T / ! 0 as T ! 1, the solution T of this equation satisﬁes the condition T ! 0 as ! 0 and is continuous in . We set T d T 2 C M 1 C C ".T / C M 1 D . /: L 1 e T Thus, . / ! 0 as ! 0, and the ﬁrst part of the lemma is proved. Estimate (5.75) follows from inequality (5.74) in view of the fact that f .t; x/, for t ¤ i ; is a function differentiable with respect to t. The lemma is proved. In the system of equations (5.71), we now set x D x0 C b, where x0 is the equilibrium position of the averaged system and b are new variables. This enables us to rewrite the system in the form db D "Ab C "B.t; b/ dt bj tDi D "Ii .x0 / C b;

for t ¤ i ;

(5.76)

where B.t; b/ D X.t; x0 C b/ Ab: In the system of equations (5.76), we pass from the variable b to the variable y as follows: b D y C "v.t; y; "/; (5.77)

197

Section 5.2 Systems with Fixed Times of the Pulse Action

where the function v.t; y; "/ coincides with the function f .t; y/ if, in the last function, we choose D ."/ as a function of " such that ."/ ! 0 as " ! 0 and " 1 ."/ ! const as " ! 0. In view of conditions (a)–(c) and the properties of the function f .t; y/ established by Lemma 4, by the change of variables (5.77), we pass from system (5.76) to a system of the form dy D "Ay C "Q.t; y; "/; dt yj tDi D "Ii0 .y; "/:

t ¤ i ;

(5.78)

Replacing y with "y in system (5.78), we ﬁnally get the following system of equations: dy D "Ay C P .t; y; "/; dt yj tDi D Ii .y; "/;

t ¤ i ;

(5.79)

where P .t; y; "/ D Q.t; "y; "/;

Ii .y; "/ D Ii0 ."y; "/:

It is easy to see that the system of equations (5.79) satisﬁes the following conditions: (1) the functions P .t; x; "/ and Ii .y; "/ are deﬁned in the domain ° ± 1 < ; 0 < " "0 I t 2 R; y 2 D1 D y 2 Rn W kyk " " (2) the following inequalities hold for t 2 R, 0 < " "0 , and i 2 Z: kP .t; 0; "/k C kIi .0; "/k M."/; where M."/ ! 0 as " ! 0; (3) for any positive < 1 and all i 2 Z, the inequality kP .t; y 0 ; "/ P .t; y 00 ; "/k C kIi .y 0 ; "/ Ii .y 00 ; "/k ."; /ky 0 ; y 00 k; where "1 ."; / ! 0 as " ! 0 and ! 0, holds in the domain t 2 R, y 0 ; y 00 2 D , 0 < " "0 ; (4) the real parts of the eigenvalues of the matrix A are not equal to zero. Note that if the original system of equations (5.71) is periodic in t with period !, then the system of equations (5.79) is also periodic with the same period. Indeed, the periodicity of system (5.71) in t with period ! means that X.t C !; x/ D X.t; x/ and there exists a natural number p such that iCp i D !;

IiCp .x/ D Ii .x/

198

Chapter 5 Method of Averaging in Systems with Pulse Action

for all i 2 Z. To prove the periodicity of the system of equations (5.79), it sufﬁces to establish the periodicity of the function v.t; y; "/ in t with period !. This follows from the deﬁnition of the function v.t; y; "/ in view of the identity X X X Ii .x/e " .t C!i / D IiCp .x/e " .ti / D Ii .x/e " .ti / : i
i
i
The system of equations (5.71) is called almost periodic in t uniformly with respect to x 2 D if X.t; x/ is continuous in t 2 R and x 2 D except, possibly, the points t D i at which it has discontinuities of the ﬁrst kind and, for any ı > 0, one can ﬁnd an integer p and a positive number l D l.ı/ for which any interval of length l contains a point s such that kX.t C s; x/ X.t; x/k < ı; kIiCp .x/ Ii .x/k < ı;

jiCp i sj < ı;

(5.80)

for all t 2 R and x 2 D. If the system of equations (5.71) is almost periodic, then we can seek its solutions almost periodic in a generalized sense, namely: a solution of the system of equations (5.71) is called almost periodic if, for any ı > 0, one can ﬁnd l D l.ı/ > 0 for which any time interval of length l contains a point s such that kx.t C s/ x.t /k < ı for all t 2 R, jt i j > ı, and i 2 Z. We now show that if system (5.71) is almost periodic in t , then the function v.t;y;"/ is almost periodic in the above-mentioned sense. Indeed, let the system of equations (5.71) be almost periodic. We ﬁx ı > 0 and choose l; s, and p such that inequalities (5.80) hold. Then Z t v.t C s; x"/ v.t; x; "/ D e " .t/ ŒX. C s; x/ X.; x/d 1

C

X

Ii .x/e " .tCsi /

i
X

Ii .x/e " .t i / :

i
Further, we have kv.t C s; x; "/ v.t; x; "/k

X ı C M je " .i CsiC1 / 1j e " .t i / " i
for all jt i j > ı, i 2 Z, where 1

s D e " L.1 d /

1 X mD1

e "

Lm d

:

199

Section 5.2 Systems with Fixed Times of the Pulse Action

This means that v.t; x; "/ is an almost periodic function for any ﬁxed 0 < " "0 . Hence, the transformed system of equations (5.79) is also almost periodic for any ﬁxed ", 0 < " "0 . Lemma 5 ([140]). Assume that the system of equations dx D "Ax C P .t /; dt

t ¤ i ;

j t Di D Ii ;

(5.81)

satisﬁes the following conditions: (1) the real parts of the eigenvalues of the matrix A are not equal to zero; (2) the function P .t / is piecewise continuous with points of discontinuity of the ﬁrst kind at t D i and the functions Ii are bounded for all t 2 R and i 2 Z; (3) the sequence ¹i º1 iD1 satisﬁes condition (b). Then (1) the system of equations (5.81) possesses a unique solution x .t / bounded in the entire axis; (2) there exists a positive constant c D c."; L; d / such that ° ± kx .t /k c max sup kP .t /k; sup kIi k t 2R

(5.82)

i2Z

for any ﬁxed 0 < " "0 ; (3) if the system of equations (5.81) is periodic in t with period T or almost periodic in t , then the solution x .t / is also periodic with the same period or almost periodic, respectively. Proof. Without loss of generality, we can assume that A D diag.AC ; A /, where AC is a matrix whose eigenvalues have positive real parts and A is a matrix whose eigenvalues have negative real parts. We deﬁne a matrix JA .t / by the formula ´ diag .e "AC t ; 0/; t > 0; JA .t / D t < 0: diag .0; e "A t /; Since the real parts of the eigenvalues of the matrix A are not equal to zero, there exist positive constants K and ˛ independent of " such that kJA .t /k Ke "˛jtj

(5.83)

for all t 2 R. By using the matrix JA .t /, we now deﬁne the function x .t / as follows: Z 1 X x .t / D JA ./P .t C /d C JA .i t /Ii : (5.84) 1

1
200

Chapter 5 Method of Averaging in Systems with Pulse Action

The right-hand side of the last equality is deﬁned because, according to inequality (5.83), both the integral and the sum are uniformly convergent. Indeed, the right-hand side of equality (5.84) can be estimated as follows: 1

X 1 ˛Lm" 2K sup kP .t /k C 2Ke "˛L.1 d / e d sup kIi k "˛ t2R i 2Z mD0 ± ° c."/ max sup kP .t /k; sup kIi k :

kx .t /k

t 2R

(5.85)

i2Z

By direct veriﬁcation, we show that the function x .t / is a bounded solution of the system of equations (5.81). The uniqueness of this solution follows from the fact that the values of pulses for any t D i are constant for all solutions and, hence, the difference of two bounded solutions of the system of equations (5.81) is a bounded solution of the system of equations dx D Ax. However, under the assumptions made dt above, this system possesses solely the trivial bounded solution. Estimate (5.82) follows from (5.85) if we set

1 1 "˛L.1 1 / X ˛Lm" d ;e c."/ D 4K max e d : "˛ mD0

If the system of equations (5.81) is periodic in t with period !, then representation (5.84) readily implies that the function x .t / is also periodic with period !. Now let the system of equations (5.81) be almost periodic. It is necessary show that, in this case, the function x .t / is also almost periodic. Let ı be an arbitrarily small positive number and let constants s > 0 and l D l.ı/ > 0 and an integer number p be such that kP .t C s/ P .t /k < ı;

kIiCp Ii k < ı;

jiCp i sj < ı

for all t 2 R, jt i j > ı, and i 2 Z. We estimate the difference x .t C s/ x .t / for jt i j > ı. Indeed, X X 2K ıC J . t s/I J . t /I kx .t C s/ x .t /k i i A i A i "˛ i2Z i2Z X 2K ıC D ŒJA .iCp i s/ JA .i s/Ii Cp "˛ i 2Z X C JA .i t /ŒIiCp Ii i 2Z

"kAkı 1 X 1 ˛Lm" e 1 2K ı C 2Ke "˛L.1 d / C 1 < ı: e d "˛ ı mD0

201

Section 5.2 Systems with Fixed Times of the Pulse Action

This means that x .t / is an almost periodic function for any ﬁxed 0 < " "0 . We now prove the existence of a bounded solution of the system of equations (5.78). To this end, we construct a sequence of functions xn .t / each of which is a bounded solution of the system of equations dx D "Ax C P .t; xn1 .t /; "/; dt xj tDi D Ii .xn1 .i /; "/;

t ¤ i ;

(5.86)

where the matrix A, the functions P .t; x; "/ and Ii .x; "/, and the sequence ¹i º satisfy conditions (1)–(4). As the initial function, we take x0 D 0. According to Lemma 5, for any n D 0; 1; 2; : : : , a bounded solution of the system of equations (5.86) exists and is given by the formula Z 1 xn .t / D JA . t /P .; xn1 ./; "/d 1

C

X

JA .i t /Ii .xn1 .i /; "/:

(5.87)

1
Moreover, for any ﬁxed 0 < " "0 we have

0<

and

1 < ; " "

2K C 2Ks M."/; "˛ 2 2K kx2 .t / x1 .t /k ."; / C 2Ks M."/; "˛

kx1 .t /k

where 1

s D e "˛L.1 d /

1 X

e

˛Lm" d

:

mD0

By induction, we can readily show that kxkC1 .t /k kxkC1 .t / xkC1 .t /k for all k D 0; 1; 2; : : : and D

M."/ ; 1 ."; /

(5.88)

. ."; //k M."/ 1 ."; /

(5.89)

2K C 2Ks: "˛

202

Chapter 5 Method of Averaging in Systems with Pulse Action

Indeed, for k D 0, estimates (5.88) and (5.89) are proved. Under the assumption that they are true for k D 1; n 1, we ﬁnd kxn .t /k kxn .t / x1 .t /k C kx1 .t /k Z 1 Ke "˛jsj ."; /kxn1 .t C s/kd 1

C

X

Ke

˛"ji t j

."; /kxn1 .i /k C

1

2K C 2Ks M."/ "˛

2 M."/ M."/

."; / C M."/ D

."; /; 1 ."; / 1 ."; /

i.e., inequality (5.88) is proved for all k D 1; 2; 3; : : : . Estimate (5.89) is checked similarly. We now choose 0 < " "0 and 0 < 0 < 1 sufﬁciently small such that the inequalities M."/ < ; ."; / < 1 (5.90) 1 ."; / " hold for all 0 < " " and 0 < 0 . This choice of " and 0 is always possible because 1

."; / ! 0 and M."/ ! 0 as " ! 0 and ! 0: " Thus, each function xk .t / takes values from the domain kxk " for ﬁxed " 2 .0; "0 / and, hence, the iterative process can be continued. For any ﬁxed "; 0 < " "0 ; in view of inequalities (5.90), relations (5.89) imply that the sequence xk .t / is uniformly convergent (in t 2 R/. Moreover, the limit function x .t / D lim xk .t / k!1

takes values from the domain

0 " and, in addition, as follows from (5.88), satisﬁes the inequality kxk

kx .t; "/k

D."/ ; "

t 2 R; D."/ ! 0; " ! 0; D."/ < 0 :

Passing to the limit as n ! 1 in (5.87), we conclude that, for i < t < i C1 , the limit function x .t / satisﬁes the equation dx D "Ax C P .t; x; "/; dt

t ¤ i :

203

Section 5.2 Systems with Fixed Times of the Pulse Action

At the same time, for t D i , this function satisﬁes the condition xj tDi D Ii0 .x .i /; "/: Therefore, x .t / is a solution of the system of equations (5.78). The function x .t /, as the uniform limit of periodic (almost periodic) functions is also periodic (almost periodic) for any ﬁxed 0 < " " if this is true for system (5.78). The properties of stability of the solution x .t / are established by analogy with the same properties for systems of equations without pulse action [27, 88]. As a result, we conclude that, in the case where the real parts of the eigenvalues of the matrix A are negative, the solution is asymptotically stable. At the same time, if r, 0 r n, eigenvalues have negative real parts and the other n r eigenvalues have positive real parts, then, in the neighborhood of the point x .t0 /, there exists an r-dimensional point manifold M t0 for which the inclusion x .t0 / 2 M t0 implies that the difference kx.t / x .t /k exponentially vanishes. Thus, we arrive at the following analog of the famous Bogolyubov theorem on substantiation of the method of averaging for an inﬁnite time interval. Theorem 6 ([140]). Assume that the system of differential equations with pulse action (5.71) satisﬁes conditions (a)–(c). Suppose that the system of averaged Eqs. (5.73) possesses a quasistatic equilibrium position x D x0 for which the real parts of all roots of the characteristic equation det . E Fx0 .x0 // D 0 differ from zero. Then one can ﬁnd positive constants "0 , 1 , D."/, D."/ 1 < 1 , D."/ ! 0, " ! 0, such that the following assertions are true for any positive " < "0 : 1. The system of equations (5.71) possesses a unique solution x D x .t / deﬁned for all t 2 R and satisfying the inequality kx .t / x0 k < 0 : 2. If the system of equations (5.71) is periodic (almost periodic) in t , then the solution x .t / is also periodic (almost periodic). 3. Let x.t / be an arbitrary solution of the system of equations (5.71) satisfying the inequality kx.t0 / x0 k < 0 for some t D t0 . In this case, if the real parts of all roots of the characteristic Eq. (5.73) are negative, then kx.t / x .t /k < ce ˛".tt0 / ;

t t0 ;

(5.91)

where c and ˛ are positive constants. At the same time, if the real parts of all roots of the characteristic Eq. (5.73) are positive, then there exists t 0 > t0 such that (5.92) kx.t 0 / x0 k > 1 :

204

Chapter 5 Method of Averaging in Systems with Pulse Action

Finally, if the real parts of r roots are negative and the real parts of the remaining n r toots are positive, then the 0 -neighborhood of the point x0 contains an r-dimensional manifold M t0 such that the inclusion x.t0 / 2 M t0 yields inequality (5.91), whereas the relation x.t0 / … M t0 yields inequality (5.92).

5.3

Systems with Nonﬁxed Times of the Pulse Action

The problem of qualitative correspondence between the exact solutions of Eqs. (5.71) and the solutions of the corresponding averaged system (5.73) in an inﬁnite time interval is studied in [137, 138, 140]. We now discuss the results obtained in the cited works. Theorem 7 ([137]). Assume that the right-hand side of the system dx D "X.t; x/; t ¤ i .x/; dt xj tDi .x/ D "Ii .x/; i D 1; 2; : : : ; i .x/ < iC1 .x/;

(5.93)

satisfy the following conditions: (1) there exist positive constants M and K such that @i .x/ @x C kX.t; x/k C kIi .x/k M; ˇ ˇ ˇ @i .x 0 / @i .x 00 / ˇ ˇ CkIi .x 0 /Ii .x 00 /k Kkx 0 x 00 k; kX.t; x 0 /X.t; x 00 /kC ˇˇ @xj @xj ˇ @i .x/ @x K for t 2 .0; 1/, x, x 0 , x 00 2 D, j D 1; n, i D 1; 2; : : : ; (2) the ﬁnite limits Z 1 t CT X.s; x/ds D X0 .x/; T !1 T t P t
exist uniformly in t and x for t 0 and x 2 D.

(5.94)

205

Section 5.3 Systems with Nonﬁxed Times of the Pulse Action

(3) the averaged system dx D "ŒX0 .x/ C I0 .x/ (5.95) dt possesses a solution x D x."t; x0 /, x.0; x0 / D x0 ; for " D 1, t 2 Œ0; L, L < 1, this solution belongs to D together with its certain -neighborhood and satisﬁes the inequalities @i .x."t; x0 // Ii .x."t; x0 // ˇ < 0; @x or

ti0 < t < ti00 ;

(5.96)

@i .x/ 0; @x

where ti0 D infx2D i .x/, ti00 D supx2D i .x/, i D 1; d , and d <

L "

< d C1 .

Then, for any > 0, there exists "0 > 0 such that, for all " < "0 , the system of equations (5.93) possesses a solution x t .x0 /, x0 .x0 / D x0 deﬁned for t 2 Œ0; L" and such that L kx t .x0 // x."t; x0 /k for t 2 0; : " Theorem 7 is an analog of the well-known Bogolyubov result concerning the substantiation of the method of averaging. Proof. It follows from relations (5.94) that there exists a function '.t / vanishing as t ! 1 and such that Z t CT '.T / ŒX.t; x/ X0 .x/dt (5.97) 2 T; t X '.T / Ii .x/ I0 .x/T 2 T: t
We choose a sufﬁciently large but ﬁxed T and consider system (5.93) for t 2 Œ0; T . Assume that exactly d1 points belong to the half interval Œ0; T /: 1 .x0 / D t10 ; : : : ; d1 .x0 / D td01 0 , t 0 < T , i D 1; d 1. and, in addition, 0 < t10 , ti0 < tiC1 1 d1 By x.t; ; c/, x.; ; c/ D c, we denote a solution of the system Z t X.t; x.t; ; c//dt: x.t; ; c/ D c C "

It is clear that

Z x.t; ; c/ D c C "

t

X.t; c/dt C R.t; "; T /;

(5.98)

206 where

Chapter 5 Method of Averaging in Systems with Pulse Action

Z t 2 2 kR.t; "; T /k D " ŒX.t; x.t; ; c// X.t; c/dt " KM T

for 0 < t T . The solution x t .x0 / of system (5.93) is formed by pieces of functions (5.98). Thus, to within the quantities of order "2 , x t .x0 / is given by the formula Z t X.t; c/dt .0 t T /: x1 .t; ; c/ D c C "

In the construction of x t .x0 /, we restrict ourselves to the interval 0 < t < T . This enables us to establish an approximate expression for x t .x0 /. If the subsequent relations are understood as true to within the quantities of order "2 , then we can write x t .x0 / D x1 .t; 0; x0 /

for 0 t t1 ;

where t1 is the solution of the equation Z t X.t; x0 /dt t D 1 .x1 .t; 0; x0 // 1 x0 C " 0

@1 .x0 / 1 .x0 / C " @x

Z

t

X.t; x0 /ds:

0

It follows from Eq. (5.99) that t1 D

t10

@t 0 C" 1 @x

Z

t10

X.t; x0 /dt D t10 C " 1 :

0

Thus, x t .x0 / D x1 .t; 0; x0 /

for 0 < t < t10 C " 1 D t1 :

Further, we have x tC1 .x0 /

Z D x1 .t1 ; 0; x0 / C

"I10

D x0 C "

t1 0

X.t; x0 /dt C "I10

and, hence, Z x t .x0 / D x0 C " Z D x0 C "

t1 0 t 0

Z X.t; x0 /dt C

"I10

C"

t t1

X.t; x0 /dt

X.t; x0 /dt C "I10 D x1 .t; 0; x0 / C "I10

(5.99)

207

Section 5.3 Systems with Nonﬁxed Times of the Pulse Action

for t1 < t < t10 , where t10 is a solution either of the equation t D 1 .x1 .t; 0; x0 / C "I10 /

(5.100)

t D 2 .x1 .t; 0; x0 / C "I10 /

(5.101)

or of the equation if (5.100) has no solutions for t1 < t < t20 . As a result of the solution of Eq. (5.100), we conclude that its root t10 is given by the formula t10

D

t10

@t 0 C" 1 @x

Z

t10 0

X.t; x0 /dt C

I10

D t1 C "W1

and, therefore, t10 D t10 provided that W1 D

@t10 0 I ˛ > 0: @x 1

(5.102)

Thus, if inequality (5.102) is true, then Z x t .x0 / D x0 C "

t10 0

X.t; x0 /dt C I10

for t1 < t < t1 C "W1 D t10 . It is clear that Z x t01 C"W1 .x0 / D x0 C "

t10 0

X.t; x0 /dt C 2I10 ;

whence it follows that Z x t .x0 / D x0 C "

t10 0

X.t; x0 /dt C

D x1 .t; 0; x0 / C 2"I10

2I10

Z C"

t t10

X.t; x0 /dt

for t10 < t < t100 ;

where t100 is a solution either of the equation t D 1 .x1 .t; 0; x0 / C 2"I10 /

(5.103)

or of the equation t D 2 .x1 .t; 0; x0 / C 2"I10 / if (5.103) has no solutions for t10 < t < t20 . However, the root t1 of Eq. (5.103) is equal to t100

D

t10

@t 0 C" 1 @x

Z

t10 0

X.t; x0 /dt C

2I10

D t1 C 2"W1 D t10 C "W1 :

208

Chapter 5 Method of Averaging in Systems with Pulse Action

Therefore, t100 D t100 and, hence, x t .x0 / D x1 .t; 0; x0 / C 2"I10 for t1 C "W1 < t < t1 C 2"W1 . Similarly, we conclude that, for any ﬁxed k, the function x t .x0 / undergoes exactly k 1 instant changes in the time interval from t1 to t1 C "kW1 and takes the values x1 .x0 / D x0 C ".ˆ.t10 ; x0 / C k1 I0 / for t1 C .k1 1/"W1 < t < t1 C k1 "W1 ;

k1 D 1; k:

Thus, the system is characterized by the presence of beating upon the surface t D 1 .x/ in the vicinity of the point t10 ; x0 . Hence, to cross the surface t D 1 .x/, i.e., pass through the beating, the curve x t .x0 / must undergo about 1" instant changes for a ﬁnite period of time. In this case, the solution x t .x0 / naturally “moves” along the surface t D 1 .x/ and strongly differs from the solution of the averaged system. Therefore, it remains to assume that the point t10 ; x0 is not a point of beating upon the surface t D 1 .x/. Hence, one must set W1 D

@t10 0 I ˇ < 0 or @x 1

@1 .x/ 0: @x

(5.104)

In this case, the root t10 becomes smaller than t1 and, thus, t10 should be found from Eq. (5.101). By using this equation, we ﬁnd t10 D t20 C "

@t20 @x

Z

t20 0

X.t; x0 /dt C I10 D t20 C " 2 :

Since t20 > t10 , we have t20 C " 2 > t10 and x t .x0 / D x1 .t; 0; x0 / C "I10

for t1 < t < t20 C " 2 D t2 :

Further, x tC2 .x0 / D x1 .t2 ; 0; x0 / C ".I10 C I20 / D x1 .t20 ; 0; x0 / C ".I10 C I20 / and, hence, Z x t .x0 / D x1 .t2 ; 0; x0 / C

".I10

C

I20 /

D x1 .t; 0; x0 / C ".I10 C I20 /

C"

t t2

X.t; x0 /dt

for t2 < t < t3 :

209

Section 5.3 Systems with Nonﬁxed Times of the Pulse Action

To avoid beating upon the surface t D t2 .x/, we assume that @t20 0 I ˇ<0 @x 2

@2 .x/ 0: @x

or

Thus, the quantity t3 is determined from the equation t D 2 .x1 .t; 0; x0 / C ".I10 C I20 // Z t0 3 @t30 0 0 0 X.t; x0 /dt C I1 C I2 t30 C " 3 ; t3 C " @x 0 etc. If we now assume that @ti0 0 I ˇ < 0 or @x 2

@i .x/ 0 @x

(5.105)

for i D 1; d1 , then we get the following relation for x t .x0 /: x t .x0 / D x1 .t; 0; x0 / C "

k X

Ii0

0 for tk0 C " k < t < tkC1 C " kC1 ;

(5.106)

iD0

where k D

@tk0

Z

@x

tk0 0

X.t; x0 /dt C

k1 X i D0

Ik0 ;

(5.107)

t0 D 0 D d1 C1 D I00 D 0;

k D 1; d1 ;

td01 C1 D T:

Thus, if conditions (5.105) are satisﬁed, then x t .x0 / exists for t 2 Œ0; T and is determined, to within the quantities of order "2 , by relations (5.106) and (5.107). This enables us to write X xT .x0 / D x1 .T; 0; x0 / C " Ii0 C "2 Z D x0 C "

0

0
X.t; x0 /dt C "

X

Ii0 C "2

0
Z

T

ŒX.t; x0 / X0 .x0 /dt D x0 C "ŒX0 .x0 / C I0 .x0 /T C " 0 X Ii0 I0 .x0 /T C "2 : (5.108) C" 0
210

Chapter 5 Method of Averaging in Systems with Pulse Action

We set A0 x0 D x0 C "T ŒX0 .x0 / C I0 .x0 /:

(5.109)

Thus, in view of (5.97), inequality (5.108) yields the following estimate: kxT .x0 / A0 x0 k "'.T /T C "2 M1 ; where M1 D M.T; d1 / is a constant depending on T and d1 . Let x."t; x0 / be a solution of the averaged system (5.95). We set Ax0 D x."T; x0 /: Thus, it is clear that Z kAx0 A0 x0 k "

T 0

kX0 .x/ C I0 .x/ X0 .x0 / I0 .x0 /kdt "2 KM T 2 (5.110)

and, hence, kxT .x0 / Ax0 k "'.T /T C "2 .M1 C KM T 2 /:

(5.111)

Inequalities (5.110) and (5.111) show that xT .x0 / belongs to the domain D together with its 1 -neighborhood, where 1 D "Œ'.T /T C ".M1 C KM T 2 /; and A0 x0 belongs to the same domain together with its 10 -neighborhood, where 10 D "2 RM T 2 : Assume that the half interval ŒT; 2T / contains exactly d2 points, i.e., T < d1 C1 .Ax0 /; : : : ; d1 Cd2 .Ax0 / < 2T: Hence, it follows from estimate (5.111) and the continuity of the functions i .x/ that the segment ŒT; 2T contains the following d2 points: .1/

.1/

T < d1 C1 .xT .x0 // D t1 ; : : : ; d1 Cd2 .xT .x0 // D td2 < 2T: Thus, condition (5.96), estimate (5.111), and the fact that the functions are continuous imply that @ti.1/ @t .1/ .1/ IiCd1 .xT .x0 // D i Ii ˇ1 < 0 @x @x for i D 1; d2 , ˇ1 D ˇ "2 .

@i @x

and Ii .x/

211

Section 5.3 Systems with Nonﬁxed Times of the Pulse Action

We extend the solution x t .x0 / of system (5.93) constructed for 0 t < T to the segment ŒT; 2T . This yields Z t x t .x0 / D x.t; T; xT / D xT .x0 / C " X.t; x.t; T; xT //dt Z D xT C "

T

t

T

X.t; xT /dt C "2 D x1 .t; T; xT / C "2

for T t < td1 C1 , where td1 C1 is the solution of the equation Z t 2 X.t; xT /dt C " t D d1 C1 xT C " T

Z @d1 C1 .xT / t C" X.t; xT /dt C "2 : @x T It follows from Eq. (5.112) that Z .1/ @t1.1/ t1 .1/ td1 C1 D t1 C " X.t; x1 /dt C "2 @x T .1/ t1

.1/

(5.112)

.1/

D t1 C " 1 C "2 : Hence, to within the quantities of order "2 , x t .x0 / D x1 .t; T; xT / Further,

for T t < t1.1/ C " 1.1/ :

x C.1/ .x0 / D x1 .t1 ; T; xT / C "I1 .1/

.1/

t1

and, hence, x t .x0 / D

x1 .t1.1/ ; T; xT / Z

D xT C " .1/

for t1

t

T

C

"I1.1/

Z C"

t .1/

t1

X.t; xT /dt

X.t; xT /dt C "I1.1/ D x1 .t; T; xT / C "I1.1/

.1/

< t < t2 , etc. In general, we can write x t .x0 / D x1 .t; T; xT / C "

k X

.1/

Ii

(5.113)

iD0 .1/ .1/ C " kC1 , where for tk.1/ C " k.1/ < t < tkC1 .1/ k

@t .1/ D k @x

Z

k D 1; d2 ;

tk.1/ T .1/

t0

X.t; xT /dt C

k X i D0

.1/

D 0

.1/

Ii.1/

;

.1/

D d2 C1 D I0

D 0;

(5.114) .1/

td2 C1 D 2T:

212

Chapter 5 Method of Averaging in Systems with Pulse Action

Thus, under conditions (5.110), x t .x0 / is determined for all t 2 ŒT; 2T , to within the quantities of order "2 , by relations (5.113) and (5.114). This enables us to conclude that Z 2T X X.t; xT /dt C " Ii.1/ C "2 x2T .x0 / D xT C " T

T
Z

D xT C "ŒX0 .xT / C I0 .xT /T C " C"

X .1/

T
.1/

ŒIi

2T

T

ŒX.t; xT / X0 .xT /dt

I0 .xT /T C "2 :

<2T

This yields kx2T .x0 / A0 xT k "'.T /T C "2 M.T; d2 /:

(5.115)

Further, in view of (5.109), (5.110), and (5.115), we get 2

kx2T .x0 / x.2"T; x0 /k D kx2T A x0 k kx2T A0 xT k C kA0 xT A0 .Ax0 /k C kA0 .Ax0 / A.Ax0 /k "'.T /T C "2 M.t; d2 / C .1 C "K T /kxT Ax0 k C "2 KM T 2 "Œ1 C .1 C "K T /Œ'.T /T C "M ;

(5.116)

where M D KM T 2 C maxiD1;2 M.T; di /. It follows from (5.116) that x2T .x0 / belongs to D together with its 2 -neighborhood, where 1 X .1 C "K T /i .'.T /T C "M /: 2 D " iD0

By using x2T .x0 /, we can construct the solution x t .x0 / for t 2 Œ2T; 3T . Thus, we obtain kx3T .x0 / A0 x2T k "Œ'.T /T C "M.T; d3 /; (5.117) where d3 is the number of points i .x.2"T; x0 // lying in Œ2T; 3T : 2T < d1 Cd2 C1 .x.2"T; x0 //; : : : ; d1 Cd2 Cd3 .x.2"T; x0 // < 3T: Inequality (5.117) implies that 3

kx3T .x0 / x.3"T; x0 /k D kx3T A x0 k 2

2

2

kx3T A0 x2T k C kA0 x2T A0 .A x0 /k C kA0 .A x0 / A.A x0 /k

Section 5.3 Systems with Nonﬁxed Times of the Pulse Action

213

2

"Œ'.T /T C "M.t; d3 / C .1 C "K T /kx2T A x0 k C "2 KM T 2 "

2 X .1 C "K T /i Œ'.T /T C "M ;

(5.118)

iD0

where M D KM T 2 C maxiD1;3 M.T; di /. It follows from inequality (5.118) that x3T .x0 / belongs to D together with its 3 neighborhood, where 3 D "

2 X .1 C "K T /i .'.T /T C "M /: iD0

The quantity x3T .x0 / can be used to construct the solution x t .x0 / for t 2 Œ3T; 4T , etc. In the kth stage, we construct the solution x t .x0 / for t 2 Œ.k1/T; kT , kT L" , and conclude that k

kxkT .x0 / x."kT; x0 /k D kxkT A x0 k "

k1 X

.1 C "K T /i Œ'.T /T C "M ;

iD0

where M D KM T 2 C maxiD1;k M.T; di /. According to conditions (5.94), we have di c < 1. Therefore, max M.T; di / M0 .T / < 1 i

and, hence, kxkT .x0 / x."kT; x0 /k "

k1 X

.1 C "K T /i Œ'.T /T C "M0 .T /:

iD0

The last inequality yields .1 C "K T /k 1 Œ'.T /T C "M0 .T / "K T L M0 .T / '.T / "T .1 C "K T / C" K KT M '.T / 0 .T / C" : .e kL C 0."// K KT

kxkT .x0 / x."kT; x0 /k "

We now ﬁx T and "0 from the conditions e kL

'.T / ; K 4

".e kL C 0."//

'.T / M0 .T / C 0."/ : KT K 4

(5.119)

214

Chapter 5 Method of Averaging in Systems with Pulse Action

Thus, it follows from (5.119) that kxkT .x0 / x."kT; x0 /k

2

(5.120)

L L for k D 0; 1; : : : ; Œ "T , where Œ "T is the integer part of the number In the half interval Œ.k 1/T; kT /, the difference

L "T

.

x."t; x0 / x.".k 1/T; x0 / satisﬁes the following estimate: Z kx."t; x0 / x.".k 1/T; x0 /k

kT .k1/T

ŒX0 .x/ C I0 .x/dt "M T:

(5.121)

At the same time, the difference x t .x0 / x.k1/T .x0 / satisﬁes the estimate Z kx t .x0 / x.k1/T .x0 /k "

kT

.k1/T

C"

kX.t; x.t; ; c//kdt X

kIi .x/k

.k1/t
"M T C "M dk "M.T C c/:

(5.122)

For

and "M.T C c/ < ; 4 4 inequalities (5.121) and (5.122), together with (5.120), yield the required estimate L kx t .x0 / x."t; x0 /k for t 2 0; : " "M T <

We now study the problem of qualitative correspondence between the exact solutions of system (5.93) and its approximations. Assume that the averaged system (5.95) possesses an isolated equilibrium position x D x 0 : X0 .x 0 / C I0 .x 0 / D 0: Thus, we want to establish the correspondence between the existence of this equilibrium position and certain properties of the exact system (5.93). To do this, we formulate several assertions. Theorem 8 ([137]). Let conditions (1) and (2) of the previous theorem be satisﬁed. If the equilibrium position x D x 0 of the averaged system is asymptotically stable and @i .x/ Ii .x/ ˇ < 0 or @x

@i .x/ 0 @x

(5.123)

215

Section 5.3 Systems with Nonﬁxed Times of the Pulse Action

for all i D 1; 2; : : : and all x from a certain 0 -neighborhood of the position x 0 , then there exist a -neighborhood D . 0 / of the point x 0 and a number "0 > 0 such that, for all " < "0 and x 2 D , the solutions x t .x/, x0 .x/ D x of system (5.93) are uniformly bounded for t 2 .0; 1/. Proof. Let x.t; x/, x.0; x/ D x be a solution of the averaged system for " D 1. Since x.t; x 0 / D x 0 is an asymptotically stable solution of the averaged system, the fact that the dependence on the initial conditions is continuous implies the existence of 0 such that kx.t; x/ x 0 k 0 for kx x 0 k 0 ; t 0: (5.124) In view of (5.123), it follows from inequality (5.124) that, for x 2 T0 D ¹xW kx x 0 k 0 º, the solutions x."t; x/ satisfy conditions (3) of Theorem 7 for all t > 0. By using this theorem, one can ﬁnd "0 D "0 .L; / such that, for all " < "0 and t 2 Œ0; L" / and some . 0 , 0 /, we have kx tC .x; / x."t; x/k

; 2

(5.125)

where x tC .x; / is a solution of system (5.93) that passes through the point x 2 T for t D . We choose such that T belongs to the domain of asymptotic stability of the solution x D x 0 and L such that kx.t; x 2 T / x 0 k for t L: (5.126) 2 In this case, inequalities (5.124)–(5.126) yield the estimates kx t C .x 2 T ; / x 0 k kx t C .x 2 T ; / x."t; x 2 T /k C kx."t; x 2 T / x 0 k L C 0 for t 2 0; ; 2 "

(5.127)

kx L .x 2 T ; 0/ x 0 k : "

The last of these estimates means that x L .x 2 T ; 0/ 2 T : "

(5.128)

By using (5.128) and the fact that x tC .x; 0/ D x t C .x .x; 0/; /; we get x tC L .x 2 T ; 0/ D x tC L "

"

L L 0 D x tC L x 2 T ; : x L .x 2 T ; 0/; " " " "

216

Chapter 5 Method of Averaging in Systems with Pulse Action

In view of (5.127), this yields kx tC L .x 2 T ; 0/ x k C 0 " 2 0

L ; for t 2 0; "

kx 2L .x 2 T ; 0/ x 0 k :

(5.129)

"

The last inequality in (5.129) means that x 2L .x 2 T ; 0/ 2 T "

and leads to the estimates C 0 ; 2 kx .kC1/L .x 2 T ; 0/ x 0 k kx t Ck L .x 2 T ; 0/ x 0 k "

(5.130)

"

for t 2 Œ0; L" /, k D 0; 1; 2; : : : . Inequalities (5.130) mean that kx t .x 2 T / x 0 k

C 0 2

for all t 2 Œ0; 1/, Theorem 8 guarantees the existence of solutions bounded for t 0. The following theorem solves the problem of solutions bounded for t 2 .1; 1/: Theorem 9 ([137]). Assume that system (5.93) satisﬁes conditions (i) and (ii) of Theorem 7 both for t > 0 and for t < 0. We set P t
(5.131)

averaged for t 0 possesses an asymptotically stable equilibrium position x D x 0 satisfying inequalities (5.123) for x from a certain 0 -neighborhood of this position. Assume that, in the -neighborhood D of the solution x 0 mentioned in Theorem 8, the system d x1 D "ŒX0 .x 1 / C I 0 .x 1 / (5.132) dt

Section 5.3 Systems with Nonﬁxed Times of the Pulse Action

217

averaged over t 0 possesses an equilibrium position x1 D x10 for which either @i .x/ Ii .x/ ˇ < 0 or @x

@i .x/ 0 @x

for all i D 1; 2; : : : and all x from a certain 00 -neighborhood of the position x10 . Under these conditions, the following assertions are true: 1. If the equilibrium position x10 of system (5.132) is asymptotically unstable (asymptotically stable for t < 0/, then there exist "0 > 0 and a domain D1 containing x 0 and x10 such that, for " < "0 , all solutions x t .x/ of system (5.93) for which x 2 D are uniformly bounded for t 2 .1; 1/. 2. If the equilibrium positions x10 of system (5.132) are asymptotically stable, then one can ﬁnd "0 > 0 and x such that the solution x t .x / of system (5.93) is uniformly bounded for t 2 .1; 1/. Proof. Assume that the equilibrium position x10 of system (5.132) is asymptotically unstable. Applying Theorem 8 to the intervals t 0 and t 0, we conclude that there exist - and -neighborhoods of the points x 0 and x10 such that kx t .x 2 T /k c1

for t 2 .0; 1/;

kx t .x 2 T /k c2 for t 2 .1; 0/: T Since x10 2 D , the set D T D D1 is nonempty. Therefore, kx t .x 2 D1 /k c D max.c1 ; c2 /

for t 2 .1; 1/:

Now let x10 be an asymptotically stable solution of system (5.132). Applying Theorem 7 to (5.93), we show that estimates of the form (5.127) are true for x t .x 2 T10 ; /, i.e., 0 kx tC .x 2 T10 ; / x10 k 1 C 10 2 L L for t 2 Œ0; " / and " and x0 x 2 T0 ; L x 0 0 ; (5.133) 1 1 1 " where T10 is a 10 -neighborhood of the point x10 . Inequalities (5.133) yield the following estimates: 10 L 0 x 0 ; k C 10 ; x 2 T x L 1 1 tk " " 2 x0 x 2 T0 ; k L x 0 0 1 1 1 " for t 2 .k L" ; 0, k D 1; 2; : : : .

218

Chapter 5 Method of Averaging in Systems with Pulse Action

In the set D2 D T10 \ D , we choose a convergent subsequence of points yk D x0

L xk 2 D2 ; k ; "

k D 1; 2; : : : ;

lim yk D y0

k!1

and consider the solution x t .y0 /. We show that x t .y0 / is deﬁned for t 2 .1; 0/ and bounded. Indeed, the sequence of solutions x t .yk / deﬁned for k L" t 0 is uniformly bounded for k D 1; 2; : : : because it coincides with the sequence xt k L .xk ; k L" / for t 2 Œk L" ; 0/. Assume that kx t .y0 /k > D for some t D t0 < 0, "

0

D > 21 C 10 C kx10 k. Since x t .y0 / is a piecewise continuous function of t in any interval, we have 0 kx t .y0 /k > D1 > 1 C 10 C kx10 k 2 0 for an interval .t0 ; t0 /, which is the interval of continuity for x t .y0 /. Since the points of discontinuity of the solution x t .x/ continuously depend on x in a ﬁnite time interval, we can choose 1 2 .t0 ; t00 / such that x1 .yk / ! x1 .y0 /

as k ! 1:

(5.134)

Relation (5.134) implies the inequality x1 .yk / >

10 C 10 C kx10 k 2

(5.135)

for all sufﬁciently large k. However, (5.135) contradicts (5.133) because L x1 .yk / D xk L xk ; k : " " This contradiction proves that x t .y0 / is a solution bounded on .1; 0/. Since y0 2 D , the solution x t .y0 / is also bounded for t 0 and, therefore, kx t .y0 /k c < 1; which completes the proof of Theorem 9. Theorems 8 and 9 essentially use the asymptotic stability of equilibrium positions. However, in the problem of qualitative correspondence between the exact and approximate solutions, parallel with stability, one can also use some other properties of the equilibrium positions. Thus, in view of the fact that, for systems periodic for t 0, the right-hand side of the averaged system (5.131) coincides, to within "."/, where ."/ ! 0 as " ! 0, with the operator of shift along the trajectories of system (5.93), we can formulate the following theorem:

Section 5.3 Systems with Nonﬁxed Times of the Pulse Action

219

Theorem 10 ([137]). Assume that conditions (1) of Theorem 7 are satisﬁed for system (5.93) T -periodic for t 0. If, in addition, the averaged system (5.131) possesses an isolated singular point x D x 0 : X0 .x 0 / C I.x 0 / D 0; lying, together with a certain -neighborhood, in the domain D, its index under the mapping x ! X0 .x/ C I.x/ is not equal to zero, and @i .x/ @i .x 0 / 0; i D 1; d ; Ii .x/ ˇ < 0 or @x 0 @x then there exists "0 > 0 such that, for all " < "0 , system (5.93) possesses a periodic solution x D x t0 , t > t10 , with period T for which lim"!0 x t0 D x 0 . Moreover, if the functions X0 .x/ and I.x/ are continuously differentiable in a neighborhood of the equilibrium position x D x 0 and the real parts of the eigenvalues of the matrix @ŒX0 .x 0 / C I.x 0 / @x are negative, then the solution x t0 is asymptotically stable. In [138], it is shown that if the averaged system (5.73) possesses an asymptotically orbitally stable periodic solution, then, for t 0, the solutions of Eqs. (5.71) starting from a sufﬁciently small vicinity of the trajectory of this solution are uniformly bounded. The sufﬁcient conditions of generation of the integral set of the original Eqs. (5.71) by an asymptotically orbitally stable periodic solution of the averaged system (5.73) are also established in the cited work. An analog of the second Bogolyubov’s theorem on substantiation of the method of averaging for systems with pulse action is proved in [140]. In this work, a system of equations in the standard form (5.71) is investigated under the assumption that the hypersurfaces t D i .x/ are hyperplanes t D i . Under the condition that the corresponding averaged system possesses an isolated equilibrium position x D x0 and the real parts of the eigenvalues of the matrix ˇ @.X0 .x/ C I0 .x// ˇˇ ˇ @x xDx0 are not equal to zero, it is shown that the unique solution of the original system bounded on the entire axis exists in the vicinity of the solution x D x0 . The properties of this solution and solutions starting in a sufﬁciently small vicinity of the bounded solution are also investigated. The presented results concerning the applicability of the method of averaging to systems of the form (5.71) were generalized to systems of differential equations subjected to pulse actions with “slow” and “fast” variables in [12] and to a class of functional-differential equations with pulses in [13].

Chapter 6

Averaging of Differential Inclusions

6.1

Averaging of Inclusions with Pulses at Fixed Times

In the present section, we consider the problem of substantiation of the method of complete and partial averaging on ﬁnite and inﬁnite intervals for differential inclusions subjected to pulse actions at ﬁxed times. Method of Averaging for a Finite Interval. Consider a differential inclusion with multivalued pulses xP 2 "X.t; x/;

t ¤ i ; x.0/ 2 X0 ;

(6.1)

xj tDi 2 Ii .x/: If, for any x 2 D, there exists a limit Y .x/ D lim

T !1

1 T

Z

tCT t

1 X.t; x/dt C T

X

Ii .x/ ;

(6.2)

t i
then inclusion (6.1) can be associated with the averaged inclusion yP 2 "Y .y/;

y.0/ 2 X0 :

(6.3)

Theorem 1. Assume that the following conditions are satisﬁed in the domain Q¹t 0; x 2 D Rn º: (1) the set-valued mappings X W Q ! conv.Rn / and Ii W D ! conv.Rn / are continuous, uniformly bounded by a constant M , and satisfy the Lipschitz condition with respect to x with constant : (2) limit (6.2) exists uniformly in t 0 and x 2 D and, in addition, 1 i.t; t C T / d < 1; T where i.t; t C T / is the number of points of the sequence i in the interval .t; t C T ; (3) for all x0 2 D 0 D and t 2 Œ0; L"1 , the solutions of inclusion (6.3) belong to the domain D together with a certain -neighborhood.

221

Section 6.1 Averaging of Inclusions with Pulses at Fixed Times

Then, for any > 0 and L > 0, there exists "0 . ; L/ > 0 such that the following assertions are true for " 2 .0; "0 and t 2 Œ0; L"1 : (1) for any solution y.t / of inclusion (6.3), there exists a solution x.t / of inclusion (6.1) such that kx.t / y.t /k < I (6.4) (2) for any solution x.t / of inclusion (6.1), there exits a solution y.t / of inclusion (6.3) such that inequality (6.4) is true. Proof. By conditions (1) and (2) of the theorem, the set-valued mapping Y W D ! conv.Rn / is bounded by a constant M1 D M.1 C d / and satisﬁes the Lipschitz condition with constant 1 D .1 C d /. Let y.t / be a solution of inclusion (6.3). We split the segment Œ0; L"1 into subsegments with steps ."/ such that ."/ ! 1 and "."/ ! 0 as " ! 0. Then there exists a measurable branch v.t / of the mapping Y .y.t // such that Z t v.s/ds; y.0/ D x0 ; t 2 Œtj ; tj C1 ; (6.5) y.t / D y.tj / C " tj

where tj D j ."/ and m."/ L"1 < .m C 1/."/, j D 0; m. Consider a function y 1 .t / D y 1 .tj / C "vj .t tj /; where the vectors vj are such that Z tj C1 ."/vj v.s/ds D tj

min

t 2 Œtj ; tj C1 ; y 1 .0/ D x0 ;

(6.6)

v.s/ds :

(6.7)

v2Y.y 1 .tj //

Z ."/v

tj C1 tj

The vector vj exists and is unique by virtue of compactness and convexity of the set Y .y 1 .tj // and strict convexity of the minimized function. Denote ıj D ky.tj / y 1 .tj /k: For t 2 Œtj ; tj C1 , by virtue of (6.5) and (6.6), we ﬁnd (6.8) ky.t / y.tj /k M1 "."/; ky 1 .t / y 1 .tj /k M1 "."/: Hence, for t 2 Œtj ; tj C1 , we get ky.t / y 1 .tj /k ky.tj / y 1 .tj /k C ky.t / y.tj /k ıj C "M1 .t tj /; h.Y .y.t //; Y .y 1 .tj // 1 ky.t / y 1 .tj /k 1 .ıj C "M1 .t tj //:

(6.9)

It follows from relations (6.7) and (6.9) that Z tj C1 Z tj C1 .v.s/ vj /ds h.Y .y.s//; Y .y 1 .tj ///ds tj

tj

2 ."/ 1 ıj ."/ C "M1 : 2

(6.10)

222

Chapter 6 Averaging of Differential Inclusions

In view of (6.5) and (6.6), we can write 2 ."/ "2 2 ."/ ıj C1 ıj C " 1 ıj ."/ C "M1 : D .1 C 1 "."//ıj C 1 M1 2 2 (6.11) Since ı0 D 0, inequality (6.11) implies that ı1 1 M1

"2 2 ."/ ; 2

ı2 .1 C 1 "."//ı1 C 1 M1 1 M1

"2 2 ."/ 2

"2 2 ."/ ..1 C 1 "."// C 1/ 2

etc., "2 2 ."/ ..1 C 1 "."//i C .1 C 1 "."//i1 C C 1/ 2 M1 "."/ ..1 C 1 "."//iC1 1/ D 2 L M1 "."/ ..1 C 1 "."// "."/ 1/ 2 M1 "."/ 1 L .e 1/: (6.12) 2

ıj C1 1 M1

Thus, by virtue of inequalities (6.8), we arrive at the following estimate: ky.t / y 1 .t /k ky.t / y.tj /k C ky.tj / y 1 .tj /k C ky 1 .tj / y 1 .t /k 2M1 "."/ C

M1 "."/ 1 L .e 1/ 2

M1 "."/ 1 L .e C 3/: 2

(6.13)

It follows from condition (2) of the theorem that, for any 1 > 0, there exists "1 . 1 / > 0 such that the following inequality is true for " "1 . 1 /: Z tj C1 X 1 1 1 1 1 h Y .y .tj //; X.s; y .tj //ds C Ii .y .tj // < 1 : ."/ tj ."/ t
i

j C1

(6.14) Hence, one can ﬁnd vectors uj .t / 2 X.t; y 1 .tj // and pij 2 Ii .y 1 .tj // such that Z tj C1 X vi 1 u .s/ds C p (6.15) j ij < 1 : ."/ tj t
i

j C1

Section 6.1 Averaging of Inclusions with Pulses at Fixed Times

Consider a family of functions Z t X x 1 .t / D x 1 .tj / C " uj .s/ds C " pij ; t 2 .tj ; tj C1 : tj

223

(6.16)

tj i
By using relations (6.6), (6.16), and (6.15), in view of the fact that x 1 .0/ D y 1 .0/, for j D 1; m, we obtain kx 1 .tj / y 1 .tj /k kx 1 .tj 1 / y 1 .tj 1 /k C 1 "."/ j 1 "."/ L 1 :

(6.17)

Note that, for t 2 .tj ; tj C1 , we have kx 1 .t / x 1 .tj /k M.1 C d /"."/ D M1 "."/: Thus, in view of inequality (6.8), we ﬁnd kx 1 .t / y 1 .t /k L 1 C 2M1 "."/; 1

(6.18)

1

kx .t / y .tj /k L 1 C M1 "."/: We now show that there exists a solution Z t X x.t / D x.tj / C " u. / d C " qi ; tj

x.0/ D x0 ;

tj i
t 2 .tj ; tj C1 ;

of inclusion (6.1) sufﬁciently close to x 1 .t /. Let 1 ; : : : ; p be the times of pulses i , from the half interval .tj ; tj C1 . For the sake of convenience, we denote 0 D tj and pC1 D tj C1 . Further, let C D kx 1 . k C 0/ x. k C 0/k k

and

1 k D kx . k / x. k /k;

k D 0; p:

In view of the Lipschitz condition, we conclude that .xP 1 .t /; "X.t; x 1 .t /// h."X.t; y 1 .tj //; "X.t; x 1 .t /// " kx 1 .t / y 1 .tj /k " .M1 "."/ C L 1 / D ; .x 1 j t D k ; "Ii .x 1 . k /// h."Ii .y 1 .tj //; "Ii .x 1 . k /// " ky 1 .tj / x 1 . k /k " .M1 "."/ C L 1 / D : According to the Filippov theorem, between the points of the pulses, there exists a solution x.t / of inclusion (6.1) such that the following inequality is true for t 2 . k ; kC1 : Z t C " .t k / 1 C" e " .t s/ ds: kx.t / x .t /k k e

k

224

Chapter 6 Averaging of Differential Inclusions

Denote k D kC1 k ."/, 0 C C p D ."/. Then C " k C kC1 k e

"."/ .e 1/:

(6.19)

In passing through the point of the pulse, we ﬁnd 1 C kC1 C "h.Ii .y .tj //; Ii .x. kC1 /// kC1 1 kC1 C "h.Ii .x . kC1 //; Ii .x. kC1 ///

C "h.Ii .y 1 .tj //; Ii .x 1 . kC1 /// 1 1 kC1 C " kC1 C "h.Ii .y .tj //; Ii .x . kC1 /// .1 C " / kC1 C :

Relations (6.19) and (6.20) imply that C .1 C " /e " k C C ˇ; kC1 k ˇD

.1 C " /.e "."/ 1/ C :

Therefore,

"0 C 0 C ˇ .1 C " /e "."/ C C 1 .1 C " /e 0 C ˇ; " 1 C C 1 C ˇ 2 .1 C " /e " 1 .1 C " /2 e " .0 C1 / C Cˇ 0 C ˇ.1 C " /e

"."/ .1 C " /2 e "."/ C C 1/; 0 C ˇ..1 C " /e

etc., C .1 C " /kC1 e " ."/ C 0 kC1 C ˇ.e "."/ ..1 C " /k C C .1 C " // C 1/ D .1 C " /kC1 e "."/ C 0 k

"."/ .1 C " / 1 .1 C " / C 1 Cˇ e "

.1Cd /"."/ C 1 C " "."/ e .e 0 C 1/ C 1

d "."/ 1

"."/ e e .1 C " / C 1 "

D ˛C 0 C ˇ1 ;

(6.20)

225

Section 6.1 Averaging of Inclusions with Pulses at Fixed Times

where ˛ D e "."/.1Cd / ;

1 C " "."/ .e ˇ1 D ."."/M1 C L 1 / 1/ C 1

.e "."/ .e d "."/ 1/.1 C " / C " /: Thus,

ıjCC1 D kx.tj C1 / x 1 .tj C1 /k ˛ıjC C ˇ1 :

As a result, we arrive at the chain of inequalities ı0C D 0;

ı1C ˇ1 ;

ı2C ˛ˇ1 C ˇ1 D .˛ C 1/ˇ1 ; : : : ;

ıjCC1 .˛ j C C 1/ˇ1 D

˛ j C1 1 ˇ1 ˛1

e L.1Cd / 1 .M1 "."/ C L 1 / e .1Cd /"."/ 1 1 C " "."/ .e 1/ C 1

"."/

d "."/ e 1 .1 C " / C " : e

Since lim "#0

1 C " "."/ .e 1/ C 1 D 1

and lim "#0

e "."/ .e d "."/ 1/.1 C " / C "

e .1Cd /"."/ 1

D lim

"!0

we have

e "."/ e

d "."/ 1

"."/

C

1 ."/

e .1Cd /"."/ 1

"."/

D

d ; 1Cd

ıjCC1 C.M1 "."/ C L 1 /

for " "2 . Hence, for t 2 .tj ; tj C1 , we get the inequality kx.t / x 1 .t /k kx.t / x.tj /k C kx.tj / x 1 .tj /k C kx 1 .t / x 1 .tj /k M.1 C d /"."/ C M1 "."/ C C.M1 "."/ C L 1 / D M1 .2 C C /"."/ C CL 1 :

(6.21)

226

Chapter 6 Averaging of Differential Inclusions

By virtue of inequalities (6.13), (6.18), and (6.21), we conclude that kx.t / y.t /k can be made smaller than any given by the proper choice of " "0 and 1 . The second assertion of the theorem is proved similarly. The following statement is obtained as a corollary of Theorem 1: Theorem 2 ([103]). Assume that the following conditions are satisﬁed in the domain Q¹t 0, x 2 D Rn º: (1) the set-valued mappings X W Q ! conv.Rn / and Ii W D ! conv.Rn / are continuous, uniformly bounded by a constant M , and satisfy the Lipschitz condition with constant with respect to x; (2) limit (6.2) exists uniformly in x and, in addition, 1 i.t; t C T / d < 1I T (3) the R-solutions of inclusion (6.3) belong to the domain D together with a certain -neighborhood for all x0 2 D 0 D and t 2 Œ0; L"1 . Then, for any > 0 and L > 0, there exists "0 . ; L/ > 0 such that the following assertions are true for " 2 .0; "0 and t 2 Œ0; L"1 : (1) for any R-solution Y .t / of inclusion (6.3), there exists an R-solution X.t / of inclusion (6.1) such that h.X.t /; Y .t // < I (6.22) (2) for any R-solution X.t / of inclusion (6.1), there exits an R-solution Y .t / of inclusion (6.3) such that inequality (6.22) is true. We now consider a scheme of partial averaging. Together with impulsive differential inclusion (6.1), we consider an impulsive differential inclusion e .t; y/; yP 2 "X

t ¤ j ; y.0/ 2 X0 ;

(6.23)

yj tDj 2 Kj .y/; where the limit 1 h T !1 T

Z

tCT

lim

Z

t

exists for any x 2 D.

X t i
tCT t

X.t; x/dt C e .t; x/dt C X

X

t j
Ii .x/; Kj .x/ D 0

(6.24)

Section 6.1 Averaging of Inclusions with Pulses at Fixed Times

227

Theorem 3 ([112]). Assume that the following conditions are satisﬁed in the domain Q¹t 0, x 2 D Rn º: e W Q ! conv.Rn / Ij ; Kj : D ! conv.Rn / are (1) the set-valued mappings X , X continuous, uniformly bounded by a constant M , and satisfy the Lipschitz condition with constant with respect to x; (2) limit (6.24) exists uniformly in x and, in addition, 1 i.t; t C T / d < 1; T

1 j.t; t C T / d < 1; T

where i.t; t C T / and j.t; t C T / are, respectively, the numbers of points of the sequences ¹tj º and ¹ j º in the segment Œt; t C T ; (3) the solutions of inclusion (6.23) belong to the domain D together with a certain -neighborhood for all x0 2 D 0 D and t 2 Œ0; L"1 . Then, for any > 0 and L 2 .0; L , there exists "0 . ; L/ 2 .0; such that the following inequality is true for " 2 .0; "0 and t 2 Œ0; L"1 : kx.t; "/ y.t; "/k < ;

(6.25)

where x.t; "/ and y.t; "/ are solutions of inclusions (6.1) and (6.23), respectively, and x.0; "/ D y.0; "/ D x0 2 X0 . Theorem 4 ([112]). Assume that conditions (1) and (2) of Theorem 3 are satisﬁed in the domain Q¹t 0, x 2 D Rn º and, in addition, that (3) for all X0 D 0 D and t 2 Œ0; L"1 , the R-solutions of inclusion (6.23) lie in the domain D together with a certain -neighborhood. Then, for any > 0 and L 2 .0; L , there exists "0 . ; L/ 2 .0; such that the following inequality is true for " 2 .0; "0 and t 2 Œ0; L"1 : h.X.t; "/; Y .t; "// < ;

(6.26)

where X.t; "/ and Y .t; "/ are R-solutions of inclusions (6.1) and (6.23), respectively, and X.0; "/ D Y .0; "/ D X0 . Method of Averaging in an Inﬁnite Interval. For an inclusion xP 2 "X.t; x/;

t ¤ i ; x.0/ 2 X0 ;

(6.27)

xj tDi 2 Ii .x/; we consider the averaged differential inclusion P 2 "X./;

.0/ D x0 ;

(6.28)

228

Chapter 6 Averaging of Differential Inclusions

where 1 X.x/ D lim T !1 T

Z

tCT t

X

X.t; x/dt C

t tj
j Ii .x/

:

(6.29)

Theorem 5 ([112]). Assume that the following conditions are satisﬁed in the domain Q¹t 0, x 2 D Rn º: (1) the mappings X.t; x/ and Ij .x/ are nonempty convex compact sets; moreover, they are continuous and uniformly bounded and satisfy the Lipschitz condition with respect to x with constant ; (2) limit (6.29) exists and, in addition, the inequality 1 i.t; t C T / d < 1 T is true uniformly in x 2 D and t 0; (3) for any x0 2 D 0 D and t 0, the R-solutions of inclusion (6.28) are uniformly asymptotically stable and belong to the domain D together with a -neighborhood. Then, for any > 0, there exists "0 . / > 0 such that the following inequality is true for " 2 .0; "0 and t 0: N // ; h.R.t; "/; R."t where R.t; "/ is the R-solution of inclusion (6.27), R."t / is the R-solution of inclusion N (6.28), and R.0; "/ D R.0/ D x0 . Theorem 6 ([112]). Assume that the following conditions are satisﬁed in the domain Q¹t 0, x 2 D Rn º: (1) conditions (1) and (2) of Theorem 5; N / whose tra(2) the averaged inclusion (6.28) possesses a periodic R-solution R. jectory C is asymptotically orbitally stable and lies in the domain D together with a certain ı-neighborhood. Then, for any 2 .0; ı; there exist "0 > 0 and 0 2 .0; such that the following inequality is true for " 2 .0; "0 , 1 2 .0; 0 ; and t 0: h.R.t /; C / < ; where R.t / is an R-solution of inclusion (6.27) satisfying the following initial condition: h.R.0/; C / < 1 .

Section 6.2 Krasnosel’skii–Krein Theorem for Differential Inclusions

229

Theorem 7 ([112]). Assume that the following conditions are satisﬁed in the domain Q: (1) conditions (1) and (2) of Theorem 5; (2) inclusion (6.28) has an asymptotically stable equilibrium position RN 0 lying in the set D together with a certain 0 -neighborhood. Then, for any 2 .0; 0 /; there exist "0 > 0 and 0 2 .0; such that the following inequality is true for " 2 .0; "0 , 1 2 .0; 0 ; and t 0: h.R.t /; RN 0 / < ; where R.t / is an R-solution of inclusion (6.27) satisfying the following initial condition: h.R.0/; RN 0 / < 1 . The proofs of Theorems 5–7 are similar to the proofs of the corresponding theorems for differential equations in [44, 45, 88, 89, 108]. However, the ﬁrst Bogolyubov’s theorem in these proofs should be replaced by Theorem 1.

6.2

Krasnosel’skii–Krein Theorem for Differential Inclusions

In the previous section, the Lipschitz condition for the original or averaged inclusions has been essentially used for the substantiation of the method of averaging for differential inclusions. For functional-differential equations satisfying the Lipschitz condition, the method of averaging was substantiated in [59]. In [40], the Lipschitz condition was replaced by a unilateral Lipschitz condition. For ordinary differential equations, similar results were obtained without using the Lipschitz condition in [70]. In what follows, we prove an analog of the Krasnosel’skii–Krein theorem for differential inclusions (with and without pulse action) in terms of ordinary solutions and R-solutions. Theorem 8 ([122]). Assume that the following conditions are satisﬁed for a differential inclusion xP 2 F .t; x; /; (6.30) where F .t; x; / is a set-valued mapping taking values in conv.Rn / and deﬁned for 0 t T and x 2 D, D is a bounded domain in Rn , 2 ƒ, and ƒ is a set of values of the parameter with limit point 0 2 ƒ: (a) the set-valued mapping F .t; x; / is uniformly bounded, continuous in t, uniformly continuous in x, and uniform in t and ;

230

Chapter 6 Averaging of Differential Inclusions

(b) the set-valued mapping F .t; x; / is integrally continuous in at the point 0 , i.e., the following condition is satisﬁed for any 0 t1 < t2 T and x 2 D: Z t2 Z t2 F .s; x; /ds; F .s; x; 0 /ds D 0I (6.31) lim h

! 0

t1

t1

(c) the solutions x.t; 0 / of the inclusion xP 2 F .t; x; 0 /

(6.32)

satisfying the initial condition x.0; 0 / D x0 2 D 1 D are deﬁned for 0 t T and lie in the domain D together with a certain -neighborhood. Then, for any > 0, there exists a neighborhood U. 0 / of the point 0 such that, for 2 U. 0 / and any solution x.t; / of inclusion (6.30) deﬁned for 0 t T and satisfying the initial condition x.0; / D x0 , there exists a solution x.t; 0 / of inclusion (6.32) satisfying the inequality kx.t; / x.t; 0 /k < ;

0 t T:

Proof. It follows from conditions (a) and (b) of the theorem and boundedness of the domain D that the convergence in (6.31) is uniform in t1 ; t2 , and x. Let x.t; / . ! 0 , 2 ƒ/ be a uniformly convergent sequence of solutions of (6.30) satisfying the initial condition x.0; / D x0 . Hence, there exists a continuous function y.t / such that lim

max kx.t; / y.t /k D 0:

! 0 t2Œ0;T

We now show that the following equality is true for any 0 < t T : Z t Z t lim F .s; x.s; /; /ds D F .s; y.s/; 0 /ds:

! 0

It is clear that Z t Z t h F .s; x.s; /; /ds; F .s; y.s/; 0 /ds I1 C I2 C I3 C I4 ;

where

Z I1 D h

t

F .s; x.s; /; /ds; Z

Z

Z

I2 D h

Z

t

t

F .s; y.s/; /ds;

I3 D h I4 D h

Z

t

F .s; y.s/; 0 /ds ;

Z

t

F .s; y.s/; 0 /ds;

t

F .s; y.s/; /ds;

F .s; y.s/; /ds ; Z

t

F .s; y.s/; /ds ;

t

F .s; y.s/; 0 /ds ;

(6.33)

Section 6.2 Krasnosel’skii–Krein Theorem for Differential Inclusions

231

and y.t / is a piecewise continuous function such that max ky.t / y.t /k < ı;

t 2Œ0;T

where ı is chosen from the condition of uniform continuity of the right-hand side to guarantee that the inequality h.F .s; x; /; F .s; y; // <

" 4T

holds for kx yk < ı. Each term is estimated separately. The neighborhood U. 0 / is chosen to guarantee the validity of the inequality kx.s; / y.s/k < ı for 2 U. 0 / and all s 2 Œ0; T . Then Z t " I1 h.F .s; x.s; /; /; F .s; y.s/; //ds ; 4 Z t " I2 h.F .s; y.s/; /; F .s; y.s/; //ds < ; 4 Z t " h.F .s; y.s/; 0 /; F .s; y.s/; 0 //ds < : I4 4 By using condition (b) of the theorem, we make the neighborhood U. 0 / somewhat smaller, such that the inequality Z t Z t " I3 D h F .s; y.s/; /ds; F .s; y.s/; 0 /ds < 4 holds for 2 U. 0 /. Thus, we have proved the limit equality (6.33). The set of ordinary solutions of the differential inclusion (6.30) coincides with the set of generalized solutions [34] deﬁned as the set of continuous functions x.t; / satisfying the inclusion Z t F .s; x.s; /; /ds x.t; / x.; / 2

for all t and . Hence, passing to the limit as ! 0 , we get Z t F .s; y.s/; 0 /ds; y.t / y. / 2

i.e., y.t / is a generalized solution of (6.32) [34]. Therefore, it is also an ordinary solution of this inclusion.

232

Chapter 6 Averaging of Differential Inclusions

Thus, we proved that the limit of any uniformly convergent sequence of solutions of (6.30) is a solution of inclusion (6.32). We now show that, for any ; there exists a neighborhood U. 0 / such that, for any solution x.t; /, 2 U. 0 / of inclusion (6.30) satisfying the initial condition x.0; / D x0 , one can ﬁnd a solution x.t; 0 / of inclusion (6.32) such that kx.t; / x.t; 0 /k < ;

0 t T:

Assume the contrary. Then there exist 0 and a sequence of solutions x.t; k /,

k 2 U. 0 /, k ! 0 , k ! 1, of inclusion (6.30) such that max kx.t; k / x.t; 0 /k 0

t2Œ0;T

(6.34)

for all solutions x.t; 0 / of inclusion (6.32). The family x.t; k / is uniformly bounded and equicontinuous. Hence, by the Arzelà theorem, it contains a uniformly convergent subsequence. As shown above, the limit of this subsequence must be a solution of (6.32), which contradicts (6.34). The theorem is thus proved. Remark 1. If x.t; 0 / is a solution of the differential inclusion (6.32), then it is not always possible to construct a sequence of solutions of inclusion (6.30) convergent to x.t; 0 / as ! 0 . Example 1. Consider an inclusion p xP 2 Œ1; 2. x C 2 /;

x.0; / D 0:

Then, for 0 D 0, inclusion (6.32) takes the form p xP 2 Œ1; 2 x; x.0; 0/ D 0:

(6.35)

(6.36)

It is clear that the solutions x.t; / of inclusion (6.35) converge to solutions of (6.36) that belong to the integral funnel Œt 2 =4; t 2 . At the same time, there is no sequence x.t; / convergent to the trivial solution of inclusion (6.36). Remark 2. If inclusion (6.32) possesses a unique solution, then any sequence of solutions x.t; / of inclusion (6.30) converges to this solution as ! 0 . This statement is similar to Theorem 1 in [34]. We now consider the proof of an analog of the Krasnosel’skii–Krein theorem for differential inclusions in terms of R-solutions. Theorem 9 ([122]). Assume that the differential inclusion (6.30) satisﬁes conditions (a) and (b) of Theorem 8 and, in addition,

233

Section 6.2 Krasnosel’skii–Krein Theorem for Differential Inclusions

(c) the R-solutions R.t; 0 / of inclusion (6.32) satisfying the initial condition R.0; 0 / D x0 2 D 1 D are deﬁned for 0 t T and, together with a certain -neighborhood, lie in the domain D. Then, for any > 0, there exists a neighborhood U. 0 / of the point 0 such that, for 2 U. 0 / and any R-solution R.t; / of inclusion (6.30) deﬁned for 0 t T and satisfying the initial condition R.0; / D x0 , there exists an R-solution R.t; 0 / of inclusion (6.32) for which h.R.t; /; R.t; 0 // < ;

0 t T:

Proof. Let R.t; / . ! 0 , 2 ƒ/ be a uniformly convergent sequence of Rsolutions of (6.30) satisfying the initial condition R.0; / D x0 . The limit of this sequence is a continuous function denoted by R.t / 2 comp.Rn /, i.e., lim

max h.R.t; /; R.t // D 0:

(6.37)

! 0 t2Œ0;T

By virtue of the deﬁnition of an R-solution the following relation is true everywhere on Œ0; T : h R.t C ; /;

²

[

Z xC

x2R.t; /

³

tC

F .s; x; /ds t

D 0./;

# 0:

(6.38)

In this relation, we pass to the limit as ! 0 . To this end, we show that lim h

! 0

[

²

Z xC

x2R.t; /

t C t

³ [ ² Z F .s; x; /ds ; xC x2R.t/

t C t

³ F .s; x; 0 /ds D 0:

We ﬁx arbitrary " > 0. By using conditions (a) and (b) of the theorem and relation (6.37), we choose U. 0 / such that Z

Z

tC

h

F .s; x; /ds; t

tC t

F .s; x; 0 /ds <

" 3

and h.R.t; /; R.t // < ı; where ı is determined from the condition of uniform continuity of the function F , namely, h.F .s; x; /; F .s; y; // < for kx yk < ı < 3" .

" 3

234

Chapter 6 Averaging of Differential Inclusions

Then [ h

²

Z xC

x2R.t; /

h

tC t

² Z xC

[ x2R.t; /

tC t

Z x2R.t/

x2R.t/

tC t

F .s; x; 0 /ds

t

F .s; x; /ds t

t

F .s; x; 0 /ds

h.F .s; x; /; F .s; y; //ds

x2R.t;/;y2R.t/; kxyk<ı

t

Z

tC

F .s; x; /ds; t

³

t C

t C

max

tC

³

tC

³ [ ² Z F .s; x; /ds ; xC x2R.t/

³

t C

³ [ ² Z F .s; x; /ds ; xC

Z

h.R.t; /; R.t // C C max h

Z [ ² F .s; x; /ds ; xC x2R.t/

[ ² Z Ch xC x2R.t/

³

t

F .s; x; 0 /ds < ":

Hence, passing to the limit as ! 0 in (6.38), we get ³ Z tC [ ² F .s; x; 0 / ds D o./; h R.t C /; xC t

x2R.t/

# 0:

Therefore, R.t / is an R-solution of inclusion (6.32). The ﬁnal part of the proof is similar to the ﬁnal part of the proof of Theorem 8 and uses the properties of uniform boundedness and equicontinuity of the set of Rsolutions of inclusion (6.30), which follow from condition (a) of the theorem. Remark 3. If R.t; 0 / is an R-solution of the differential inclusion (6.32), then it is not always possible to construct a sequence of R-solutions of (6.30) that converges to R.t; 0 / as ! 0 . Thus, for inclusion (6.35), the sequences of R-solutions R.t; / converge to the 2 R-solution R.t; 0/ D Œ t4 ; t 2 but there is no sequence R.t; / convergent, e.g., to the R-solution R1 .t; 0/ D Œ0; t 2 . Remark 4. If inclusion (6.32) possesses a unique R-solution, then any sequence of R-solutions R.t; / of inclusion (6.30) converges to this R-solution as ! 0 . Remark 5. If inclusion (6.32) possesses a unique R-solution, then, for any ordinary solution of inclusion (6.32), there exists a sequence of ordinary solutions of (6.30) that converges to this solution as ! 0 . Example 2. Let t

t

xP 2 Œ1; 2.1 e /x C .1 e /;

x.0; / D 0:

(6.39)

235

Section 6.2 Krasnosel’skii–Krein Theorem for Differential Inclusions

Then the R-solutions have the form 1 2 R.t; / D e 1; .e 1/ ; 2

t

D t C .e 1/

and converge as # 0 to the unique R-solution 1 2t t R.t; 0/ D e 1; .e 1/ 2 of the inclusion xP 2 Œ1; 2x C 1;

x.0; 0/ D 0:

(6.40)

Moreover, despite the nonuniqueness of an ordinary solution of inclusion (6.40), for any solution x.t; 0/, there exists a sequence of solutions x.t; / of inclusion (6.39) that converges to this solution as # 0. Just as in the case of ordinary differential equations, the Bogolyubov theorem on substantiation of the method of averaging in a ﬁnite interval for differential inclusions is directly connected with the theorems on continuous dependence of the solutions of differential inclusions with right-hand sides integrally continuous as functions of the parameter. We now formulate the Bogolyubov theorems for differential inclusions corresponding to Theorems 8 and 9. Theorem 10. Assume that, in a domain Q D ¹t 0, x 2 D Rn , D is a bounded domainº, the differential inclusion xP 2 "F .t; x/

(6.41)

satisﬁes the following conditions: (a) the set-valued mapping F W RC Rn ! conv.Rn / is uniformly bounded, continuous in t , and uniformly continuous in x uniformly with respect to t; (b) for all x 2 D, the following limit exists: 1 lim T !1 T

Z

T 0

F .s; x/ds D F0 .x/I

(c) the solutions y. / of the inclusion dy 2 F0 .y/; d

y.0/ D x0 2 D 1 D;

D "t;

(6.42)

are deﬁned for 0 L and lie in the domain D together with a -neighborhood.

236

Chapter 6 Averaging of Differential Inclusions

Then, for any > 0, there exists "0 > 0 such that, for 0 < " "0 and any solution x.t; "/ of inclusion (6.41) satisfying the condition x.0; "/ D x0 , there exists a solution of inclusion (6.42) for which the inequality kx.t; "/ y."t /k < holds on the segment Œ0; L"1 . Theorem 11. Assume that the differential inclusion (6.41) satisﬁes conditions (a) and (b) of Theorem 10 and, in addition, (c) the R-solutions Y . / of inclusion (6.42), Y .0/ D x0 2 D 1 D, are deﬁned for 0 L and belong to the domain D together with a certain -neighborhood. Then, for any > 0, there exists "0 > 0 such that, for 0 < " "0 and any R-solution X.t; "/, X.0; "/ D x0 , of inclusion (6.41), there exists an R-solution of inclusion (6.42) for which the inequality h.X.t; "/; Y ."t // < holds on the segment Œ0; L"1 . We now substantiate the method of averaging in the form of Bogolyubov’s theorem for impulsive differential inclusions xP 2 "X.t; x/; xj t Di 2 "Ii .x/;

t ¤ i ; x.0/ D x0 ;

(6.43)

where t 0 is time, x 2 Rn is the phase vector, " > 0 is a small parameter, X W RC Rn ! conv.Rn / and Ii W Rn ! conv.Rn / are set-valued mappings, and i are the times of pulses such that 0 i < iC1 . If the limit exists Z tCT X 1 1 lim h X.s; x/ds C Ii .x/; X.x/ D 0; (6.44) T !1 T t T t i
then inclusion (6.43) is associated with the inclusion yP 2 "X.y/;

y.0/ D x0 :

(6.45)

Inclusion (6.45) is called the averaged inclusion. The proposed averaging scheme was substantiated in [112] and [114] under the assumption that the functions X.t; x/ and Ii .x/ satisfy the Lipschitz condition with respect to x. In what follows, we study the problem of closeness of solutions of systems (6.43) and (6.45) under weaker conditions.

Section 6.2 Krasnosel’skii–Krein Theorem for Differential Inclusions

237

Theorem 12 ([123]). Assume that the following conditions are satisﬁed in the domain Q¹t 0; x 2 D Rn º: (a) the set-valued mappings X.t; x/ and Ii .x/ are uniformly bounded by a constant M , X.t; x/ is continuous in t and uniformly continuous in x uniformly with respect to t, and Ii .x/ are equicontinuous; (b) the limit of (6.44) exists uniformly in x 2 D and t 0 and, in addition, 1 i.t; t C T / d < 1; T where i.t; t C T / is the number of points of the sequence ¹i º in the interval .t; t C T ; (c) the solutions of inclusion (6.45) with " D 1 are deﬁned for t 2 Œ0; L and lie in the domain D together with a certain -neighborhood; (d) the modulus of continuity of the set-valued mapping X.x/ is the Kamke function [23]. Then, for any > 0 and any L 2 .0; L , there exists "0 D "0 . ; L/ > 0 such that, for 0 < " "0 and any solution x.t; "/ of inclusion (6.43) satisfying the condition x.0; "/ D x0 , one can ﬁnd a solution of inclusion (6.45) for which the inequality kx.t; "/ y.t; "/k < is true on the segment Œ0; L"1 . Proof. We ﬁx L 2 .0; L . Let x.t / D x.t; "/ be an arbitrary solution of inclusion (6.43). We split the segment Œ0; L" into subsegments by points tk D kh."/, k D 0; m, L where m D Œ "h."/ and tmC1 D L" . Further, we choose steps h."/ such that h."/ ! 1 and "h."/ ! 0 as " ! 0. Since x.t / is a solution of inclusion (6.43), there exist a measurable selector v.t / 2 X.t; x.t // [23] and pulse vectors pi 2 Ii .x.i // such that Z t X x.t / D x.tk / C " v.s/ds C " pi ; x.0/ D x0 ; (6.46) tk

tk i
t 2 Œtk ; tkC1 ;

k D 0; m:

For each k D 0; m, for t 2 Œtk ; tkC1 , we deﬁne an Euler quasibroken line as follows: Z t w.s/ds x1 .t / D x1 .tk / C " C"

X tk i
tk

ri C "˛k .t tk /;

x1 .0/ D x0 ;

(6.47)

238

Chapter 6 Averaging of Differential Inclusions

where w.t / 2 X.t; x1 .tk //W kw.t / v.t /k D ri 2 Ii .x1 .tk //W kri pi k D

min

u2X.t;x1 .tk //

min

r2Ii .x1 .tk //

ku v.t /k;

kr pi k;

w.t / is a measurable function [23], and ˛k 2 Rn are constants chosen from the condition x.tkC1 / D x1 .tkC1 /; k D 0; m: We now show that these ˛k exist. Indeed, by using equalities (6.46) and (6.47), for t D tkC1 , we get Z "˛k .tkC1 tk / D "

tkC1 tk

X

.v.s/ w.s//ds C "

.pi ri /:

tk i
Therefore, Z "k˛k k.tkC1 tk / "

tkC1 tk

Z "

X

kv.s/ w.s/kds C "

kpi ri k

tk i
tkC1 tk

C"

h.X.s; x.s//; X.s; x.tk ///ds X

h.Ii .x.i //; Ii .x.tk ///:

tk i
In view of the uniform continuity of the function X.t; x/ and equicontinuity of the functions Ii .x/, for any 1 > 0, one can ﬁnd ı > 0 such that the inequalities h.X.t; x/; X.t; y// < 1 and h.Ii .x/; Ii .y// < 1 are true for kx yk < ı. We have kx.t / x.tk /k "M h."/ C "dM h."/ D M "h."/.1 C d /: Further, we choose "0 such that the inequality M "h."/.1 C d / < ı is true for " "0 . This yields "k˛k k.tkC1 tk / " 1 .tkC1 tk / C "d 1 .tkC1 tk / D 1 ".1 C d /.tkC1 tk /: Then k˛k k 1 .1 C d /. We now show that, for any > 0, there exists "0 such that the estimate kx.t / x1 .t /k < 3 is true for " "0 and t 2 Œ0; L" .

Section 6.2 Krasnosel’skii–Krein Theorem for Differential Inclusions

239

For t 2 Œtk ; tkC1 , we ﬁnd kx.t / x1 .t /k kx.t / x.tk /k C kx.tk / x1 .tk /k C kx1 .tk / x1 .t /k .M "h."/ C dM "h."// C .M "h."/ C dM "h."/ C k˛k k"h."// D .1 C d /.2M C 1 /"h."/: We choose "0 such that the inequality .1 C d /.2M C 1 /"h."/ < is true for " "0 . For each k D 0; m, we deﬁne

Z

y1 .t / D y1 .tk / C " y1 .0/ D x0 ;

t tk

D

min z2X.x1 .tk //

Z

tk i
tkC1 tk

z.s/ds C "˛k .t tk /;

t 2 Œtk ; tkC1 ;

where z.s/ zk for t 2 Œtk ; tkC1 and Z tkC1 X w.s/ds C " tk

3

ri zk .tkC1 tk /

w.s/ds C "

X tk i
ri z.tkC1 tk / :

It is necessary to show that, for any > 0, there exists "0 such that the estimate ky1 .t / x1 .t /k < 3 is true for " "0 and t 2 Œ0; L" . Denote ık D ky1 .tk / x1 .tk /k. In view of the uniform convergence of (6.44), there exists a monotonically decreasing function F .t / ! 0 as t ! 1 such that Z tkC1 Z tkC1 X w.s/ds C " ri z.s/ds ıkC1 ık C " tk

tk i
tk

Z tkC1 1 ık C "h."/h X.s; x1 .tk //ds h."/ tk X 1 C Ii .x1 .tk //; X .x1 .tk // h."/ t
i

kC1

ık C "h."/F .h."//: Thus, we get the following sequence of estimates: ı0 D 0;

:::;

ıkC1 ık C "h."/F .h."//;

240

Chapter 6 Averaging of Differential Inclusions

whence it follows that ıkC1

kC1 X

"h."/F .h."// LF .h."//:

iD0

Hence, for t 2 Œtk ; tkC1 , we obtain ky1 .t / x1 .t /k ky1 .t / y1 .tk /k C ky1 .tk / x1 .tk /k C kx1 .tk / x1 .t /k 2ŒM "h."/ C k˛k k"h."/ C M d "h."/ C LF .h."//

D "h."/ M.2 C d / C 2 1 .1 C d / C LF .h."//: We choose "0 such that the estimate "h."/ŒM.2 C d / C 2 1 .1 C d / C LF .h."// <

3

is true for " "0 . For t 2 Œtk ; tkC1 , we get kx1 .tk / y1 .t /k ky1 .t / y1 .tk /k C ky1 .tk / x1 .tk /k "h."/ŒM C 1 .1 C d / C LF .h."// ."/: Then .yP1 .t /; "X.y1 .t /// .yP1 .t /; "X.x1 .tk /// C "h.X.x1 .tk //; X.y1 .t /// "k˛k k C "!.."// " 1 .1 C d / C "!.."//; where ! is the modulus of continuity of the function X.x/. By the Plis theorem [105, 106], there exists a solution y.t / D y.t; "/ of inclusion (6.45) such that ky1 .t / y.t /k r."t /; where r. / is the upper solution of the Cauchy problem dr D !.r/ C ˇ."/; d

r.0/ D 0; 0 L;

(6.48)

ˇ."/ D 1 .1 C d / C !.."//. Since ˇ."/ can be made arbitrarily small and the equation rP D !.r/;

r.0/ D 0

possesses solely the trivial solution, by the Hartman theorem [57], we conclude that all solutions of (6.48) uniformly converge to the solution of the limit problem, i.e.,

241

Section 6.3 Averaging of Inclusions with Pulses at Nonﬁxed Times

r. / can be made smaller than =3 by the choice of "0 . Thus, for any solution x.t /, there exists a solution y.t / such that the estimate kx.t / y.t /k kx.t / x1 .t /k C kx1 .t / y1 .t /k C ky1 .t / y.t /k < holds for " "0 . The theorem is proved. Remark 6. Let X.t; "/ be the set of solutions of the impulsive differential inclusion (6.43) and let X.t; 0/ be the set of solutions of the differential inclusion (6.45). Then the theorem establishes the upper semicontinuity (by inclusion) of the set-valued mapping X.t; "/ at the point " D 0.

6.3

Averaging of Inclusions with Pulses at Nonﬁxed Times

Consider differential inclusions with pulse actions xP 2 F 1 .t; x; "/;

x.0/ D x0 ; t ¤ "i1 .x/; t ¤ i1 .x/;

(6.49)

xj tD" 1 .x/ 2 "Ii1 .x/;

(6.50)

xj tDp1 .x/ 2 Kp1 .x/:

(6.51)

i

Inclusion (6.49)–(6.51) is associated with the following differential inclusion: yP 2 F 2 .t; y; "/;

y.0/ D x0 ; t ¤ "i2 .x/; t ¤ i2 .y/;

(6.52)

"Ii2 .y/;

(6.53)

yj tDp2 .y/ 2 Kp2 .y/;

(6.54)

yj tD" 2 .y/ 2 i

where t 2 Œ0; L is time, x 2 D Rn is the phase vector, " is a small parameter, ij , ij W D ! R are pulse surfaces, and F j W Œ0; L D RC ! conv.Rn / and Iij , j Kp W D ! conv.Rn /, i D 1; k, p D 1; r, j D 1; 2, are set-valued mappings. First, we consider the differential inclusions (6.49) and (6.52) solely with asymptotically small pulse actions (6.50) and (6.53). Assume that Z t C X 1 1 1 lim h F .t; x; "/ dt C " Ii .x/ ; "!0 t 1

t<"i .x/
Z

t C t

F 2 .t; x; "/ dt C "

X

Ii2 .x/

D 0:

(6.55)

t<"i2 .x/
Let Jj .t; t C /, j D 1; 2, be the number of asymptotically stable pulses of the solutions of inclusions (6.49), (6.50) and (6.52), (6.53), respectively, in the interval .t; t C (0 t; t C L).

242

Chapter 6 Averaging of Differential Inclusions

Theorem 13. Assume that the following conditions are satisﬁed in the domain Q¹t 2 Œ0; L; x 2 Dº: (1) the set-valued mappings F j .t; x; "/ and Iij .x/ and the functions ij .x/ satisfy the Lipschitz condition with respect to x with constant ; they are continuous in j t , and, in addition, F j .t; x; "/ TD .x/ and x C Ii .x/ D; (2) the set-valued mappings F j .t; x; "/ and Iij .x/ are uniformly bounded by a constant M ; (3) limit (6.55) exists uniformly in .t; x/ 2 Q; (4) the numbers Jj .t; t C /, j D 1; 2, satisfy the inequalities 1 A Jj .t; t C / < 1I " (5) the surfaces t D "ij .x/ do not intersect and the inequalities j

j

i .x/ i .x C z/;

j

j

jiC1 .x/ i .x/j M

hold for any x 2 D and z 2 Iij , j D 1; 2. Then, for any > 0, there exists "0 . / 2 .0; "1 / such that the following statements are true: (1) if y.t / D y.t; "/ is a solution of (6.52), (6.53) with " 2 .0; "0 /, then there exists a solution x.t / D x.t; "/, x.0/ D y.0/ of inclusion (6.49), (6.50) such that [ (6.56) kx.t / y.t /k ; t 2 Œ0; L n Œti2 i ; ti2 C i ; where ti2 D i .y..ti2 // and, in addition,

P

i

i

i < C ;

(2) if x.t / D x.t; "/ is a solution of (6.49), (6.50) with " 2 .0; "0 /, then there exists a solution y.t / D y.t; "/, x.0/ D y.0/ of inclusion (6.52), (6.53) such that inequality (6.56) is true. Proof. Let y.t / be an arbitrary solution of inclusion (6.52), (6.53). It follows from conditions (1) and (2) of the theorem that there exist solutions of inclusions (6.49), (6.50) and (6.52), (6.53) that can be extended to Œ0; L and all these solutions belong to the domain D. We now show that, for sufﬁciently small " "1 , every solution x.t / and y.t / crosses the surfaces t D ij .x/ exactly once. Assume the contrary, i.e., that the solution y j .t / of inclusion (6.49), (6.50) or, respectively, of (6.52), (6.53) suffers beating upon the surface t D ij .x/. Let t0 D "ij .y j .t0 // and let the solution y j .t / with the initial condition j

y j .t0 / C z 2 y j .t0 / C "Ii .y j .t0 //

243

Section 6.3 Averaging of Inclusions with Pulses at Nonﬁxed Times

cross the same surface at time t , i.e., t D "ij .y j .t //. Further, let .t0 ; t / be the interval of continuity of the function y j .t /. Then we get j

Z

j

y .t / D y .t0 / C z C

t

u. / d ; t0

We choose 0 < " < "1 <

u.t / 2 F j .t; y j .t /; "/:

1 :

M

By using the Lipschitz condition, we ﬁnd t t0 D "ij .y j .t // "ij .y j .t0 // D "ij .y j .t // "ij .y j .t0 / C z/ C "ij .y j .t0 / C z/ "ij .y j .t0 // Z t "

u. / d C "ij .y j .t0 / C z/ "ij .y j .t0 // t0

" M.t t0 / C "ij .y j .t0 / C z/ "ij .y j .t0 //: Thus,

.1 " M /.t t0 / ".i .y j .t0 / C z/ i .y j .t0 ///; j

j

which contradicts condition (5) of the theorem. j We now suppose that the solution y j .t / crosses another surface t D "k .x/ at time t 2 .t0 ; t /. In this case, we get t0 D "ij .y j .t0 // > "ij .y j .t //; t D t D

"kj .y j .t // > "ij .y j .t //:

"kj .y j .t //;

t0 < t t I t < t t I

For any arbitrarily chosen continuous function z.t / such that z.t / D y j .t / and z.t / D y j .t /; there exists tN 2 .t ; t for which ij .z.tN// D kj .z.tN//. The last equality contradicts the condition that the surfaces t D "ij .x/ and t D "kj .x/ are disjoint. Hence, each solution crosses each surface at most once. L , k D 0; m. We split the segment Œ0; L into subsegments by points tk D k m Since y.t / is a solution of (6.52), (6.53), there exist a measurable selector v.t / 2 F 2 .t; y.t /; "/ and pulse vectors pi 2 Ii2 .y."i2 // such that Z y.t / D y.tk / C y.0/ D x0 ;

t tk

v. / d C "

t 2 .tk ; tkC1 :

X tk "i2 .y/
pi ;

244

Chapter 6 Averaging of Differential Inclusions

For any k D 1; m, we deﬁne a function Z t v. N / d C " y 1 .t / D y 1 .tk / C tk

y 1 .0/ D x0 ;

X

pNi ;

tk "i2 .y/
t 2 .tk ; tkC1 ;

where v. N / 2 F 2 .; y 1 .tk /; "/ and pNi 2 Ii2 .y 1 .tk // are such that Z tkC1 Z tkC1 v. N /d v. /d tk

D kpNi pi k D

tk

min

z./2F 2 .;y 1 .tk

min

r2Ii2 .y 1 .tk //

Z /;"/

tkC1 tk

Z z. /d

tkC1 tk

v. /d ;

kr pi k:

Denote ık D ky.tk / y 1 .tk /k. Thus, in view of condition (5) of the theorem, we get ky.t / y 1 .tk /k ky.t / y.tk /k C ky.tk / y 1 .tk /k ık C M.1 C A/.t tk /: For ıkC1 D ky.tkC1 / y 1 .tkC1 /k, we can write Z tkC1 Z tkC1 C " ıkC1 ık C v. N / d v. / d tk

Z ık C

tkC1 tk

C"

tk

(6.57)

X

.pNi pi /

tk "i2 .y/
h.F 2 .; y. /; "/; F 2 .; y 1 .tk /; "// d

X

h.Ii2 .y 1 .tk //; Ii2 .y."i2 ///

tk "i2 .y/
Z

ık C

tkC1 tk

ky.t / y 1 .tk /k dt C "

X

ky."i2 / y 1 .tk /k

tk "i2 .y/
L L2 L L ık C ık C M.1 C A/ 2 C A ık C M.1 C A/ : m 2m m m Let a D .1 C A/ and let b D M.1 C A/. 12 C A/. Hence, we have ıkC1

L bL2 ık C 2 : 1Ca m m

245

Section 6.3 Averaging of Inclusions with Pulses at Nonﬁxed Times

Thus, we get the following sequence of inequalities: ı0 D ky 1 .0/ y.0/k D 0; 2 bL L k L k1 L C 1Ca C C 1 C a ; ıkC1 1 C a C1 m m m m2 i.e., ıkC1

L kC1 / 1 bL2 .1 C a m bL 2 L m am am

L 1Ca m

m

1

ML. 12 C A/ .1CA/L b L aL .e 1/ D .e 1/: am m

(6.58)

It follows from (6.57) and (6.58) that ky.t / y 1 .t /k ky.t / y 1 .tk /k C ky 1 .tk / y 1 .t /k ık C 2M.1 C A/

L m

ML. 12 C A/ .1CA/L L .e 1/ C 2M.1 C A/ m m ML.1 C A/ .1CA/L .e C 1/; t 2 Œ0; L: m

(6.59)

By virtue of condition (3) of the theorem, for any 1 > 0, there exist ". 1 / > 0, u. N / 2 F 1 .t; y 1 .tk /; "/; and qN i 2 Ii1 .y 1 .tk / such that the following estimate is true for 0 < " < ". 1 /: Z t X u. N / d C " q N i tk

Z

tk "i1 .y/
X

v. N / d C "

ti "i2 .y/
L pNi < 1 m :

In each interval .tk ; tkC1 ; we deﬁne a function Z t X 1 1 u. N / d C " x .t / D x .tk / C tk

qN i ;

x 1 .0/ D x0 :

tk "i1 .y/
For t 2 .tk ; tkC1 , k D 0; m 1, we ﬁnd kx 1 .t / y 1 .t /k kx 1 .tk / y 1 .tk /k C 1

L L D ık C 1 : m m

246

Chapter 6 Averaging of Differential Inclusions

Since ı0 D kx 1 .0/ y 1 .0/k D 0, we have kx 1 .t / y 1 .t /k 1 L;

t 2 Œ0; L:

We now show that there exists a solution Z t u. / d C " x.t / D x.tk / C tk

x.0/ D x0 ;

X

(6.60)

qi ;

tk "i1 .x/
t 2 Œtk ; tkC1 /;

of inclusion (6.49), (6.50) sufﬁciently close to x 1 .t /. By using the Lipschitz condition, we get .xP 1 .t /; F 1 .t; x 1 .t /; "// h.F 1 .t; x 1 .tk /; "/; F 1 .t; x 1 .t /; "// L 1 1 kx .tk / x .t /k M C " m M

X

qN i

tk "i1 .y/
AL

M.1 C A/L L C "M D : m "m m

Let t 2 .tk ; tkC1 and let si and siC be times at which x.t / and x 1 .t / reach the surfaces t D i1 .x/. Denote i D kx 1 .si / x.si /k

and

iC D kx 1 .siC C 0/ x.siC C 0/k:

Consider the intervals .siC1 ; si . By the Filippov theorem [23, 49], there exists a solution x.t / of inclusion (6.52), (6.53) such that Z t C C C C e .si 1 s/ ds: kx.t / x 1 .t /k e .tsi1 / i1 C si1

Therefore,

C si 1 /

i e .si

C i1 C

.s sC / .e i i1 1/:

C By using conditions (1) and (5) of the theorem, we estimate si si1 . Without C C 1 1 1 loss of generality, we can assume that si D i .x .si // and si1 D i1 .x.si1 //: C C 1 D "Œi1 .x 1 .si // i1 .x.si1 // si si1 C 1 1 1 .x 1 .si // C i1 .x 1 .si // i1 .x.si1 // "Œi1 .x 1 .si // i1 C C M.si siC1 //: "ŒM C . i1

Section 6.3 Averaging of Inclusions with Pulses at Nonﬁxed Times

247

Thus, we get

C ".M C i1 / : 1 " M Since the right-hand sides of inclusions (6.49), (6.50), (6.52), and (6.53) are C bounded, there exits a constant K1 > 0 such that i1 K1 . Hence, C si si1

C si si1 K";

This yields

where K D

C si 1 /

i e .si

M C K1 : 1 " M

K" .e 1/:

C i1 C

(6.61)

We estimate the difference between x 1 .t / and x.t / at the times of pulses. Without loss of generality, we can assume that si is the time of crossing the surface t D "i1 .x/ by x 1 .t / and siC is the time of crossing this surface by x.t /. Further, we get kx 1 .si / x.siC /k kx 1 .si / x.si /k C kx.si / x.siC /kº i C M.siC si /; siC si D "Œi1 .x.siC // i1 .x 1 .si // " . i C M.siC si //; i.e., siC si

" i : 1 " M

(6.62)

We now choose x.siC C 0/ such that iC D kx 1 .siC / x.siC C 0/k D

min

x2x.siC /CIi1 .x.siC //

Z 1 i 1 h x .s1 / C "I1 .x .si // C x.si /

C

"Ii1 .x.siC //

Z C

Z kx 1 .si / x.si /k C

s1C

s1

s1C s1

siC si

kx 1 .siC / xk

u.s/ds; N

u.s/ ds

.u.s/ N u.s// ds

C "h.Ii1 .x 1 .si //; Ii1 .x.siC /// i C 2M.siC si / C " . i C M.siC si // D

1 C " C " M i : 1 " M

248

Chapter 6 Averaging of Differential Inclusions

Thus, we get iC a1 i ;

1 C " C " M : 1 " M

where a1 D

(6.63)

By virtue of inequalities (6.61) and (6.63), we obtain C si 1 /

iC a1 e .si

C i1 C ˇ;

ˇD

a1 K" .e 1/:

Denote ık D kx.tk C 0/ x 1 .tk C 0/k. Thus, we arrive at the following sequence of estimates (here, we set s0C D tk /: 0C D ık ;

C

1C a1 e .s1 s0 / 0C C ˇ;

C

C

2C a1 e .s2 s1 / .a1 e .s1 s0 / 0C C ˇ/ C ˇ L

L

a1 e m 0C C ˇ.a1 e m 1 C 1/; L

:::;

L

iC a1i e m 0C C ˇ.e m .a1i1 C C a1 / C 1/ ai 1 1 L L D a1i e m 0C C ˇ e m a1 1 C1 : a1 1 Separately, we perform the following estimates: L

L A.1C2M /

L

a1i e m e i.a1 1/ e m e m . 1"M C1/ D K2 1 ; m ! 1I D1CO m AL.1C2M / ai1 1 L L e m.1"M / 1 ˇ e m a1 1 C 1 ˇ e m a1 " .1C2M / C1 a1 1 1" M

AL.1C2M / a1 e K" 1 L e m.1"M / 1 a1 K" m e a1 .e 1/ D C

.1C2M /

"

1" M 1 " DO ; m ! 1: CO 2 m m

Thus, there exist positive constants K3 and K4 such that the following estimate is true for m m1 : iC K2 0C C

K3 K4 " : C m2 m

249

Section 6.3 Averaging of Inclusions with Pulses at Nonﬁxed Times

Hence, K3 K4 " : C 2 m m As a result, we get the following sequence of estimates: ıkC1 K2 ık C

K3 K4 " ; :::; C 2 m m K3 K4 " k .K2 C C 1/ C m2 m K kC1 1 K3 K4 " C D 2 2 K2 1 m m K3 K3 K4 " K5 .k C 1/ C K C " !0 K 5 4 m2 m m

ı0 D 0; ıkC1

ı1 C

(6.64)

for m m1 and " "1 . We now choose 0 < " "0 and m m0 . Thus, inequalities (6.59), (6.60), and (6.64) imply inequality (6.56). Moreover, by virtue of (6.62), we obtain X .siC si / i

A : 1 " M

Similarly, we get the second statement of the theorem. The theorem is proved. Remark 7. Let X.t / and Y .t / be the graphs of the solutions x.t / and y.t /. At the times of pulses, the vertical segments Œx.ti /; x.tiC / and Œy.ti /; y.tiC / belong to these graphs. Then the statement of the theorem can be rewritten in the form h X.t /; Y .t / < , t 2 Œ0; L. Remark 8. Assume that the conditions of the theorem are satisﬁed. We consider inclusion (6.52), (6.53) with y.0/ D y0 and denote ı D kx0 y0 k. Then, for any > 0 and each solution y.t / of inclusion (6.52), (6.53), there exist "0 . / > 0, ı0 . / > 0, a constant C , and a solution x.t / of inclusion (6.49), (6.50) such that kx.t / y.t /k < C ı C ;

t 2 Œ0; L; 0 < " < "0 ; 0 < ı < ı0 :

We now formulate the theorem of the method of averaging for impulsive differential inclusions (6.49)–(6.51). Theorem 14. Assume that all conditions of Theorem 13 are satisﬁed in the domain Q and, in addition,

250

Chapter 6 Averaging of Differential Inclusions

(5) the mappings Kij .x/ and the functions ji .x/ satisfy the Lipschitz condition with j

constant and, in addition, x C Ki .x/ D, x 2 D; (6) the surfaces t D ji .x/ are disjoint and ji .x/ ji .x C z/ for any x 2 D and j

z 2 Ki , j D 1; 2; (7) the following inequalities hold: M < 1;

h.Ki1 .x/; Ki2 .x// ;

ji1 .x/ i2 .x/j ;

kx0 y0 k ı:

Then, for any > 0, there exist > 0 and ı > 0 such that, for each solution x.t / of inclusion (6.49)–(6.51), one can ﬁnd a solution y.t / of inclusion (6.52)–(6.54) satisfying the estimate kx.t / y.t /k ; ²[ ³ [ 2 2 2 2 t 2 Œ0; L n Œsp ıp ; sp C ıp Œti i ; ti C i ; p

i

where sp2 D p .y.sp2 // and ti2 D i .y.ti2 // and, in addition,

P

i ıi

C

P i

i < C .

Remark 9. By analogy with procedure used for differential equations [88], it can be shown that Theorems 13 and 14 lead to theorems on substantiation of the method of averaging for impulsive differential inclusions.

6.4

Averaging of Impulsive Differential Equations with Hukuhara Derivative

In the present section, we consider the method of complete averaging of impulsive differential equations with Hukuhara derivative of the form Dh X.t / D "F .t; X /;

t ¤ i ; X.0/ D X0 ;

X j tDi D "Ii .X /:

(6.65) (6.66)

Assume that the set-valued mapping F .t; X / is 2-periodic in t and there exists p 2 N such that the equalities iCp D i C 2 and IiCp .X / Ii .X / are true for all i 2 N . Equations (6.65), (6.66) are associated with the following averaged equation: Dh Y .t / D "F 0 .Y .t //;

Y .0/ D X0 ;

(6.67)

Section 6.4 Averaging of Impulsive Differential Equations with Hukuhara Derivative

where 1 F .Y / D 2

Z

2

0

0

F .s; Y /ds C

X

1 2

Ii .Y /:

251

(6.68)

0i <2

Theorem 15. Assume that the following conditions are satisﬁed in the domain Q D ¹t 0, X 2 D conv.Rn /º: (1) the set-valued mappings F W R conv.Rn / ! conv.Rn / and Ii W conv.Rn / ! conv.Rn / are uniformly bounded by a constant M and satisfy the Lipschitz condition with constant with respect to the variable X ; moreover, the mapping F .t; X / is continuous in t ; (2) there exists a domain D 0 D such that, for any X0 2 D 0 , the solutions of Eq. .6:67/ belong to the domain D together with a certain -neighborhood. Then there exist constants C > 0 and "0 > 0 such that the estimate h.X.t /; Y .t // C "

(6.69)

holds for " 2 .0; "0 and t 2 Œ0; L"1 . Proof. The differential equations with Hukuhara derivative (6.65), (6.66), and (6.67) are equivalent to the integral equations [25] Z X.t / D X0 C "

0

Z Y .t / D X0 C "

t

X

F .s; X.s//ds C "

Ii .X.i //;

(6.70)

0i
F 0 .Y .s//ds;

(6.71)

0

where the integral is understood in Hukuhara’s sense. We now show that the set-valued mapping F 0 .X / is uniformly bounded and satisﬁes the Lipschitz condition jF 0 .Y /j D h.F 0 .Y /; ¹0º/ Z 2 1 1 F .s; Y /ds C Dh 2 0 2 ˇ Z ˇ 1 ˇˇ 2

ˇ 2 ˇ 1 F .s; Y /ds ˇˇ C 2 0

0i <2

X 0i <2

p

p M DM 1C ; M C 2 2

Ii .Y /; ¹0º

X

jIi .Y /j

252

Chapter 6 Averaging of Differential Inclusions

h.F 0 .Y 0 /; F .Y 00 // D h

1 2

1 2

Z

2 0

Z

2 0

F .s; Y 0 /ds C

1 2

1 F .s; Y /ds C 2

X

Ii .Y 0 /;

0i <2

X

00

00

Ii .Y /

0i <2

Z 2 Z 2 1 1 0 00 h F .s; Y /ds; F .s; Y /ds 2 0 2 0 1 X p C h.Ii .Y 0 /; Ii .Y 00 // h.Y 0 ; Y 00 / C

h.Y 0 ; Y 00 / 2 2 i p

h.Y 0 ; Y 00 /: D 1C 2

The established inequalities mean that the set-valued mapping F 0 .Y / is uniformly p bounded by a constant M1 D M.1 C 2 / and satisﬁes the Lipschitz condition with p constant 1 D .1 C 2 /. We now estimate h.X.t /; Y .t //. By virtue of relations (6.70) and (6.71), we get Z t 0 F .Y .s//ds h.X.t /; Y .t // D h X.t /; X0 C " 0

Z t 0 F .X.s//ds h X.t /; X0 C " 0

Z t Z t 0 0 F .X.s//ds; " F .Y .s//ds Ch " 0

Z " 1

0

t

h.X.s/; Y .s//ds 0

Z t F 0 .X.s//ds : C h X.t /; X0 C "

(6.72)

0

By using the Gronwall–Bellman lemma and inequality (6.72), we ﬁnd Z t F .s; X.s//ds h.X.t /; Y .t // e " 1 t h " 0

C"

X 0i
Z Ii .X.i //; "

t 0

F 0 .X.s//ds :

(6.73)

Section 6.4 Averaging of Impulsive Differential Equations with Hukuhara Derivative

253

Let t 2 Œ2 m; 2.m C 1//. Then Z t Z t X F .s; X.s//ds C Ii .X.i //; F 0 .X.s//ds h 0

Dh

m1 XZ

C

2.j C1/

2j

j D0

0

0i t

m1 X

Z F .s; X.s//ds C

X

t

F .s; X.s//ds 2 m

j D0 2j i <2.j C1/ m1 X Z 2.j C1/ iD0

Dh

2j

m1 X

j

F .X.s//ds C

F CF ;

j D0

t

0

m1 X

Ii .X.i //;

2 mi
Z

m

X

Ii .X.i // C

0

F .X.s//ds 2 m

F

0j

CF

0m

;

(6.74)

j D0

where Z Fj D

2j

Z Fm D F j0 D D

Ii .X.i //;

j D 0; m 1;

2j i <2.j C1/

X

F .s; X.s//ds C

Ii .X.i //;

2 mi
2.j C1/

t

X

F .s; X.s//ds C

F 0 .X.s//ds;

j D 0; m 1;

2j

Z F

t 2 m

Z

0m

2.j C1/

F 0 .X.s//ds:

2 m

We estimate h.F j ; F j 0 /: h.F j ; F j 0 / Z 2.j C1/ F .s; X.s//ds C Dh 2j

Z h

2.j C1/

2j

Z

2.j C1/ 2j

X

Z Ii .X.i //;

2j i <2.j C1/

X

F .s; X.s//ds C

2 i

Ii .X.i //;

2j i <2.j C1/

F .s; X.2j //ds C

2.iC1/

X 2j i <2.j C1/

Ii .X.2j //

F .X.s//ds 0

254

Chapter 6 Averaging of Differential Inclusions

Z Ch

2.j C1/

2j

Z

2.j C1/

Ch Z

Ii .X.2j //;

2j i <2.j C1/

F 0 .X.2j //ds

2j

Z

X

F .s; X.2j //ds C

Z

2.j C1/

0

F .X.2j //ds; 2j

2.j C1/

0

F .X.s//ds 2j

2.j C1/

h.F .s; X.s//; F .s; X.2j ///ds 2j

X

C

h.Ii .X.i //; Ii .X.2j ///

2j i <2.j C1/

Z

C Z

2.j C1/

h.F 0 .X.2j //; F 0 .X.s///ds

2j 2.j C1/

h.X.s/; X.2j //ds 2j

Z

X

C

h.X.i /; X.2j // C 1

2j i <2.j C1/

X

D

2.j C1/

h.X.s/; X.2 i//ds 2j

h.X.i /; X.2j //

2j i <2.j C1/

Z p 2.j C1/ C 2C h.X.s/; X.2j //ds: 2 2j

(6.75)

On the other hand, we ﬁnd h.X.s/; X.2j // Z D h X.2j / C " Z "

s 2j

s 2j

F .; X. //d C "

jF .; X. //jd C "

X

Ii .X.i //; X.2j /

2j i <s

X

jIi .X.i //j

2j i <s

"M.s 2j / C "Mp: It follows from (6.75) and (6.76) that p

h.F j ; F 0j / " 2 C 2 Z 2.j C1/ ŒM.s 2j / C Mpds C " p Œ2M C Mp 2j

(6.76)

Section 6.4 Averaging of Impulsive Differential Equations with Hukuhara Derivative

p

Œ2 2 M C 2Mp C " pŒ2M C Mp " 2 C 2 h p i p

. C p/ C p 1 C D 2" C1 ; D 2" M 2 C 2 2

255

(6.77)

where C1 D M

h

2C

p i p

. C p/ C p 1 C : 2 2

Thus, we get h.F m ; F 0m / Z t F .s; X.s//ds C Dh 2 m

Z

t 2 m

Z

X

Ii .X.i //;

2 mi
X

jF .s; X.s//jds C

2M C pM C M 1 C

0

F .X.s//ds 2 m

Z

jIi .X.i //j C

2 mi

t

t 2 m

jF 0 .X.s//jds

p

p

2 D 4M 1 C : 2 2

(6.78)

By using (6.74), (6.77), and (6.78), we obtain Z

t

h 0

m1 X

F .s; X.s//ds C

X

Z Ii .X.i //;

0i t

t

0

F .X.s//ds 0

h.F j ; F j 0 / C h.F m ; F 0m /

j D0

m1 X j D0

p

p

C1 L C 4M 1 C : 2" C1 C 4M 1 C 2 2

It follows from (6.73) and (6.79) that h.X.t /; Y .t // C "; where

h p i 1 L C D C1 L C 4M 1 C e : 2 The value of "0 is found from the condition C " ;

i.e., "0 D

e 1 L C1 L C 4M.1 C

p : 2 /

(6.79)

256

Chapter 6 Averaging of Differential Inclusions

Remark 10. Since the operation of subtraction is not always deﬁned in the space conv.Rn /, the condition of pulse action (6.66) in the general case should be rewritten in the form X.i C 0/ D

i .X.i /; "/;

where h.X.i C 0/;

i .X.i /; "//

M ":

In this case, the formula of averaging (6.68) takes the form 1 F .Y / D 2

Z

2

0

0

F .s; Y /ds C

1 2

p . p1 .

2 . 1 .Y // //:

Remark 11. If, in Eqs. (6.65), (6.66), the set-valued mappings F .t; X / and Ii .X / are not 2-periodic, then it is possible to prove an analog of Theorem 15 under the assumption that the following limit exists: 1 F .Y / D lim T !1 T

Z

T

0

0

F .s; Y /ds C

X

Ii .Y / :

0i
Under this assumption, the statement of the theorem takes the form: Then, for any > 0, there exists "0 > 0 such that the estimate h.X.t /; Y .t // is true for " 2 .0; "0 and t 2 Œ0; L"1 .

Chapter 7

Differential Equations with Discontinuous Right-Hand Side

7.1

Motions and Quasimotions

In analyzing positional differential games, Krasovskii (see e.g., [71]) assumed that one of the gamblers may use an arbitrary discontinuous function as his strategy, e.g., v.t; x/. The differential equation obtained in this case may have no solutions in the ordinary sense. We generalize the results obtained in [154] to differential equations with discontinuous right-hand side. Note that the solutions x.t / constructed in this case and substituted in the differential equation may violate this equation on a set of nonzero measure. Consider the dynamical system described by a system of ordinary differential equations xP D f .t; x; v.t; x//; (7.1) where x 2 Rn is the phase vector, t 2 Œt0 ; T is time, and v 2 Rq . In what follows, we suppose that the following conditions are satisﬁed: Conditions 1. A vector function f .t; x; v/ is continuous in the collection of its arguments; – v 2 Q 2 comp.Rq /; – for any bounded domain G Rn , there exists a constant .G/ > 0 such that kf .t; x 1 ; v/ f .t; x 2 ; v/k .G/kx 1 x 2 k for any x 1 ; x 2 2 G uniformly in t 2 Œt0 ; T and v 2 Q; – there exists a constant > 0 such that the inequality kf .t; x; v/k .1 C kxk/

(7.2)

is true for all t 2 Œt0 ; T , v 2 Q. If an arbitrary measurable function v.t / such that v.t / 2 Q is taken as v.t; x/, then the validity of conditions 1 guarantees the existence, uniqueness, and extendability of the solutions x.t / of system (7.1) to the segment Œt0 ; T for x.t0 / D x0 . Assume that the function v.t; x/ is such that f .t; x; v.t; x// does not satisfy the conditions of existence theorems.

258

Chapter 7 Differential Equations with Discontinuous Right-Hand Side

We now determine the motion [71, 154] of system (7.1). Suppose that the initial point .t0 ; x0 / is ﬁxed. We cover the segment Œt0 ; T by a system of half intervals j t < j C1 , j D 0; m./ 1; 0 D t0 , m. / D T . Deﬁnition 1. A stepwise motion x. ; / D ¹x.t; t0 ; x0 ; /; t0 t T º of system (7.1) is deﬁned as an absolutely continuous solution of the integral equation Z t x.t; / D x.j ; / C f .s; x.s; /; v.j ; x.j ; /// ds; j

x.0 ; / D x0 ;

j t j C1 ; j D 0; m./ 1:

(7.3)

The existence, uniqueness, and extendability of this stepwise motion x. ; / to Œt0 ; T are established by using the classic theorems of the theory of differential equations. Note that each speciﬁc partition of the segment Œt0 ; T is associated with “its own” unique stepwise motion x. ; /. We also note that the validity of inequality (7.2) is sufﬁcient for the extendability of the stepwise motion to the left up to the time t 2 Œ0; t0 . Deﬁnition 2 ([71]). A function x.t /; t0 t T , is called a motion x. / D ¹x.t; t0 ; x0 /; t0 t T º of system (7.1) generated from the initial point .t0 ; x0 / if, for this function, one can ﬁnd a sequence of stepwise motions x.t; t0 ; x k ; k /;

t0 t T;

(7.4)

uniformly convergent to x.t / on the segment Œt0 ; T under the condition lim supŒjkC1 jk D 0;

k!1 j

lim kx k x0 k D 0:

k!1

(7.5)

A ﬁxed initial point .t0 ; x0 / 2 Œ0; T / Rn generates not a single motion but, generally speaking, a set of motions called a bundle. Different motions in the bundle (denoted by X.t0 ; x0 // are obtained for different sequences of partitions and different initial values of the vector x k . In [71], it is shown that a bundle of motions X.t0 ; x0 / forms a nonempty compact set in the space of continuous functions C Œt0 ; T with standard norm. Kononenko Counterexample ([69]). As a disadvantage of the presented concept of motion, one can mention the fact that a segment of motion may be not a motion.

259

Section 7.1 Motions and Quasimotions

Assume that system (7.1) has the form xP D v; where

0 t 1; x.0/ D x0 D 0;

8 ˆ 1 ˆ ˆ ˆ <1 v.t; x/ D ˆ0 ˆ ˆ ˆ :1

for x for x for x for x

> 0; < 0; D 0; D 0;

(7.6)

t 2 Œ0; 1; t 2 Œ0; 1; t 2 Œ0; 12 / [ . 12 ; 1; t D 12 :

As one of motions of this system from the point .t0 ; x0 / D .0; 0/, we can mention the motion x.t / D 0, 0 t 1. However, a part of this motion x 3 .t / 0, 12 t 1, is not a motion on the segment Œ 12 ; 1. At the same time, segments of the straight lines x 1 .t / D t 12 and x 2 .t / D .t 12 /, t 2 Œ 12 ; 1, are motions on this segment. Thus, a part of the motion x.t / 0 is not a motion generated from the “current” initial point . 12 ; 0/. This is explained by the absence of the limit transition with respect to the initial time in the deﬁnition of motion presented above [(7.5)], i.e., initial points of the form .t0 ; x k /, where t0 is a ﬁxed initial time, are considered for a convergent sequence of stepwise motions. To avoid the “loss of a part of motions,” Kononenko proposes to complement the deﬁnition of stepwise motions of system (7.1) by the limit transition with respect to the initial time. Namely, every stepwise motion should start at the initial position .t k ; x k / but not at the initial point .t0 ; x k /. Then (for t k > t0 / it is extended to the left up to t0 [by virtue of condition (7.2), this extension is possible]. Further, in the construction of motions x. ; t0 ; x0 /, the limit transition is performed either for these extensions to the left (for t k > t0 / or for the “truncated” (for t k < t0 / stepwise motions x.t; t k ; x k ; k /; t0 t T , by adding the requirement lim jt k t0 j D 0

k!1

(7.7)

to conditions (7.5). Thus, the motion x 3 .t / D 0; 12 t 1, is the limit of the sequence of stepwise motions originating from 1 1 k k .t ; x / D C ; 0 ; k D 1; 2; : : : : 2 10k The following assertion is true for (7.5) and (7.7): Theorem 1 ([69]). Let x.t / D x.t; t0 ; x0 /; t0 t T , be the motion of system (7.1) from the initial point .t0 ; x0 /. Then, for any t 2 Œt0 ; T /, a part of motion x.t; t0 ; x0 /, t t T , is a motion of the system from the initial point .t ; x.t ; t0 ; x0 //.

260

Chapter 7 Differential Equations with Discontinuous Right-Hand Side

Subbotin Counterexample. The other disadvantage of the deﬁnition of motion is connected with the possibility of appearance of new motions absent in the original bundle of motions. This was indicated by Subbotin. In what follows, we present Subbotin’s example. Assume that system (7.1) has the form xP D v; where

0 t 2; x.0/ D x0 D 0;

8 ˆ 0 ˆ ˆ ˆ ˆ ˆ 0 ˆ ˆ ˆ <1 v.t; x/ D ˆ 0 ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ :1

for x for x for x for x for x for x

> 0; < 0; D 0; D 0; > 0; < 0;

(7.8)

t 2 Œ0; 1/; t 2 Œ0; 1/; t 2 Œ0; 1/; t 2 Œ1; 2; t 2 Œ1; 2; t 2 Œ1; 2:

The bundle of motions from the initial point .t0 ; x0 / D .0; 0/ consists of two motions ´ 0 for t 2 Œ0; 1/; x 1 .t / D t 1 for t 2 Œ1; 2; and

´ 0 for t 2 Œ0; 1/; x 2 .t / D t C 1 for t 2 Œ1; 2:

Three motions x 1 .t /, x 2 .t /, and x 3 .t / 0; 1 t 2, are generated from the point .1; 0/. Thus, as the current point .t; x.t // moves along the motions x 1 .t / or x 2 .t /, an additional motion x 3 .t / absent in the original bundle X.t0 ; x0 / appears in this bundle at time t D 1. Thus, in the course of time, we may observe not only the “loss” of parts of the original motions but also the appearance of new motions absent in the original bundle of motions (as in the considered example). To exclude this possibility, it is necessary to modify the notion of stepwise motions by assuming that stepwise motions with discontinuities (jumps) at the points of partition are possible. Thus, in the analyzed example, the motion x.t / 0, 0 t 2, is the limit of a sequence of stepwise quasimotions ´ x k for t 2 Œ0; 1/; x k .t / D x k ! 0 as k ! 1: 0 for t 2 Œ1; 2; These quasimotions have discontinuities of the ﬁrst kind at the point t D 1 and are continuous at the other points of the segment Œ0; 2.

261

Section 7.1 Motions and Quasimotions

In this case, the convergence of x k .t / to x.t / 0 should be studied in the space M Œt0 ; T of bounded n-dimensional functions with norm kx. /kM Œt0 ;T D

sup kx.t /k:

t0 tT

For this deﬁnition of motion, the appearance of new motions is impossible. In what follows, we substantiate the proposed approach (the possibility of application of piecewise continuous stepwise motions). Given an initial point .t0 ; x0 / and a number ˛ 2 Œ0; 1, we construct a covering of the segment Œt0 ; T with a system of half intervals j t < j C1 ; j D 0; m./ 1, 0 t0 , m. / D T . Deﬁnition 3. A stepwise quasimotion of system (7.1) generated from the initial point .t0 ; x0 / by a partition of by a number ˛ is deﬁned as an arbitrary function x. ; ; ˛/ D ¹x.t; 0 ; x0 ; x 0 ; ; ˛/; 0 t T º satisfying the equation Z x.t; ; ˛/ D xj C

t j

f .s; x.s; ; ˛/; v.j ; xj // ds

(7.9)

for j t < j C1 under the condition m. /1 X

kxj x0 .j ; ; ˛/k ˛;

(7.10)

j D0

where x0 .j ; ; ˛/ is the value of the solution x.t; ; ˛/ of Eq. (7.9) extended to the right on the segment j 1 t j for t D j , i.e., Z j f .s; x.s; ; ˛/; v.j 1 ; xj 1 // ds x0 .j ; ; ˛/ D xj 1 C j 1

and, in addition, x.0 ; ; ˛/ D x 0 ;

x0 .0 ; ; ˛/ D x0 :

The stepwise quasimotions may have ﬁnite jumps at points of the partition j . The sum of these jumps should not exceed a given number ˛. For ˛ D 0, a stepwise quasimotion turns into an ordinary stepwise motion. By X .t0 ; x0 / we denote the set of stepwise quasimotions of system (7.1) deﬁned as indicated above and generated from the initial point .t0 ; x0 / by all possible partitions of and all numbers ˛ 2 Œ0; 1. We now present several theorems similar to the corresponding theorems in [154]. The proofs of these theorems in terms of the theory of differential equations with

262

Chapter 7 Differential Equations with Discontinuous Right-Hand Side

discontinuous right-hand sides are somewhat simpler than the corresponding proofs in [154] and illustrate the basic idea of the approach to the investigation of differential equations with discontinuous right-hand sides. Theorem 2. The set of all stepwise quasimotions X .t0 ; x0 / of system (7.1) is bounded in the norm of the space of bounded functions M Œt0 ; T , i.e., there exists a constant K such that kx. ; ; ˛/kM Œt0 ;T K for all x. ; ; ˛/. Proof. First, we show that the inequality kx.t; ; ˛/k .1 C kx0 k/e .j C1 t0 / C

j X

kx l x0 .l ; ; ˛/ke .j C1 t0 / 1

(7.11)

lD0

is true for j t < j C1 , j D 0; m./ 1. Moreover, for x0 .j C1 ; ; ˛/, it is necessary to show that kx0 .j C1 ; ; ˛/k .1 C kx0 k/e .j C1 t0 / C

j X

kxl x0 .l ; ; ˛/ke .j C1 t0 / 1:

(7.12)

lD0

In the proof, we use the inequality kxj k kxj x0 .j ; ; ˛/k C kx0 .j ; ; ˛/k;

(7.13)

which follows from the triangle axiom, and the relation kx.t /k .1 C kxj k/e .tj / 1;

t 2 Œj ; j C 1;

(7.14)

valid for the solution x. / of system (7.1) under condition (7.2). Inequalities (7.11) and (7.12) are proved by induction. First, we check the validity of these inequalities for j D 0. For t 2 Œ0 ; 1 /, inequality (7.14) implies that kx.t; ; ˛/k .1 C kx 0 k/e .tt0 / 1: By using inequality (7.13) and the value x0 .0 ; ; ˛/, we get kx.t; ; ˛/k .1 C kx 0 x0 .0 ; ; ˛/k C kx0 k/e .tt0 / 1 .1 C kx0 k/e .1 t0 / C kx 0 x0 .0 ; ; ˛/ke .1 t0 / 1;

263

Section 7.1 Motions and Quasimotions

i.e., inequality (7.11) is true for j D 0. Similarly, we check the validity of inequality (7.12) for j D 0. Assume that inequalities (7.11) and (7.12) are true for j D k, i.e., for k t < kC1 , kx.t; ; ˛/k .1 C kx0 k/e .kC1 t0 / C

k X

kx l x0 .l ; ; ˛/ke .kC1 t0 / 1;

lD0

kx0 .kC1 ; ; ˛/k .1 C kx0 k/e .kC1 t0 / C

k X

kx l x0 .l ; ; ˛/ke .kC1 t0 / 1:

(7.15)

lD0

It is necessary to prove inequalities (7.11) and (7.12) for j D k C 1. For t 2 ŒkC1 ; kC2 /, inequality (7.14) implies that kx.t; ; ˛/k .1 C kx kC1 k/e .tkC1 / 1:

(7.16)

Further, by using (7.13) (for j D k C 1/, we conclude that kx kC1 k kx kC1 x0 .kC1 ; ; ˛/k C kx0 .kC1 ; ; ˛/k:

(7.17)

Substituting (7.15) and (7.17) in (7.16), we obtain kx.t; ; ˛/k Œ1 C kx kC1 x0 .kC1 ; ; ˛/k C .1 C kx0 k/e .kC1 t0 / C

k X

kx l x0 .l ; ; ˛/ke .kC1 t0 / 1e .tkC1 / 1

l D0

.1 C kx0 k/e

.kC2 t0 /

C

kC1 X

kx l x0 .l ; ; ˛/ke .kC2 t0 / 1:

lD0

Similarly, we prove the corresponding inequalities for kx0 .kC2 ; ; ˛/k. Thus, inequalities (7.11) and (7.12) are true. By using the chain of inequalities .1 C kx0 k/e .1 t0 / C kx 0 x0 .0 ; ; ˛/ke .1 t0 / 1 .1 C kx0 k/e

.2 t0 /

C

1 X lD0

kx l x0 .l ; ; ˛/ke .2 t0 / 1

264

Chapter 7 Differential Equations with Discontinuous Right-Hand Side

.1 C kx0 k/e .m. / t0 / C

m. /1 X

kx l x0 .l ; ; ˛/ke .m. / t0 / 1;

l D0

the inequality kx0 .T; ; ˛/k .1 C kx0 k/ C

m. /1 X

kx l x0 .l ; ; ˛/k e .T t0 / 1;

lD0

and inequalities (7.11), (7.12), and (7.10), we get the following estimates for all t 2 Œt0 ; T : kx.t; ; ˛/k .1 C kx0 k/ C

m. /1 X

kx l x0 .l ; ; ˛/k e .T t0 / 1

lD0

.˛ C 1 C kx0 k/e

.T t0 /

1 .2 C kx0 k/e .T t0 / 1 D K:

Thus, it is shown that the set of all stepwise quasimotions X .t0 ; x0 / is bounded. Note that the set X .t0 ; x0 / is a subset of the space M Œt0 ; T of all bounded vector functions z W Œt0 ; T ! Rn . We now consider the notion of quasimotions [154], i.e., the limits of constructed sequences of stepwise quasimotions in M Œt0 ; T . Deﬁnition 4. The quasimotion x. / D ¹x.t; t0 ; x0 /; t0 t T º of system (7.1) from the initial point .t0 ; x0 / is deﬁned as an arbitrary function continuous on the segment Œt0 ; T for which there exists a sequence of stepwise quasimotions (extended to the left up to t0 if 0r > t0 ) convergent (as r; m ! 1/ to this function (in the metric of the space M Œt0 ; T / x. ; r ; ˛ m / D ¹x.t; 0r ; x0r ; x r0 ; r ; ˛ m /; t0 t T º for diam r ! 0 and j0r t0 j C kx0r x0 k ! 0;

˛ m ! 0 as r; m ! 1:

Here, diam r D maxŒjrC1 jr ; j

0 ˛ m 1 .r; m D 1; 2; : : :/:

(7.18)

Section 7.1 Motions and Quasimotions

265

Thus, for a given sequence ¹x. ; r ; ˛ m /º, we have sup kx.t / x.t; r ; ˛ m /k ! 0

t0 tT

provided that condition (7.18) is satisﬁed. By X.t0 ; x0 / we denote the bundle of constructed quasimotions of system (7.1). Different quasimotions of the bundle are realized for different possible sequences r ; ˛ m that can be used for the construction of stepwise quasimotions convergent to quasimotions x.t; t0 ; x0 //. Consider the properties of the bundles of quasimotions of system (7.1) deﬁned as indicated above. Theorem 3. The bundle X.t0 ; x0 / is a nonempty bounded (in norm) closed subset of the space C Œt0 ; T . Proof. The set X.t0 ; x0 / contains motions [71] (they are obtained for ˛ D 0/. Hence, by virtue of [71], we conclude that X.t0 ; x0 / ¤ ¿. It follows from Theorem 2 that the set X .t0 ; x0 / of all stepwise quasimotions x. ; ; ˛/ is bounded in the norm of the space M Œt0 ; T . Then its closure X .t0 ; x0 / is also bounded in this space. Hence, the bundle of quasimotions X.t0 ; x0 / is bounded (in the norm of the space M Œt0 ; T / as a subset of X .t0 ; x0 /. Since the metric of the space C Œt0 ; T is induced by the metric of M Œt0 ; T and X.t0 ; x0 / C Œt0 ; T , the bundle of quasimotions X.t0 ; x0 / is a bounded (in norm) subset of the space C Œt0 ; T . We now show that the bundle X.t0 ; x0 / is closed in C Œt0 ; T , i.e., prove that X.t0 ; x0 / X .t0 ; x0 /, where X .t0 ; x0 / is the closure of the set X.t0 ; x0 / in C Œt0 ; T . Thus, we choose y. / 2 X.t0 ; x0 / and show that y. / 2 X.t0 ; x0 /. Since y. / is the limit point of a sequence of quasimotions ¹x l . /º X.t0 ; x0 /, the function y.t /, as the limit of a sequence of continuous functions x l .t / uniformly convergent on Œt0 ; T , is continuous on Œt0 ; T . Further, we choose a sequence of positive numbers "l .l D 1; 2; : : :/ convergent to zero. For any l D 1; 2; : : : ; we construct a ball B.y. /; "l / D ¹z. / 2 M Œt0 ; T W kz. / y. /kM Œt0 ;T < "l º: in the space M Œt0 ; T . Since y. / 2 X.t0 ; x0 /, where X .t0 ; x0 / is the closure of X.t0 ; x0 /, there exist quasimotions x l . / 2 X.t0 ; x0 / \ B.y. /; "l /: By the deﬁnition of quasimotions, for any l D 1; 2; : : : ; there exists a sequence of stepwise quasimotions l ¹x. ; lr ; ˛m /º convergent to x l . / in the space M Œt0 ; T [under condition (7.18)]. For any l and l / from “sufﬁciently large” r and m, we select a single “representative” x l . ; lr ; ˛m l l each sequence ¹x. ; r ; ˛m /º such that

266

Chapter 7 Differential Equations with Discontinuous Right-Hand Side

l / 2 B.y. /; " /, (1) x l . ; lr ; ˛m l l <" . (2) diam lr < "l ; ˛m l

As l ! 1, the sequence

l ¹x. ; lr ; ˛m /º

of stepwise quasimotions constructed from these “representatives” satisﬁes conditions (7.18) and converges (in the metric of the space M Œt0 ; T / to a continuous function y.t /; t0 t T . Hence, y. / is a quasimotion, i.e., y. / 2 X.t0 ; x0 /. The theorem is proved. The following statement directly follows from Theorem 3: Theorem 4. For any point .t0 ; x0 /, the section of the bundle of quasimotions by the hyperplane t D T is a nonempty compact set in Rn . Theorem 5. For any point .t ; x.t //, where t 2 Œt0 ; T /; x. / is a quasimotion from the bundle X.t0 ; x0 /, the following inclusion is true: X.T; t ; x.t // X.T; t0 ; x0 /:

(7.19)

Proof. Let x. / D ¹x.t /; t0 t T º be a quasimotion of system (7.1) from the initial point .t0 ; x0 / at time t 2 Œt0 ; T /. We take an arbitrary quasimotion x . / D ¹x .t /; t t T º but starting from the initial point .t ; x.t //, where x.t / is the value of x.t / at time t . To prove inclusion (7.19), it is necessary to show that the function ´ x.t / for t0 t t ; x.t Q /D (7.20) x .t / for t t T is a quasimotion of system (7.1) from the initial point .t0 ; x0 /. To establish this fact, we ﬁx a numerical sequence "l 2 .0; 1/ .l D 1; 2; : : :/ such that lim "l D 0: (7.21) l!1

For any l D 1; 2; : : : ; we construct two balls B.x. /; "l / D ¹z. / 2 M Œt0 ; T W kz. / x. /kMn Œt0 ;T < "l º and

B.x . /; "l / D ¹z. / 2 M Œt ; T W kz. / x . /kMn Œt ;T < "l º:

According to the deﬁnition of quasimotions, there exist two sequences of stepwise quasimotions ¹x. ; r ; ˛ m /º M Œt0 ; T ;

¹x . ; ; ˛ /º M Œt ; T p

q

267

Section 7.1 Motions and Quasimotions

convergent in the metric of the indicated spaces to x. / and x . /, respectively, under the conditions diam r ! 0; diam

p

! 0;

˛m ! 0

as r; m ! 1;

˛q

as p; q ! 1:

!0

(7.22)

Here, r r W t0 0r < 1r < < s. r / D T; p p p p W t Q0 < Q1 < < Qs . p / D T:

(7.23)

Then, for any l D 1; 2; : : : , the sequences q ¹x. ; r ; ˛ m /º and ¹x . ; p ; ˛ /º;

contain stepwise quasimotions x l . ; l ; ˛ l /

x l . ; l ; ˛l /

and

such that ˛ l C ˛l C 2"l 1; x l . ; l ; ˛ l / 2 B.x. /; "l /; diam l < "l ;

x l . ; l ; ˛l / 2 B.x . /; "l /; diam l < "l :

(7.24) (7.25) (7.26)

Indeed, inequalities (7.24) and (7.26) follow from (7.22) and inclusion (7.25) folq lows from the convergence of the sequences ¹x. ; r ; ˛ m /º and ¹x . ; p ; ˛ /º to x. / and x . /; respectively. Note that, according to (7.22), ˛ l ! 0;

˛l ! 0;

diam l ! 0;

and

diam l ! 0 as l ! 1: (7.27)

We introduce the functions

´ l l l Q l ; ˛Q l / D x .t; ; ˛ / xQ l .t; x l .t; l ; ˛l /

for t0 t t ; for t t T:

(7.28)

By using (7.20), (7.25), and (7.28), we get Q l ; ˛Q l / 2 B.x. /; xQ l . ; Q "l /:

(7.29)

Q l ; ˛Q l / is a stepwise motion with the following diameter We now show that xQ l . ; of partition: Q l max¹diam l ; diam l º; (7.30) diam

268

Chapter 7 Differential Equations with Discontinuous Right-Hand Side

and the number ˛ is equal to ˛Q l ˛ l C ˛l C 2"l :

(7.31)

Indeed, by using (7.28) and (7.23), we conclude that Q l W t0 0l < 1l < < l t Q0l < Q1l < < Q l l D T: s. /

(7.32)

Since the diameter of partition cannot increase as a result of the addition of a point t , relation (7.32) yields (7.30). Q l ; ˛Q l / according to (7.10) and, hence, it folThe number ˛Q l is constructed for xQ l . ; Q l ; ˛Q l / lows from (7.32) and (7.28) that the left-hand side of inequality (7.10) for xQ l . ; does not exceed ˛ l C ˛l C kx l .t; l ; ˛ l / x l .t ; l ; ˛l /k: By virtue of (7.29), the last term in this sum cannot be larger than 2"l . This yields inequality (7.31). Note that, according to (7.31) and (7.24), the number ˛Q l 1. Finally, we prove that the function x. / Q deﬁned by relation (7.20) is a quasimotion. Let l ! 1. Then, by virtue of (7.21), we have "l ! 0. In this case, it follows from (7.30) and (7.27) that Q l ! 0 as l ! 1: diam Q l ; ˛Q l / converge Moreover, it follows from (7.29) that the stepwise quasimotions xQ l . ; Q Thus, as l ! 1 (in the metric of the space M Œt0 ; T / to a continuous function x. /. by deﬁnition, the function x. / Q is a quasimotion of system (7.1) from the initial point .t0 ; x0 /. The theorem is proved. This means that if the initial point moves along any quasimotion (from a given initial point), then the appearance of new parts of quasimotions absent in the original bundle of quasimotions (from the same initial point) is impossible. Example 1 ([154]). Consider a system xP D v; where

0 t 2; x.0/ D x0 D 0;

8 ˆ 1 for x > 0; t 2 Œ0; 2; ˆ ˆ ˆ ˆ ˆ 1 for x D 0; t D 1; < v.t; x/ D 0 for x D 0; t 2 Œ0; 1/; ˆ ˆ ˆ 0 for x D 0; t 2 .1; 2; ˆ ˆ ˆ : 1 for x < 0; t 2 Œ0; 2:

269

Section 7.1 Motions and Quasimotions

The stepwise motions starting at the point .t0 D 0; x k /, where x k ! x0 D 0, generate the motions x 1 .t / 0; x 2 .t / D t; x 3 .t / D t; ´ 0 for t 2 Œ0; 1; x 4 .t / D t 1 for t 2 Œ1; 2:

t 2 Œ0; 2;

If motions start at the point .t D 1; x.t / D 0/, then a part of the previous motion D 0; 0 t 2, disappears and a new motion x 5 .t / D 1 t , 1 t 2, x appears. We now apply the proposed approach to the construction of quasimotions described above. First, the “new” motion x 5 .t / is a part of the quasimotion ´ 0 for t 2 Œ0; 1/; x.t / D x 5 .t / for t 2 Œ1; 2; 1 .t /

because x. / is the limit of the sequence of stepwise quasimotions ´ 0 for t 2 Œ0; 1/; x.t / D ık .t / for t 2 Œ1; 2; x where x ık .t / D 1 t ık ;

0 < ık < 1;

and

lim ık D 0:

k!1

Second, the “piece” of motion x 1 . / is the limit of the sequence of stepwise quasimotions ¹x.t; k ; 0; 0; k ; 0/; t 2 Œ1; 2º for k # 1 as k ! 1. In this case, the role of quasimotions is played by the functions ´ 0 for t 2 Œ0; ˛; C x˛ .t / D t ˛ for t 2 .˛; 2; ´ 0 for t 2 Œ0; ˛; x˛ .t / D ˛ t for t 2 .˛; 2: We now prove one more theorem necessary in what follows. Theorem 6. The set-valued mapping X.t0 ; x/ is upper semicontinuous with respect to x. Proof. Assume the contrary. Let the mapping X.t; x/ be not upper semicontinuous at the point x0 . Then there exists "0 > 0 such that, for any ı > 0, one can ﬁnd a quasimotion x.t; x/ 2 X.t0 ; x/, where kx x0 k < ı, such that the inequality kx.t; x/ x.t; x0 /k "0 holds for any quasimotion x.t; x0 / 2 X.t0 ; x0 /.

270

Chapter 7 Differential Equations with Discontinuous Right-Hand Side

We ﬁx a sequence ıi ! 0 as i ! 1 and construct a sequence of quasimotions x.t; xi /, where xi ! x0 as i ! 1, such that kx.t; xi / x.t; x0 /k "0

(7.33)

for any quasimotion x.t; x0 / 2 X.t0 ; x0 /. By the deﬁnition of quasimotions, for any i, there exists a sequence of stepwise i /º convergent to x.t; x / in the space M Œt ; T [under conquasimotions ¹x.t; ir ; ˛m i 0 ditions (7.18)]. For any i and “sufﬁciently large” r and m, we select a single “reprei / from each sequence such that sentative” x i .t; ir ; ˛m diam ir ! 0;

i ˛m ! 0 as i ! 1

and i / x.t; x0 /k kx i .t; ir ; ˛m

"0 2

for any quasimotion x.t; x0 / 2 X.t0 ; x0 /. The constructed sequence is a sequence of stepwise quasimotions from the initial point .t0 ; x0 /. By using the Filippov lemma [50], we select a subsequence of this sequence uniformly convergent to the function x .t; x0 / which is continuous and, hence, x .t; x0 / 2 X.t0 ; x0 /. This result contradicts inequality (7.33), which completes the proof of the theorem.

7.2

Impulsive Motions and Quasimotions

Consider an impulsive differential system of the form xP D f .t; x; v.t; x//; xj tDi D Ii .x/;

t ¤ i ;

i D 1; m;

(7.34) (7.35)

where t 2 Œt0 ; T , t0 D 0 < 1 < < mC1 D T , x 2 Rn , v 2 Rq , and Ii W Rn ! Rn are bounded continuous functions. Deﬁnition 5 ([126]). An impulsive quasimotion x. / D ¹x.t; t0 ; x0 /; t0 t T º of system (7.34), (7.35) from an initial point .t0 ; x0 / is deﬁned as a piecewise continuous function that coincides with a quasimotion x.t; i ; x.i C 0// in the intervals .i ; iC1 , i D 0; m, and satisﬁes the condition of jump (7.35) at the points of pulses. By X.t0 ; x0 / we denote a bundle of impulsive quasimotions of system (7.34), (7.35).

271

Section 7.2 Impulsive Motions and Quasimotions

Theorem 7 ([126]). The bundle X.t0 ; x0 / is a nonempty bounded (in norm) closed subset of the space M Œt0 ; T . Proof. By virtue of Theorem 3 for quasimotions, the theorem is true for t 2 .i ; iC1 . Since the functions Ii .x/ are continuous and bounded, we conclude that, at the time of a pulse, a nonempty bounded closed subset of the space M Œt0 ; T transforms into a nonempty bounded closed subset of the same space. Theorem 8 ([126]). For any points .t0 ; x0 /, the section of the bundle of impulsive quasimotions by the hyperplane t D T is a nonempty compact set in Rn . Theorem 9 ([126]). For any point .t ; x.t //, where t 2 Œt0 ; T /, x. / is an impulsive quasimotion from the bundle X.t0 ; x0 /, the following inclusion is true: X.T; t ; x.t // X.T; t0 ; x0 /: Example 2. Consider an impulsive differential equation xP D 1 2 sign x; xj tDi D x C ˛;

t ¤ i; i 2 N;

0 < ˛ < 1;

(7.36)

x.0/ D x0 0: If x0 1, then x.t / D x0 t for t 2 Œ0; 1. At the time t D 1, the solution suffers the pulse action and x.1 C 0/ D ˛. For t 2 .n; n C ˛, n 2 N , the solution x.t / D ˛Cnt . For t 2 .nC˛; nC1, the solutions do not exist in the ordinary sense and, in these segments, one can consider solutions, e.g., in a sense of the Filippov deﬁnition [51] or in a sense of quasimotions. Then x.t / 0 for t 2 .n C ˛; n C 1. If 0 x0 < 1, then, in the interval .n 1; n; n 2 N , we have the following solution: ´ A C n 1 t; t 2 .n 1; n 1 C A; x.t / D 0; t 2 .n 1 C A; n; ´ x0 ; n D 1; AD ˛; n > 1: Thus, a periodic quasimotion exists for x0 D ˛=. Example 3. Consider an impulsive differential equation xP D v.t; x/; xj t D1 D x;

t ¤ 1;

272

Chapter 7 Differential Equations with Discontinuous Right-Hand Side

where v.t; x/ is the function deﬁned in Example 1. In this case, the role of impulsive quasimotions is played by the functions ´ x ˙ .t / for t 2 Œ0; 1; ˙ x˛;ˇ .t / D ˛˙ xˇ .t 1/ for t 2 .1; 2: Theorem 10 ([126]). The set-valued mapping X.t0 ; x/ is upper semicontinuous with respect to x. The proof of the theorem follows from the deﬁnition of impulsive quasimotions and Theorem 6. We now consider the problem of existence of periodic quasimotions for the impulsive differential equation of the form xP D A.t /x C f .t; x/;

t ¤ i ;

(7.37)

xj tDi D Bi x C Ii .x/; where A W R ! Rnn is a continuous T -periodic function, f W R Rn ! Rn is a vector function T -periodic in t , f .t; x/ 2 Q 2 comp.Rn /, and the constant matrices Bi , the continuous functions Ii .x/ 2 V 2 conv.Rn /, and the times i are such that, for some natural r, BiCr D Bi ;

IiCr .x/ Ii .x/;

i Cr D i C T;

i 2 N:

It is also assumed that 0 1 < < r < T and det.E C Bi / ¤ 0 for all i D 1; r. Let ˆ.t; s/ D ˆABi .t; s/ be the matrizant of the homogeneous system corresponding to (7.37). Theorem 11 ([126]). If the spectral radius of the matrix ˆ.T; 0/ is smaller than 1, then the impulsive differential Eq. (7.37) possesses a T -periodic quasimotion. Proof. The existence of a T -periodic quasimotion of Eq. (7.37) is directly connected with the existence of a ﬁxed point of the mapping ' W x ! XT .x/, where XT .x/ is the section of the bundle of quasimotions X.0; x/ by the hyperplane t D T . By virtue of Theorem 8, the set XT .x/ is a nonempty compact set in Rn . In view of Theorem 11, the mapping ' is upper semicontinuous. We now prove that there exists a set K such that the mapping ' maps the set K onto itself. Parallel with Eq. (7.37), we consider a linear differential inclusion xP 2 A.t /x C Q; xj t Di 2 Bi x C V:

t ¤ i ;

(7.38)

273

Section 7.3 Euler Quasibroken Lines

The spectral radius of the matrix ˆ.T; 0/ is smaller than 1. Thus, inclusion (7.38) has a T -periodic R-solution R.t / whose initial set R0 is deﬁned as the ﬁxed point of the mapping Z T X .R0 / D ˆ.T; 0/R0 C ˆ.T; /Qd C ˆ.T; i /V: 0

0i
By the deﬁnition of impulsive quasimotions, we have XT .x/ R.T / D R0 for any x 2 R0 . Hence, the mapping ' maps the set R0 onto itself. By the Kakutani theorem [14], there exists a ﬁxed point x0 2 R0 of the mapping ', i.e., there exists a T -periodic quasimotion of Eq. (7.37). The theorem is proved.

7.3

Euler Quasibroken Lines

We now consider continuous systems of differential equations that do not satisfy the conditions of uniqueness theorems, e.g., p xP D x; x.0/ D 0: (7.39) It is clear that, for an arbitrarily chosen convergent subsequence of Euler broken lines, the limit of this subsequence is the function x.t / 0, although Eq. (7.39) has inﬁnitely many solutions. We show that, by using the algorithm of construction of Euler quasibroken lines based on the simpliﬁed procedure of construction of quasimotions, it is possible, for any solution of the equation, to construct a subsequence of Euler quasibroken lines convergent to this solution. Consider a differential equation xP D f .t; x/;

x.t0 / D x0 ;

(7.40)

where f .t; x/ is continuous in D W t0 t t0 C a0 ; kx x0 k b. We set a D min¹a0 ; MbC1 º, where M D maxD kf .t; x/k. We split the segment Œt0 ; t0 C a into n equal parts by the points tk D t0 C khn , hn D na , k D 0; n. The Euler quasibroken lines are constructed as follows: x n .t / D x n .ti / C .t ti /.f .ti ; x n .ti // C ˛i /; i D 0; n 1;

ti t ti C1 ;

x n .t0 / D x0 ;

where k˛i k ˛ m 1, i.e., x n .tiC1 / D x n .ti / C hn f .ti ; x n .ti // C hn ˛i :

(7.41)

274

Chapter 7 Differential Equations with Discontinuous Right-Hand Side

Remark 1. The Euler quasibroken lines can also be represented in the form Z t n n f .s; x n .ti //ds C ˛i .t ti /: x .t / D x .ti / C ti

Since ˛i are chosen arbitrarily, we obtain an inﬁnite set of Euler quasibroken lines even for ﬁxed hn . We prove that family (7.41) is uniformly bounded and equicontinuous. For tk t tkC1 and any n, we get k1 X n n kx n .t / x0 k D h .f .t ; x .t // C ˛ / C .t t /.f .t ; x .t // C ˛ / i i i k k k k n iD0

hn

k X

kf .ti ; x n .ti // C ˛i k hn .M C ˛ m /.k C 1/

iD0

hn n.M C ˛ m / a.M C 1/ b: This family is equicontinuous because kf .ti ; x n .ti // C ˛i k M C 1;

i D 0; n 1:

Hence, by the Arzelà theorem, for any ˛ m , one can select a convergent subsequence from the family of quasibroken lines,. We show that, for any solution x.t / of Eq. (7.40), there exists a sequence of quasibroken lines convergent to this solution. Let x.t / be an arbitrary solution of (7.40). We construct a sequence of inscribed broken lines xQ n .t / D x.ti / C

t ti .x.tiC1 / x.ti //; hn

ti t tiC1 ; i D 0; n 1:

Let us show that, by the choice of ˛i , one can ﬁnd a quasibroken line that coincides with xQ n .t / at the points of the partition: x.t1 / xQ n .t1 / D x.t1 / x0 hn .f .t0 ; x0 / C ˛0 / Z t1 Œf .s; x.s// f .t0 ; x0 / ds ˛0 hn : D t0

Since kf .s; x.s// f .ti ; x.ti //k ˛ m for sufﬁciently small hn (this follows from the uniform continuity of the function f .t; x.t //, there exists ˛0 such that k˛0 k ˛ m and xQ n .t1 / D x n .t1 /. Similarly, there exist ˛i , k˛i k ˛ m , such that xQ n .tiC1 / D x n .tiC1 /;

i D 0; n 1:

275

Section 7.3 Euler Quasibroken Lines

It is clear that the sequence of inscribed broken lines converges to x.t /, i.e., there exists a subsequence of quasibroken lines that converges to an arbitrary solution of Eq. (7.40). However, one can also ﬁnd subsequences of quasibroken lines that converge not to a solution of Eq. (7.40) but to a solution of the equation xP D f .t; x/ C ˇ.t /; where ˇ.t / is an arbitrary continuous function such that kˇ.t /k ˛ m . Passing to the limit as ˛ m ! 0, we exclude the superﬂuous solutions. Remark 2. Consider a differential inclusion xP 2 F .t; x/;

x.t0 / 2 X0 :

We now construct the Euler quasibroken lines for the R-solutions and t 2 .ti ; tiC1 : ³ ² Z t [ m F .s; x/ds ; (7.42) xC X .t; ˛/ D x2X˛mi .ti ;˛/

ti

h.X˛mi .ti ; ˛/; X m .ti ; ˛// ˛i : If F .t; x/ is a Lipschitz function, then, for ˛i D 0, we get ordinary Euler broken lines convergent to a unique R-solution. For ˛i ¤ 0, the quasibroken lines also converge to the unique R-solution as ˛ ! 0. If an R-solution is not unique, then, by analogy with the previous case, we can show that, for any R-solution, there exists a sequence of quasibroken lines (7.42) that converges to this solution.

Appendix A

Some Elements of Set-Valued Analysis

Let Rn be an n-dimensional real Euclidean space with norm v u n uX x2: kxk D t i D1

i

By comp.Rn /.conv.Rn // we denote the space of all nonempty compact (and convex) subsets of Rn with the Hausdorff metric h.F; G/ D min¹r 0 W F G C Br .0/; G F C Br .0/º; where Br .a/ is a ball in Rn of radius r 0 centered at a point a 2 Rn . Let .x; F / D minf 2F kx f k be the distance from a point x 2 Rn to a set F 2 comp.Rn /. The mapping . ; F / satisﬁes the Lipschitz condition j.x; F / .y; F /j kx yk

for all x; y 2 Rn :

The Hausdorff distance can also be introduced as follows: Assume that a set DF ;G consists of numbers of the form .f; G/, .g; F /, where f 2 F and g 2 G. Then h.F; G/ D sup DF ;G . In addition to the ordinary set-theoretic operations, we consider two more operations in the space comp.Rn /, namely, the operations of summation and multiplication by a scalar. Deﬁnition 1 ([22, 23]). The sum F C G of two sets F; G 2 comp.Rn / is deﬁned as the set H D F C G D ¹h D f C g W f 2 F; g 2 Gº: Example 1. Br1 .a1 / C Br2 .a2 / D Br1 Cr2 .a1 C a2 /. Deﬁnition 2 ([22, 23]). The product F of a scalar 2 R by a set F 2 comp.Rn / is deﬁned as the set G D F D ¹g D f W f 2 F º: Example 2. Br .0/ D Bj jr .0/.

277

Appendix A Some Elements of Set-Valued Analysis

Deﬁnition 3 ([22, 23]). The image of the set F 2 comp.Rn / under a linear transformation speciﬁed by a matrix A 2 Rnn is deﬁned as the set G D AF D ¹g D Af W f 2 F º: Lemma 1. If sets F; G 2 comp.Rn /.conv.Rn //, a scalar 2 R, and a matrix A 2 Rnn , then the sets F C G; F; AF 2 comp.Rn /.conv.Rn //. Lemma 2. If F; G; H 2 comp.Rn /; ˛; ˇ 2 R, and A; B 2 Rnn , then the following equalities are true: 1: .F C G/ C H D F C .G C H /I 3: F C G D G C F I 5: F C ¹0º D F I 7: ˛.F C G/ D ˛F C ˛GI 9: EF D F:

2: ˛.ˇF / D .˛ˇ/F I 4: 1 F D F I 6: A.F C G/ D AF C AGI 8: A.BF / D .AB/F I

However, the space comp.Rn /.conv.Rn // is not a linear space with respect to the operations presented above because, in the general case, it is impossible to introduce the notion of element opposite to F 2 comp.Rn /.conv.Rn // and the following property of linear spaces is not true: .˛ C ˇ/F D ˛F C ˇF:

(A.1)

In the general case, instead of this property, we have the following inclusion: .˛ C ˇ/F ˛F C ˇF: Thus, for ˛ D 1; ˇ D 1, and F D Br .0/; we ﬁnd .1 C .1// Br .0/ D ¹0º; 1 Br .0/ C .1/ Br .0/ D Br .0/ C Br .0/ D B2r .0/: However, equality (A.1) is true in the case where ˛ > 0; ˇ > 0, and the set A 2 conv.Rn /. 1 Pm Let F 2 comp.Rn / and let Mm .F / D m iD1 F . Then the following inequality holds: jF j.n C 1/ h.co F; Mm .F // ; m where jF j D maxf 2F kf k is the modulus of the set F . n Deﬁnition 4 ([127]). A sequence of sets ¹An º1 nD1 , An 2 comp.R /, n D 1; 1, is 1 called convergent to A 2 comp.Rn / if the sequence ¹h.An ; A/ºnD1 converges to zero.

Theorem 1 ([127]). The metric space comp.Rn / is a complete space.

278

Appendix A Some Elements of Set-Valued Analysis

Deﬁnition 5 ([23]). A support function of the set A 2 comp.Rn / is deﬁned as a scalar function c.A; / speciﬁed by the condition c.A; / D max.a; /; a2A

where .a; / is the scalar product of the vectors a; 2 Rn . The support set of the set F 2 comp.Rn / in the direction of the vector 0 2 Rn is deﬁned as the set of all vectors f0 2 F on which the maximum is attained in the deﬁnition of the support function U.F;

0/

D ¹f0 2 F W .f0 ;

0/

D c.F;

0 /º:

The hyperplane 0 in the space Rn speciﬁed by the relation 0 D ¹x 2 Rn W .x; 0 / D c.F; 0 /º is called the support hyperplane for the set f in the direction of the support vector 0 . The following representation is true for the support set U.F; U.F;

DF \

0/

0

0 /:

:

The hyperplane 0 splits the entire space Rn into two half spaces RC and R . The set F lies in the negative half space R relative to the vector 0 , i.e., the inequality .f;

0/

c.F;

0/

holds for all points f 2 F . Deﬁnition 6. The set F 2 comp.Rn / is called strictly convex in the direction of the vector 0 2 Rn if its support set U.F; 0 / is formed by a single point. The set F is called strictly convex if it is strictly convex in any direction. We now present the main properties of the support function. Let F; G 2 comp.Rn / and let ; 1 ; 2 2 Rn . Then (1) c.F; / D c.F; / for 0; (2) c.F;

1

C

2/

c.F;

1/

C c.F;

2 /;

(3) c.F C G; / D c.F; / C c.G; /; (4) c. F; / D c.F; /; (5) c.AF; / D c.F; AT /; (6) c.co F; / D c.F; /; T n (7) co F D 2S1 .0/ ¹x 2 R W .x; / c.F; /º; (8) if F D G, then c.F; / D c.G; / for all all 2 S1 .0/, then co F D co G;

2 Rn ; if c.F; / D c.G; / for

279

Appendix A Some Elements of Set-Valued Analysis

(9) if F G, then c.F; / c.G; / for all 2 Rn ; if c.F; / c.G; / for all 2 S1 .0/, then co F co G; T (10) if F G 6D ;, then c.F; / C c.G; / 0 for all 2 Rn ; if c.F; / C c.G; / 0 for all 2 S1 .0/, then co F \ co G 6D ;; (11) jc.F;

1 /c.G;

2 /j

(12) h.co F; co G/ D max

jF j k 2S1 .0/

1

2 kCk 1 kh.F; G/C2k 1

2

kh.F; G/;

jc.F; / c.G; /j h.F; G/;

(13) the set F is strictly convex in the direction of the vector its support function c.F; / is differentiable at the point

2 Rn if and only if 0.

0

Deﬁnition 7. A set-valued mapping is deﬁned as an arbitrary function F W Rm ! comp.Rn /, i.e., a function whose argument is a vector x 2 Rm and values are elements of the space comp.Rn /, i.e., nonempty compact sets from the space Rn . Deﬁnition 8 ([24]). A set-valued mapping F W Rm ! comp.Rn / is called measurable if, for any nonempty compact set K, the set ¹x 2 Rm W h.F .x/; K/ "º is Lebesgue measurable. Deﬁnition 9. A set-valued mapping f W Rm ! Rn is called a measurable section (a single-valued measurable branch or a measurable selector) of the set-valued mapping F W Rm ! comp.Rn / if f .x/ is measurable and f .x/ 2 F .x/ for almost all x 2 Rm . Theorem 2 ([49]). If a set-valued mapping F W Rm ! comp.Rn / is measurable, then it has a measurable single-valued branch. Theorem 3 ([48]). If F W Rm ! comp.Rn / is a measurable set-valued mapping, n 0 2 R , then there exists a measurable single-valued branch f .x/ of the mapping F .x/ that belongs to the set U.F .x/; 0 /. Theorem 4 ([49]). Let a set-valued mapping F W Rm ! comp.Rn / and a function v W Rm ! Rn be measurable. Then there exists a measurable branch f .x/ of the mapping F .x/ such that the condition .v.x/; F .x// D kv.x/ f .x/k is satisﬁed for almost all x 2 Rm . Theorem 5 ([48]). Assume that the function f W Rm Rp ! Rn is measurable with respect to x 2 Rm and continuous in u 2 Rp . Moreover, suppose that a mapping U W Rm ! comp.Rp / and a function v W Rm ! Rn are measurable and, in addition, v.x/ 2 f .x; U.x//. Then there exists a measurable branch u.x/ of the mapping U.x/ such that v.x/ D f .x; u.x//. We now ﬁx a segment I D Œt0 ; t1 and a set-valued mapping F W I ! comp.Rn /.

280

Appendix A Some Elements of Set-Valued Analysis

Deﬁnition 10 ([10]). The Aumann integral of the set-valued mapping F .t / on the segment I is deﬁned as the set ² Z t1 ³ Z t1 GD F .t /dt D f .t /dt W f .t / 2 F .t / : t0

t0

Here, the Lebesgue integral on the right-hand side is taken over all single-valued branches of the mapping F .t /, where it exists. Theorem 6 (Lyapunov [78]). Assume that the set-valued mapping F .t / is measurable and on I . Then G D R t1 satisﬁes the estimate jF .t /j k.t /, where k.t / is summable n t0 F .t /dt is a nonempty convex compact set in the space R . Note that this integral may exist even in the case where a set-valued mapping is not measurable on I because the condition of its existence is the presence of a singlevalued Lebesgue integrable branch of the set-valued mapping. Thus, the set-valued mapping ´ S1 .0/; t 2 J; F .t / D ¹0º; t 2 I n J; where J is a nonmeasurable subset of I , is not measurable on I . However, this setvalued mapping contains a Lebesgue integrable single-valued branch f .t / 0, t 2 I , and therefore, Z 02

t1

F .t / dt: t0

Theorem 7 ([10, 78]). Assume that a set-valued mapping F .t / is measurable and satisﬁes the estimate jF .t /j k.t /, where k.t / is summable on I . Then Z t1 Z t1 c F .t /dt; D c.F .t /; /dt: t0

t0

Deﬁnition 11 ([23]). A set-valued mapping F W Rm ! comp.Rn / is called upper semicontinuous at a point x0 2 Rm if, for any number " > 0, one can ﬁnd a number ı > 0 such that F .x/ F .x0 / C S" .0/ for kx x0 k < ı. Deﬁnition 12 ([23]). A set-valued mapping F W Rm ! comp.Rn / is called lower semicontinuous at a point x0 2 Rm if, for any number " > 0, there exists a number ı > 0 such that the inclusion F .x0 / F .x/ C S" .0/ holds for kx x0 k < ı.

Appendix A Some Elements of Set-Valued Analysis

281

Deﬁnition 13 ([23]). A set-valued mapping F W Rm ! comp.Rn / is called continuous at a point x0 2 Rm if it is both upper and lower semicontinuous at this point. As one of the most important properties of set-valued mappings extensively used in applications, we can mention the Michael theorem [85]. We now present one of its numerous interpretations. Theorem 8 ([8]). Let X be a metric space and let Y be a Banach space. Assume that a set-valued mapping F . / from X into a closed convex subspace of Y is lower semicontinuous. Then there exists a continuous selector f W X ! Y from F . /. Remark. The condition of convexity of F .x/ for all x 2 X is essential because if F .x/ is not convex, then even its continuity does not guarantee the existence a continuous selector for F . /. Theorem 9 ([52]). Assume that a set-valued mapping F W R Rm ! conv.Rn / is measurable with respect to t and continuous in x. Then there exists a single-valued branch f .t; x/ 2 F .t; x/ measurable with respect to t and continuous in x. Deﬁnition 14 ([51]). A set-valued mapping F W I Rm ! comp.Rn / satisﬁes the Carathéodory conditions if (a) for any ﬁxed x 2 Rm , the set-valued mapping F . ; x/ is measurable; (b) for almost all ﬁxed t 2 I , the set-valued mapping F .t; / is upper semicontinuous. Deﬁnition 15 ([51]). A set-valued mapping F .t; x/ satisﬁes the improved Carathéodory conditions if it satisﬁes condition (a) and the condition (b0 ) for almost all ﬁxed t 2 I , the set-valued mapping F .t; / is continuous. Theorem 10 ([28]). If a set-valued mapping F . ; / satisﬁes the improved Carathéodory conditions, then, for any measurable set-valued mapping Q W I ! comp.Rn /, the set-valued mapping F . ; Q. // W I ! comp.Rn / is measurable. Note that this property is not true if the improved Carathéodory conditions are replaced by the ordinary conditions. Deﬁnition 16 ([3]). A set-valued mapping F W R ! comp.Rn / is called absolutely continuous if, for any " > 0, there exists a number ı > 0 such that, for any natural N , the following inequality is true: N X

h.F .bi /; F .ai // < "

iD1

for a1 < b1 ; : : : ; aN < bN and

PN

iD1 .bi

ai / < ı.

282

Appendix A Some Elements of Set-Valued Analysis

Deﬁnition 17 ([4]). A set-valued mapping F W Rm ! comp.Rn / is called locally Lipschitz if, for any x0 2 Rm , there exist a neighborhood U.x0 / Rm and a constant L 0 such that h.F .x 00 /; F .x 0 // Lkx 00 x 0 k for any x 0 ; x 00 2 U.x0 /. Moreover, this mapping is called Lipschitz if there exists L 0 such that h.F .x 00 /; F .x 0 // Lkx 00 x 0 k for any x 0 ; x 00 2 Rm .

Appendix B

Differential Inclusions

Consider a differential inclusion xP 2 F .t; x/;

(B.1)

where t 2 I R is time, x 2 Rn is the phase vector, and F W I Rn ! comp.Rn /. Deﬁnition 1. An absolutely continuous function x.t / deﬁned on a segment (in an interval) J I is called an ordinary solution of the differential inclusion (B.1) on J if x.t P / 2 F .t; x.t // almost everywhere on J . It is known that, in the theory of differential equations, the transitions from differential equations to integral equations, and vice versa, are equivalent. For differential inclusions, this is not true, i.e., a solution of inclusion (B.1) is a solution of the integral inclusion Z t x.t / 2 x.t0 / C F .s; x.s//ds (B.2) t0

but not all solutions of the integral inclusion (B.2) are solutions of the differential inclusion (B.1). Example 1 ([34]). Let F .t; x/ Œ0; 1, x0 D 0, I D Œ0; 2 and let ´ 0; t 2 Œ0; 1/; x.t / D 2t 2; t 2 Œ1; 2: It is clear that x.t / is not a solution of the differential inclusion xP 2 Œ0; 1;

x.0/ D 0

on Œ0; 2 because x.t P / D 2 … Œ0; 1 for t 2 .1; 2/ but is a solution of the corresponding integral inclusion Z t Œ0; 1ds x.t / 2 0 C Rt

0

because 0 Œ0; 1dt D Œ0; t and 0 x.t / t for t 2 Œ0; 2. For this reason, the theory of differential equations deals with a different form of integral inclusions for which it is possible to obtain a result similar to the corresponding result in the theory of differential equations.

284

Appendix B Differential Inclusions

Deﬁnition 2 ([34]). A continuous function x.t / is called a generalized solution of inclusion (B.1) on J if the integral inclusion 00

0

Z

x.t / x.t / 2

t 00 t0

F .t; x.t //dt

(B.3)

is valid for all t 0 < t 00 W t 0 ; t 00 2 J . Theorem 1 ([34]). Assume that F W I Rn ! conv.Rn / satisﬁes the following conditions: (1) F . ; x/ is measurable for all x 2 Rn ; (2) F .t; / is continuous for almost all t 2 I ; (3) jF .t; x/j m.t /, .t; x/ 2 I Rn , m.t / is summable on I . Then the set of ordinary solutions of inclusion (B.1) coincides with the set of generalized solutions. Corollary 1. Let F W I Rn ! comp.Rn / be a set-valued mapping satisfying conditions (1)–(3) of Theorem 1. Then the set of ordinary solutions of inclusion (B.1) is contained in the set of generalized solutions. We now present some other deﬁnitions of solutions of the differential inclusion (B.1). Deﬁnition 3 ([65]). A function x.t / is called a quasisolution of the differential inclusion (B.1) if there exists a sequence of functions ¹xk .t /º1 such that kD1 (1) xk .t / is absolutely continuous on J ; (2) jxP k .t /j m.t /; t 2 J; m.t / is summable on J; k D 1; 2; : : : ; (3) limk!1 xk .t / D x.t /; t 2 J ; (4) limk!1 .xP k .t /; F .t; xk .t /// D 0 almost everywhere on J . Deﬁnition 4 ([65]). A function x.t / is a called a Riemannian solution of the differential inclusion (B.1) if x.t P / is Riemann integrable and x.t P / 2 F .t; x.t // for all t 2 J . Deﬁnition 5 ([65]). A function x.t / is called a classic solution of the differential inclusion (B.1) if x.t / is continuously differentiable on J and x.t P / 2 F .t; x.t // for all t 2 J . Since differential inclusions are obtained as a result of generalization of differential equations, all problems typical of the theory of ordinary differential equations appear in the theory of differential inclusions, namely, the problems of existence of solutions,

285

Appendix B Differential Inclusions

their extendability, boundedness, continuous dependence on the initial conditions and parameters, etc. At the same, for differential inclusions, a family of trajectories originates from every initial point. This set-valuedness leads to the appearance of various speciﬁc problems, including the closedness and convexity of the family of solutions, existence of boundary solutions, selection of solutions with given properties, and many others. First, we present some results concerning the conditions of existence of ordinary solutions to the differential inclusion (B.1) with initial condition x.t0 / D x0 . Theorem 2 ([51]). Assume that, at every point .t; x/ of the domain D D ¹t0 t t0 Ca; kxx0 k bº, a set-valued mapping F .t; x/ satisﬁes the following conditions: (1) the set F .t; x/ is nonempty and closed; (2) F . ; x/ is measurable for all x; (3) F .t; / is continuous for all t ; (4) jF .t; x/j m.t /, where m.t / is summable on Œt0 ; t0 C a. Then, for t0 t t0 C d , there exists a solution of problem (B.1), where Z t d a; '.t0 C d / b; '.t / D m.s/ds: t0

Deﬁnition 6 ([23]). A function !.t; r/ 0 .t t0 ; 0 r b/ is called a Kamke function if it is continuous in r, measurable with respect to t, !.t; r/ m0 .t /, where m0 .t / is summable on the segment Œ0; c for any c, and the function r.t / 0 is a unique solution of the problem rP .t / D !.t; r.t //;

r.t0 / D 0;

for t t0 . Thus, if the function k.t / is summable, then k.t /r is a Kamke function .0 r b/. Theorem 3 (Filippov [23, 105]). Assume that, at every point .t; x/ of the domain D D ¹t 2 Œt0 ; T ; kxx0 k bº, a set-valued mapping F .t; x/ satisﬁes the following conditions: (1) the set F .t; x/ is nonempty and closed; (2) F . ; x/ is measurable for all x; (3) the set F .t; x/ is convex; (4) for any r > 0, kx yk r, and almost all t, h.F .t; x/; F .t; y// w.t; r/; where w.t; r/ is a Kamke function.

(B.4)

286

Appendix B Differential Inclusions

In addition, let the function y.t / be absolutely continuous for t 2 Œt0 ; T , let its graph be contained in D, y.t0 / D y0 , and let, for almost all t 2 Œt0 ; T , .y.t P /; F .t; y.t /// .t /; where .t / is summable on Œt0 ; T . Then, for .t0 ; x0 / 2 D, one can ﬁnd a solution x.t / of the problem xP 2 F .t; x/;

x.t0 / D x0 ;

such that kx.t / y.t /k r.t /;

kx.t P / y.t P /k w.t; r.t // C .t /

(B.5)

for almost all t 2 Œt0 ; t , where r.t / is the upper solution of the problem rP D w.t; r/ C .t /;

r.t0 / D kx0 y0 k;

and t is an arbitrary number such that .t; x.t // 2 D for t0 t t . Remark 1. If, in Theorem 3, condition (4) is replaced by the Lipschitz condition, i.e., h.F .t; x/; F .t; y// k.t /kx yk;

k.t / is summable on Œt0 ; T ;

then condition (3) can be removed and, in inequalities (B.5), we have Z t Rt Rt k.s/ds C .s/e s k./d ds: r.t / D kx0 y0 ke t0 t0

Deﬁnition 7. An integral funnel of the point .t0 ; x0 / (of the set K/ is deﬁned as a set of points lying on the graphs of all solutions passing through this point (resp., through the points of the set K/. Deﬁnition 8. A section t D t 0 of the funnel of the point .t0 ; x0 / is deﬁned as a set of attainability at time t 0 , i.e., as the set of points ¹x.t 0 /º that can be attained at time t 0 by moving along all possible solutions originating at time t0 from the point x0 . A section of the funnel of the set K is deﬁned similarly. Theorem 4 ([23,33]). Assume that the following conditions are satisﬁed in a bounded domain D: (1) the set F .t; x/ is nonempty and closed; (2) jF .t; x/j m.t /, where m.t / is a function summable on Œt0 ; t1 ; (3) F .t; / is upper semicontinuous on D;

287

Appendix B Differential Inclusions

(4) F . ; x/ is measurable on D; (5) the set F .t; x/ is convex. If all solutions of (B.1) on the segment Œt0 ; t1 exist and are contained in D, then the set HF .t0 ; x0 / of these solutions is a compact set in the space C Œt0 ; t1 . The same is true for the set HF .K/ of all solutions with all possible initial conditions .t0 ; x0 / 2 K, where K is a compact set and K D. If K is a connected compact set (and, in particular, if K is a point), then the set HF .K/ is connected. If the set F .t; x/ is not convex, then both the funnel and the set of attainability can be nonclosed. Example 2 ([48]). Consider a system xP D y 2 C u2 ;

yP D u;

1 u.t / 1:

(B.6)

Here, the set F is an arc of the parabola v1 D v22 y 2 ;

1 v2 1;

i.e., is not convex .v1 and v2 are the projections of points of the set F onto the coordinate axes). For 0 t 1, we consider the set of solutions with initial conditions x.0/ D y.0/ D 0. If y.t / 0, then u.t / D 0 everywhere, xP D y 2 C u2 D 0, and x.t / 0. If y.t / is not identically equal to zero .0 t 1/, then xP D y 2 C u2 1. Moreover, xP < 1 in the intervals where y.t / ¤ 0. Hence, x.1/ < 1 for all solutions and the point t D 1; x D 1; y D 0 belongs neither to the graphs of solutions, nor to the segment 0 t 1 of the integral funnel. We now consider a solution xk .t /; yk .t / for which xk .0/ D 0, yk .0/ D 0, and ´ 2i 1; t < 2ikC1 ; k uD i D 0; 1; 2; : : : : t < 2ikC2 ; 1; 2iC1 k In this case, 0 yk .t /

1 ; k

xP k .t / 1

1 ; k2

xk .1/ 1

1 : k2

Thus, points of the graphs of solutions with trivial initial conditions lie arbitrarily close to the point t D 1; x D 1; y D 0, whereas the point itself does not lie on the graph of this solution. Hence, the set of these points and the segment 0 t 1 of the funnel are not closed. Equations (B.6) can be regarded as equations of a controlled system, i.e., a system whose motion can be controlled by an arbitrary choice of the function u.t / within the prescribed limits. This means that, for a unit period of time, this system cannot

288

Appendix B Differential Inclusions

be transferred from the state x D y D 0 into the state x D 1, y D 0 but can be transferred into a state arbitrarily close to x D 1; y D 0 by changing the function u.t / sufﬁciently rapidly from 1 to 1 and back (sliding mode). In the absence of the condition of convexity of the set F .t; x/, the relations between the sets of solutions of the inclusion xP 2 F .t; x/ and the inclusion xP 2 co F .t; x/

(B.7)

were studied, e.g., in [31, 157]. Theorem 5 ([31, 104]). Assume that a set-valued mapping F .t; x/ satisﬁes the conditions: (1) the set F .t; x/ is nonempty and closed; (2) jF .t; x/j m.t /, where m.t / is a summable function; (3) F .t; / is upper semicontinuous in x; (4) for any r > 0, kx yk r, and almost all t , h.F .t; x/; F .t; y// w.t; r/; where w.t; r/ is a Kamke function. Then each solution of inclusion (B.7) with the initial condition x.t0 / D x0 is the limit of a uniformly convergent sequence of solutions of the inclusion xP 2 F .t; x/ with the same initial condition. In this case, the indicated limit may be not a solution of the inclusion xP 2 F .t; x/ if the set F .t; x/ is not convex.

Figure 1.

Thus, if x 2 R and the set F .t; x/ consists of two points 1 and 1, then the sequence of solutions ¹xk .t /º uniformly converges to the function x.t / 0, which is not a solution of inclusion (B.1) (Figure 1). Condition (4) cannot be removed [104] and replaced by the Hölder condition.

Appendix B Differential Inclusions

289

Example 3 ([104]). Assume that the setpF .t; x/, t 2 R, x 2p R2 , does not depend on 2 2 t and consists of two points .1; x1 C jx2 j/ and .1; x1 C jx2 j/. Then co F .t; x/ is the segment connecting these points. The vector function x.t / 0 satisﬁes the inclusion xP 2 co F .t; x/ but does not satisfy the inclusion xP 2 F .t; x/. In [104], it is shown that none of the sequences of solutions of the inclusion x 2 F .t; x/ has the limit x.t / 0, i.e., the solution of the inclusion xP 2 co F .t; x/. We now consider inclusion (B.1), where F W D ! conv.Rn / is a set-valued mapping continuous in D. The section of the integral funnel by the plane t D const is a closed set R.t / depending on t . Thus, the funnel is the graph of the set-valued function R.t /. The following approach to the determination of this function is proposed in [99–102]: Deﬁnition 9 ([23, 101, 102, 144]). A set-valued function R.t / is called an R-solution generated by the differential inclusion (B.1) if, for any t , the set R.t / is closed, the function R.t / is continuous, and, for all t, [ 1 h R.t C /; ¹x C F .t; x/º ! 0 as # 0: (B.8) x2R.t/

Theorem 6 ([101]). Assume that, for any t; x, F .t; x/ is a convex compact set continuous in the collection of its variables as a set-valued mapping. Then there exists > 0 such that the R-solution generated by the set-valued function F .t; x/ exists in the half interval Œt0 ; t0 C /. Theorem 7 ([101]). Assume that F .t; x/ satisﬁes the Lipschitz condition in a certain neighborhood S.R0 / of the set R0 2 comp.Rn /. Then the indicated solution is unique for all t t0 for which the R-solution R.t / .R.t0 / D R0 / is deﬁned and R.t / S.R0 /. Moreover, this solution continuously depends on the initial set R0 . Theorem 8 ([101]). Let R.t / W for t 2 Œt0 ; T , where the set W is open and bounded, and let F .t; x/ be a Lipschitz function in W . Then, for t 2 Œt0 ; T , the set R.t / is the set of attainability from R.t0 / D R0 at time t . Theorem 9 ([23,144]). For any compact set K Rn , there exists an R-solution with the initial condition R.t0 / D K. The integral funnel is the graph of the R-solution R.t /. If condition (B.4) is satisﬁed, then the R-solution with the initial condition R.t0 / D K is unique and continuously depends on K, and the graph of R.t / is an integral funnel. p Example 4. Let F .t; x/ D 2Œ˛; ˇ x; x.0/ D x0 ; 0 ˛ ˇ. For x D 0, the mapping F .t; x/ does not satisfy the Lipschitz condition.

290

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Let x0 D 0. We now show that a set-valued mapping 8 ˆ for 0 t t1 ; <0 2 2 R.t / D Œ0; ˇ .t t1 / for t1 t t2 ; ˆ : 2 2 2 2 Œ˛ .t t2 / ; ˇ .t t1 / for t2 t is an R-solution of the differential inclusion p xP 2 2Œ˛; ˇ x;

x.0/ D 0;

i.e., satisﬁes the equation [ p 1 ¹x C 2Œ˛; ˇ xº D 0: lim h R.t C /; #0

(B.9)

x2R.t/

For 0 t < t1 , it is clear that R.t / 0 satisﬁes (B.9). Let t1 t < t2 . Then [ p 1 2 2 ¹x C 2Œ˛; ˇ xº lim h Œ0; ˇ .t C t1 / ; #0 2 2 x2Œ0;ˇ .t t1 /

1 h.Œ0; ˇ 2 .t t1 /2 C 2ˇ 2 .t t1 / C ˇ2 2 ; #0

D lim

Œ0; ˇ 2 .t t1 /2 C 2ˇ 2 .t t1 // D 0: Assume that t2 t . Then 1 lim h Œ˛ 2 .t C t2 /2 ; ˇ 2 .t C t1 /2 ; #0 [ p ¹x C 2Œ˛; ˇ xº x2Œ˛ 2 .t t2 /2 ;ˇ 2 .tt1 /2

1 h.Œ˛ 2 .t t2 /2 C 2˛ 2 .t t2 / C ˛ 2 2 ; #0

D lim

ˇ 2 .t t1 /2 C 2ˇ 2 .t t1 / C ˇ 2 2 ; Œ˛ 2 .t t2 /2 C 2˛ 2 .t t2 /; ˇ 2 .t t1 /2 C 2ˇ 2 .t t1 // D 0: For t1 D 0 and t2 D 1, the R-solution R.t / coincides with the integral funnel X.t /. All other R-solutions corresponding to arbitrary values t2 t1 0 are such that X.t / R.t /. If x0 > 0; then the R-solution p p x0 x0 2 2 ;ˇ t C R.t / D ˛ t C ˛ ˇ is unique and coincides with the integral funnel.

291

Appendix B Differential Inclusions

A similar example is constructed in [9]. Later, Panasyuk generalized the notion of R-solutions to the case of right-hand sides F .t; x/ measurable with respect to t and continuous in x. Deﬁnition 10 ([23,101,102,144]). A set-valued function R.t / is called an R-solution generated by the differential inclusion (B.1) if, for any t , the set R.t / is closed, the function R.t / is absolutely continuous, and Z t C [ 1 ¹x C F .s; x/dsº ! 0 . # 0/ (B.10) h R.t C /; t x2R.t/

for almost all t . There are several approaches used for the investigation of stability of differential inclusions. These approaches differ by the objects of investigation. Thus, by analogy with the theory of ordinary differential equations, the ﬁrst approach is based on the analysis of stability of separate trajectories [49, 51]. At present, there exists another approach aimed at the description of dynamics of the sets speciﬁed by differential inclusions. Within the framework of this approach, the R-solutions are used for the investigation of stability. Deﬁnition 11 ([107]). An R-solution R.t / .t0 t < C1/ of the differential inclusion xP 2 co F .t; x/ (B.11) is called Lyapunov stable if, for any " > 0; there exists ı."/ > 0 such that (1) all R-solutions X.t / of inclusion (B.11) satisfying the condition h.Y .t0 /; F .t0 // < ı

(B.12)

are deﬁned for all t > t0 ; (2) the following inequality holds for all solutions satisfying inequality (B.12): h.Y .t /; F .t // < ": Deﬁnition 12 ([107]). An R-solution R.t / .t0 t < C1/ of the differential inclusion (B.11) is called asymptotically stable if: (1) it is Lyapunov stable; (2) for any R-solution X.t / satisfying the inequality h.Y .t0 /; F .t0 // < ı; the following relation is true: lim h.Y .t /; F .t // D 0:

t !1

292

Appendix B Differential Inclusions

Deﬁnition 13 ([23]). A solution .t / .t0 t < C1/ of the differential inclusion (B.1) is called stable if, for any " > 0; one can ﬁnd ı > 0 such that, for any xQ 0 satisfying the inequality kxQ 0 .t0 /k < ı, every solution x.t Q / with the initial condition x.t Q 0 / D xQ 0 exists for t0 t < C1 and satisﬁes the inequality kx.t Q /

.t /k < ":

Deﬁnition 14 ([23]). A solution .t / .t0 t < C1/ of the differential inclusion (B.1) is called weakly stable if, for any " > 0, there exists ı > 0 such that, for any xQ 0 satisfying the inequality kxQ 0 .t0 /k < ı, a solution x.t Q / with the initial condition x.t Q 0 / D xQ 0 exists for t0 t < C1 and satisﬁes the inequality kx.t Q /

.t /k < ":

Deﬁnition 15 ([23]). A solution .t / .t0 t < C1/ of the differential inclusion (B.1) is called asymptotically stable if (1) it is stable and (2) satisﬁes the condition kx.t Q /

.t /k ! 0 as t ! 1:

Deﬁnition 16 ([23]). A solution .t / .t0 t < C1/ of the differential inclusion (B.1) is called weakly asymptotically stable if (1) it is weakly stable and (2) satisﬁes the condition kx.t Q /

.t /k ! 0 as t ! 1:

Example 5. Consider a differential inclusion xP 2 ˛x C Œ1; 1;

x.0/ D x0 ;

where ˛ is an arbitrary parameter. In this case, the R-solution can be represented in the form R.t / D Œx1 .t /; x2 .t /; 1 ˛t 1 e ; x1 .t / D x0 C ˛ ˛

1 ˛t 1 x2 .t / D x0 e C : ˛ ˛

Any solution x.t / of this differential inclusion satisﬁes the relation x1 .t / x.t / x2 .t /. For ˛ < 0, the R-solution is asymptotically stable. For any value of ˛; the stable ordinary solution x.t / does not exist. At the same time, for any value ˛ < 0, every solution x.t / is weakly asymptotically stable.

293

Appendix B Differential Inclusions

Example 6 ([23]). Consider a differential inclusion xP 2 Œ˛; ˇx;

x.0/ D x0 ;

where ˛ and ˇ are arbitrary constants. For the solution of the differential inclusion x.t /, we can write the following inequality: x0 e ˛t x.t / x0 e ˇ t ;

x0 0:

If x0 D 0, then the solution x.t / is asymptotically stable for ˛ ˇ < 0, stable for ˛ ˇ D 0, weakly asymptotically stable for ˛ < 0 < ˇ, weakly stable for ˛ D 0 < ˇ, and unstable for 0 < ˛ ˇ. The R-solution is asymptotically stable for ˛ ˇ < 0 and stable for ˛ ˇ D 0 for any x0 .

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Index

Approximation of a bundle of solutions, 131 of an integral funnel, 130 Aumann Integral, 43 Aumann integral of a set-valued mapping, 280 Autonomous oscillating system, 169 Averaged inclusion, 236 Beating of a solution of a system, 5 Bundle of motions, 258 of solutions, 67 Carathéodory condition, 281 Case nonresonance, 169 resonance, 169 Cauchy problem, 7 Cauchy–Schwarz inequality, 49 Compact set, 9 connected, 287 convex, 289 Conjugate system, 39 Connectedness property, 10 Continuous dependence, 10 Continuous selector, 281 Convex combination of corner points, 72 Counterexample Kononenko, 258 Subbotin, 260 Degree of freedom, 169 Differentiability in Hukuhara’s sense, 152 Differential inclusion, 42 Discontinuity of the ﬁrst kind, 2 Discontinuous cycles, 186 Discontinuous dynamical systems, 3

Equation integrodifferential, 193 matrix, 26 Equations averaged, 184 of the ﬁrst approximation, 184 Equivalence class of, 151 relation of, 151 Estimation of an error, 126 Euler broken line, 126 Euler quasibroken lines, 273 Evolution process, 1 Family of bounded solutions, 176 Fixed point of an operator, 2 Frequency of oscillations, 187 Function absolutely continuous, 42 almost periodic, 198 equicontinuous, 238 Green, 38 Kamke, 43 limiting, 181 matrix, 37 periodic, 177 summable, 42 uniformly continuous, 238 vector, 37 Games differential, 257 positional, 257 Hölder condition, 288 Hausdorff metric, 152 Hukuhara derivative, 124 Hyperplane, 7, 278

306 Impulsive differential equations, 1 Integral curve, 2 Integral funnel, 10 of a point, 42 of a set, 42 Iterative method, 176 Jordan cell, 93 Linear impulsive differential inclusions, 66 Linear periodic system, 36 Linear subspace, 25 Linear systems, 23 homogeneous, 23 inhomogeneous, 23 Liouville–Ostrogradskii formula, 27 Lipschitz condition, 11 Mapping bijective, 22 bounded, 228 compact, 228 convex, 228 integrally continuous, 230 isometric, 151 measurable, 42 measurable branch of, 67 set-valued, 42, 229 measurable branch of, 279 upper semicontinuous, 42 Matrix, 24 degenerate, 24 diagonal, 90 eigenvalues of, 36 extended, 110 inverse, 27 nondegenerate, 24, 90 of monodromy, 36 rank of, 25 real canonical form of, 90 spectral radius of, 90 Matrix norm, 80 Matrizant, 26 Measurable selector, 49 Method of averaging, 169, 227 “Mortal” systems, 3

Index Motion, 1, 257 stepwise, 258 Multiplier, 36 Multivalued pulses, 44, 220 Operator, 1 linear, 25 set of images of, 25 restriction of, 15 operator bijective, 3 of shift, 8 one-to-one, 2 Optimal control, 5 Oscillating process, 169 Oscillator, 184 Periodic system, 36 Phase vector, 66 Piecewise-continuous function, 7 Point accumulation, 2 limit, 229 of discontinuity of a function, 7 Polyhedron, 72 Problem of control, 131 Process of successive changes, 170 Quasimotion, 257 stepwise, 260 R-solution of a differential inclusion, 43 Representative point, 1 Section of a bundle of quasimotions, 266 Sequence of functions, 68 equicontinuous, 68 uniformly bounded, 68 Set compact, 42 connected, 42 convex, 42 integral, 185 invariant, 175 of attainability, 42, 131 projection of, 93 strictly convex, 278

307

Index support function of, 127 toroidal, 175 Set of “death” of a trajectory, 3 Set of states of a process, 1 Solution absorbed, 2 asymptotically orbitally stable, 219 bounded, 216 boundedness of, 8 extendable, 47 nontrivial, 36 nonunique, 235 of a system of equations, 2 of inclusion, 46 periodic, 37 stability of, 8, 30 stationary, 173, 184 upper, 46 weakly extendable, 47 Solutions fundamental system of, 25 linear combination of, 26 linearly dependent, 26 linearly independent, 39 Space complete, 151 Euclidean, 1 extended phase, 1 functional, 14 linear, 25

basis of, 25 metric, 151 quotient, 151 vector, 25 Stability Asymptotic, 31 in the ﬁrst approximation, 33, 86 of a solution, 8 Strategy, 257 Sufﬁcient condition for the absence of beating, 17 Switching point, 6 Switching surface, 6 System of differential equations, 1 Tangent cone, 48 Theorem Arzelà, 68 Bogolyubov, 235 Filippov, 44, 55 Krasnosel’skii–Krein, 229 Kronecker–Capelli, 41 Lyapunov, 69 Michael, 281 on existence and uniqueness, 66 Picard–Cauchy, 23 Topological product, 1 Trajectory of a motion, 4 Vector, 1 Velocity of a phase point, 169

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