Journal o.[ Mathematwal S,'iem'e.s. Vol. 83. No. 4. 1997
P R I N C I P L E S OF A F F I N I T Y IN N O N L I N E A R P R O B L E M S N. A. Bobylev and M. A. Krasnosel'sky
UDC 517.988.63
Introduction When we investigate specific problems of nonlinear analysis, we often use equations that are different in principle. For instance, when seeking the conditions for oscillation of a nonlinear system, we either try to find the fixed points of the Poincar&-Andronov operator that acts in the phase space of the system or seek the solutions of different equivalent integral equations considered in different functional spaces. A similar situation may occur in the theory of boundary-value problems, namely, we can reduce the problem of solvability and an approximate construction of the solution of a boundary-value problem for a differential equation to the analysis of operator equations with different operators that possess special properties. Problems from the calculus of variations can also serve as an example. For instance, we can either carry out the investigation of a degenerate extremal of an integral functional (say, by the methods of field theory) in the space on which the functional is defined or reduce the problem to the investigation of the critical point of a function of a finite number of variables that is defined on the subspace of the degeneration. We can give more examples of this kind. Every approach to the specific problem in question has its advantages and drawbacks, and therefore the picture we obtain when investigating a problem becomes more complete if we manage to find internal connections between different approaches and use the positive aspects of each of them. The theory of affinity that studies the relationships between the topological characteristics of solutions of various equations encountered in the process of investigating a problem is one of the instruments for discovering these connections. Below we give general theorems that serve as the basis for the affinity theory and indicate some of its applications. The article is supplied with a bibliography.
1.
R o t a t i o n of a V e c t o r Field
1.1. C o m p l e t e l y c o n t i n u o u s v e c t o r fields. This section contains all the necessary information concerning the theory of rotation of vector fields. The fields that we consider below are either finite-dimensional or completely continuous. Since every finite-dimensional field is completely continuous, we only consider the properties of completely continuous fields. Suppose that E is a real Banach space and M C E is a set in it. A completely continuous vector field is a mapping r M ~ E which has the form r = I - A, where A: M , E is a completely continuous operator. The point x0 is a zero of the field r if ~(z0) = 0. If the field ~ has no zeros on M , then it is said to be nondegenerate on M. The completely continuous vector fields r = I - A0 and r = I - A1, defined on the set M C E, are homotopicon M if there exists a one-parameter family of nondegenerate fields r A) = I - A ( . , A)(1 < A < 1) such that the operator A: M x [0,1] , E is completely continuous and r = r ~(-, 1) = r In what follows, we denote by ~Po r the superposition of the fields ~ and r so that (~P o ~)(z) = ~P[r 1.2. Rotation and its p r o p e r t i e s . Let f~ C E be a bounded domain on whose boundary 0f~ a nondegenerate completely continuous vector field r is defined. Then the rotation 7(r Of~) of the field r on 0f~ is defined. The rotation 3'(~, 011) is an integer-valued topological characteristic possessing the following properties. i
ii
i
i
i i
i
ii
i
i
i
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. T,'maticheskie Obzory, Vol. 22, Dynamical Systems-3, 1995. 1072-3374/97/8304-0485 $18.00 9
Plenum Publishing Corporation
485
1. The rotation of the completely continuous vector fields r
and r
homotopic on Off, is the same:
(r 0fl)= ~(,,,0fl). 2. Suppose that fi~ C fi are pairwise nonintersecting subdomains and the field r is nondegenerate on \ Uifii. Then
oa,).
oa) = 3. If the field ~ is defined and nondegenerate on f~, then
~(r o n ) = 0 . 4. Let Zo E fi be an isolated zero of the field ~. The topological indez ind (zo, ~) of the zero xo is the rotation of the field r on the spheres S(r, zo) of small radii r > 0. If the field r is nondegenerate on OF/and has a finite number of zeros z 1 , . . . , z~ in fl, then Irl,
0fl) - I ;
ind (xi, ~).
i=1
5. Suppose that ~ is a completely continuous field defined and nondegenerate on 0f~ and 9 is a completely continuous field that is defined on E and has a single zero zo = 0. Then the completely continuous vector field ~ o ~ is nondegenerate on 0fl and
"7(~ o r
~ , ( ~ , 0 a ) ind (0, ~).
6. Suppose that Eo is a subspace of the space E and the completely continuous operator A: F/ ; E0 does not have any fixed points on Off. Let A0 be a restriction of the operator A to the closure of the domain flo = fi gl E0. Then the vector fields = I - A and ~o = I - Ao are defined, respectively, on F / a n d flo and the equality
~(~, 0 f l ) = ~(~o,0ao) holds (if no = o, then 7(~o, 0flo) = 0). 7. Suppose that Eo and Ex are real Banach spaces and r: Eo ; Ex is an isomorphism of Eo onto El. Let the completely continuous vector field r be defined and nondegenerate on the boundary 0flo of the bounded domain flo C Eo. Then the completely continuous vector field q~l - r o r o r -1 is nondegenerate on the boundary 0f~: of the domain fl: = r(flo) C E: and
~(~0, 0flo) - ~(~,, 0fl,). 8. Suppose that 1 is not an eigenvalue of the linear completely continuous operator A: E the zero Zo = 0 of the completely continuous vector field ~ = I - A is isolated and ind(0,~) =
, E. Then
(-i)~(a),
where fl(A) is the sum of multiplicities of the real eigenvalues of the operator A which are greater than 1. 9. Suppose that the space E is the direct sum of the subspaces Eo and E1 and Po: E ~ Eo, 1='1: E ~ El are projection operators for which Pox = 0 for z E E1 and P l z = 0 for z E Eo. Suppose, furthermore, that fl0 C Eo and fll C E1 are bounded domains in the subspaces Eo and E:. The product fl = flo x fll of the domains flo and f~l is a domain in E which ~:onsists of points of the form z = Zo + x l , Xo ~_ flo, z: E ill. The direct sum ~ = ~o @ ~1 of the fields ~o and ~1 defined, respectively, on fl and fil is the vector field
r
= ~o(Po~)+ ~,(P,~),
9 e ~o x fl, = h,
defined on ~. If the fields r and q~l are nondegenerate on 0flo and 0ill, respectively, then the field 9 = ~o@~1 is nondegenerate on c3fl and 7 ( r Off) - 7(r 0flo). ~(~,, 0fl,). 486
0
T h e P r o b l e m on Forced Oscillations of N o n l i n e a r S y s t e m s
2.1. T h e shift o p e r a t o r . Let us consider the differential equation dx
= f(t,
e
a ")
(2.1)
with a continuous right-hand side which is T-periodic with respect to t. We are interested in the problem of finding T-periodic solutions of Eq. (2.1), i.e., solutions z(t) such that z(t + T) ___-z(t) and, in particular, z(0) = z(T).
(2.2)
Let us assume that every initial condition z E RN defines the unique solution p(t,z) (p(O,z) = 0) of Eq. (2.1) that can be extended to the interval [0, T]. Then, in the phase space Rtr of Eq. (2.1), the operator
(2.3) uCz) = pCT, z) is defined, which is known as the Poincarg-Andronov operator or shift operator along the trajectories of Eq. (2.1). Under these conditions, the fixed points of the shift operator (and only these points) are the initial conditions for the T-periodic solutions of Eq. (2.1). Therefore, the search for T-periodic solutions of Eq. (2.1) reduces to the search for the zeros of the vector field ~o = I - U considered in the phase space ~N. 2.2. An e q u i v a l e n t i n t e g r a l e q u a t i o n . Another approach to the problem on T-periodic solutions of Eq. (2.1) consists in the reduction of this problem to finding the solutions of equivalent integral (or integrofunctional) equations. The following equation is the simplest one:
(2.4) 0
The solutions of Eq. (2.4) coincide with the T-periodic solutions of Eq. (2.1), or, what is the same, with the zeros of the vector field (I) = I - A, where A is an operator defined by the right-hand side of Eq. (2.4). Equation (2.4) and, together with it, the field (I), are studied, in the sequel, in the space C[O, T] of the vector functions z(t), with a uniform norm, continuous on [0, T]. In this space the field (I) is completely continuous. The essential advantages of the methods of investigation of the T-periodic problem for (2.1), based on the study of the vector field ~ = I - U with the operator (2.3), consist in the finite dimensionality of the problem, in the geometric visuality, and in the possibility of carrying out effective computations of the field qo. The drawback of the method is that the operator U, in explicit form, is, as a rule, unknown. It exists only under restrictive requirements placed on the right-hand side of Eq. (2.1). The advantages of the methods that use the integral equation (2.4) are that the corresponding operator A and the field (I) are defined by explicit formulas which do not need any assumptions of the uniqueness of the solutions of the Cauchy problem or of the local continui,y of the~. oolutions for their construction. But it is difficult to carry out a direct computation or estimation of the rotation of the corresponding fields in infinite-dimensional spaces, which is the main difficulty of these methods. The theory of affinity, which establishes the relationship between the rotations of the fields ~ and (I), makes it possible to compensate for the drawbacks of every one of the indicated methods by the advantages of some other method. Already the first applications of this theory have led to new theorems on the existence of periodic oscillations in distributed argument systems (and, in particular, in time-lag systems), to new effective tests for the convergence of various computation procedures in the problem of approximate construction of periodic solutions and in the investigation of a number of problems concerning systems with hysteresis nonlinearities, etc. 2.3. T h e f u n d a m e n t a l t h e o r e m . Now we shall give the main result. We say that the bounded domains G C n~N and fl C C[0, T] have the same core if the fields tp and are nondegenerate, respectively, on OG and 0fl and if the set of initial values of the T-periodic solutions of Eq. (2.1) that lie in G coincide with the set of values for t = 0 of all T-periodic solutions of this equation lying in f/. 487
Theorem
2.1.
Let the d o m a i n s G C R Iv and fl C C[O, T] have the same core. Then the relation
~(~, 0a) = ~(r 0a)
(2.5)
holds. Proof.
We set I
Im
(2.6)
\I
~E~, O
and consider the domain f't, = {z e C[O,T]: z(O) e a , Ilxllcto,Tn < m + 1}.
(2.7)
It follows from the definition of the domain fix that the field ~ is nondegenerate on 0f/1 and the sets of zeros of the field ~ coincide on fi and ill. Therefore, the field ~ is nondegenerate in the domains fl \ f l n ill, 12a \ 9t f'l f~l and, in accordance with properties 2 and 3 of rotation, we have
~(~,o~) =~(~,on,).
(2.8)
Let us consider the completely continuous family of vector fields
r
~) =
a)u(~(0))-
z ( t ) - A z ( T ) - (1 -
ff(~,~(~))d~
(0 _< A _< 1)
(2.9)
0
on 0fla. We shall show that the fields of family (2.9) are nondegenerate on 0fix. r Ao) = 0 holds for certain Zo E 0fl~ and Ao E [0, 1], then
Indeed, if the relation
t
zo(t)
= Aozo(T)+ (1
- ao)U(,o(0))+
fj~(,,,o(,))d~.
(2.to)
0
It follows from this relation that zo(t)is a solution of Eq. (2.1). But then U(xo(0)) = xo(T) and relation (2.10) assumes the form t
Zo(t) = zo(T) +
f 0
Therefore, Xo is a zero of the field ~, and this contradicts its nondegeneracy on 0ill. Thus the fields r A) (0 _< ), _< 1) are nondegenerate on 0f/~ and, consequently, are homotopic to each other. Thercf..re, according to property 1 of rotation we have
~(~,on,) =~(~,,on,),
(2.11)
where t
r
= ~(t)- u(~(0))-
ff(~,~(~))d~.
(2.12)
0
Let us now consider on 0ffa the completely continuous family of vector fields
9 (~, a) = ~ ( t ) - u(~(0)) - s~ o ff(~,
=(~))d~
(o <_~ <_T),
(2.13)
0
where
&(~)= 488
{~(t)
ifO<_t
~(~) i f A < t < T .
(2.14)
We shall show that the fields qJ(., A) (0 < A < T) are nondegenerate on Of/,.
Indeed, if the equality qJ(zo, Ao) = 0 is satisfied for certain zo E Off, and Ao E [0, T], then either z0(0) E OG or IlZollctO,Tl
=
(2.15)
m + 1.
If the inclusion x0(0) e. OG holds, then, setting t = 0 in the relation
=o(t) =
U(=o(0)) + &o o
ff(,,
=o(~)) d~,
(2.16)
0
we find that xo(O) = U(zo(O)), and this contradicts the nondegeneracy of the field ~ on OG. But equality (2.15) is also impossible since, by virtue of Eq. (2.16), the function zo(t) is a solution of Eq. (2.1) on the interval [0,,~0], and it follows from Eqs. (2.6) and (2.14)that
II=oIIcto,T] S m. Thus the fields (2.13) are nondegenerate on oqf~l and, consequently, the field ~a is homotopic to the field (2.17)
,~(=) = =(t) - u(=(o)).
Therefore,
(2.18) But the field ~1 is generated by the operator acting from C[0, T] into the subspace Eo C C[0, T] of constant functions. Therefore, by virtue of property 6,
~(~,, o~,) = ~(~o, O~o),
(2.19)
where ~o is the restriction of the field ~1 to Eo and
flo = {= e Eo: =(o) ~ c}. We denote by r: Eo ---} R
Iv
the isomorphism that associates any constant function x E Eo with its value.
Then @o = r -1 o qa o r and f~o = r-l(G). Therefore, by virtue of property 7,
0~o) = 7(:, oc)
7(r
(2.20)
The chain of relations (2.8), (2.11), (2.18), (2.19), and (2.20)implies (2.5). We have proved the theorem. Let Zo be an isolated zero of the field ~p. Then the function z0(t) = p(t, zo) is an isolated singular point of the field ~. Theorem 2.1 implies the following corollary. Corollary
2.1.
The relation ind (Zo, :) = ind (Zo(.), ~)
(2.21)
holds. 2.4. A d d i t i o n a l r e m a r k s . There are many techniques of reducing the problem on T-periodic solutions of Eq. (2.1) to an integral equation. For instance, the following equations are equivalent to this problem: !
T
=(t) = =(o) - j(:(,, =(,)) d,, T
=(t) = =(o)+ J'/(,, =(,))d. + 0
(2.22)
t
jr:(s,=(,))ds
(2.23)
0
489
1
T
t
(2.24) 0
0
Suppose that Eq. (2.1) has the form dlg
dt
=A(t)z+g(t,z)
(x 6 R/v),
(2.25)
where A(t)is a T-periodic (N x N)-matrix and g: [0, T] x R/v ~ RN is a vector-valued function T-periodic with respect to t. If the homogeneous equation d2
dt
=A(t)z
(2.26)
has only a trivial T-periodic solution, then the finding of T-periodic solutions of Eq. (2.25) is equivalent to the finding of the solutions of the integral equation T
(2.27)
9 (t) = fa(t, 0
where G is the Green function:
X(t)(I
X(T)
'X(T)X-*(s)
if 0 < t < s < T.
(2.2s)
Here X(t) is the fundamental matrix of the linear system (2.6). In particular, if the matrix A(t)is constant, i.e., A(t) = A, then Eq. (2.27) assumes the form T
9 (t) =/c(t-
(2.29)
0
where G(t) is a T-periodic function, defined on the whole axis, for which C(t) = ( I - eAT) -' e At
(0 <_ t <_ T).
(2.30)
Statements similar to Theorem 2.1 hold for every one of the fields corresponding to Eqs. (2.22)-(2.24), (2.27), (2.29).
Q
Oscillations in A u t o n o m o u s Systems
3.1. T h e first r e t u r n f u n c t i o n and the index of a cycle. The problem of finding periodic solutions of
the autonomous equation
d2
dt
=f(z)
(z e R N)
(3.1)
has certain peculiarities. First, the period of the periodic solution being sought is a priori unknown (in the nonautonomous case it is either equal to o~, is a multiple of the period of the right-hand side). Second, the periodic solutions of autonomous systems which are different from equilibrium states are nonisolated, i.e., if zo(t) is a periodic solution of Eq. (3.1), then every function zo(t + h) is also a periodic solution. This circumstance excludes the possibility of introducing a topological index of the periodic solution of an autonomous system by means of the standard procedure described in the preceding section. Therefore, in the autonomous case we define the indez of a cycle. The construction of the introduction of this index is based on the concept of the first return function introduced by Poincar4.
690
Suppose that zo(t) is a T-periodic solution of Eq. (2.1), different from the equilibrium position, which is associated, in the phase space, with the isolated cycle F. Let us consider any hyperplane H which passes through the point x0 = z0(0) E F and is transversal to F. Let z be a point of the hyperplane I1 sufficiently close to the point z0. Since II is transversal to F, there is a unique t = t(z), close to T, for which p(t(z), z) E H, where p(t,z) is a solution of Eq. (3.1) with the initial condition p(0, z) = z. The first return function is the map P(x) = p(t(z),z). (3.2) The continuous mapping P is defined in the small neighborhood W C H of the point x0 and x0 is its isolated fixed point. The topological index ind F of the cycle F is the topological index ind (Xo , ~o) of the field
r
= x - P(z)
(z e U C 11).
This index does not depend either on the choice of thepoint x0 E F or on the hyperplane H transversal to F. Therefore, the definition of the topological index of the cycle is correct. 3.2. T h e m e t h o d of f u n c t i o n a l i z a t i o n of a p a r a m e t e r and t h e f u n c t i o n a l c h a r a c t e r i s t i c of a cycle. The definition of the topological index of the cycle of system (3.1) is based on the constructions of this system in the phase space. The reduction of the problem on the oscillations of system (3.1) to an integral equation makes it possible to introduce one more characteristic of the cycle F, namely, its functional characteristic based On the constructions in the infinite-dimensional functional space. Here is a scheme of introduction of this characteristic. Along with Eq. (3.1), we shall consider the following equation with the scalar parameter ~:
dz dt = Af(x).
(3.3)
If the function z(t) is a T-periodic solution of Eq. (3.1), then the function z(Tt) is a 1-parameter solution of Eq. (3.3) which corresponds to the value 2 = T of the parameter. The finding of the periodic solutions of Eq. (3.1) is equivalent to the finding of the values of the parameter for which Eq. (3.3) has periodic solutions with fixed period 1, and the subsequent construction of these solutions. In turn, this problem is equivalent to the finding of ~ > 0 for which the integral equation t
=(t) =
+
/ f(=(,))
ds
(3.4)
0
has solutions. From the solutions z(t) of Eq. (3.4) we can find A-periodic solutions of the equation z(t) = of system (3.1). Equation (3.4) is solvable, generally speaking, only for certain (a priori unknown) values of the parameter )~ which are the periods of the periodic solutions of Eq. (3.1). In addition, the solutions of Eq. (3.4) are not isolated, namely, with every solution, all the shifts of this solution with time enter into the set of solutions. Therefore, in order to find the topological index of the solution of Eq. (3.4), we need additional constructions, one of which, the method of functionalization of the parameter, is described below. Suppose that x0(t) is a T-periodic solution of Eq. (3.1) which is associated with the isolated cycle F in the phase space. Then the integral equation (3.4) has the solution yo(t) = xo(Tt) for )~ = T. In the space C[0, 1] of vector functions x(t), continuous on the interval [0, 1], we consider the continuous functional )~(x) satisfying the conditions ,
h(A(yo(t + h)) - T) > 0
=
T,
(h # O, Ih} <_ ~),
(3.5) (3.6)
where r is a positive number. We can make sure by a direct verification that the function y0 = y0(t) is an isolated zero of the completely continuous vector field t
~(x) = x(t)- x(1)- A(x)/f(z(s))ds
(3.7)
0
491
considered in the space C[O, 11. Therefore, the topological index ind(yo(.),r the zero yo(') of the fi.eld is defined. The index ind (yo('), r does not depend either on the choice of the functional A(z) satisfying conditions (3.5), (3.6) or on the solution zo(t) of Eq. (3.1) corresponding to the cycle F. We call this generic value of the topological index a functional characteristic of the cycle F and denote it by X(F). 3.3. T h e affinity t h e o r e m . The topological index of the cycle F and its functional characteristic are closely related. Theorem
3.1.
The relation ind r
=
x(r)
(3.8)
holds. Proof. We fix a point Xo E F and a certain hyperplane H which passes through xo and is transversal to F. On a small neighborhood U of the point z0 we define the continuous functional p (x) possessing the properties
~(zo) = T, p(,(~),~) e n
(3.9)
(3.10)
(~ e U).
Next, we consider the vector field
r
= = - p(~(~),
~)
(ze U).
Let the neighborhood U be so small that the point Zo is the only zero of the field ~ o n U . p(#(z), z) = P(z)
Since
(z e U N H),
(3.11)
where P is operator (3.2), it follows, by virtue of property 6 of rotation, that the relation ind (z0, r
= ind F
(3.12)
holds. Let us now consider the functional :~(=(t)) = s,(=(o))
(3.13)
on a small neighborhood V C C[0, 1] of the function y0(t) = p(tT, Xo). By virtue of (3.9) and (3.10), the functional A possesses properties (3.5) and (3.6), and therefore
(3.14)
X(F) = ind (y(-), (I)0), where ind (y(.), Oo) is a topological index of the zero yo(') of the completely continuous vector field t
(3~5)
(Po(X) - x(t) - x(1) - #(x(0)) f f ( x ( s ) ) d s . 0
Let us consider on V the completely continuous family of vector fields t
(I)(z, r) = x(t) - rp(#(z(O)),x(O)) - (1 - r ) x ( 1 ) - # ( z ( 0 ) ) f f ( ~ ( ~ ) ) d ~
(~ e V,
0 <_ ~ <_ 1).
(a.~6)
0
A direct verification shows that the function y0(') is a uniformly isolated, with respect to r E [0, 1], zero of the fields of family (3.16). Therefore, t
(3.17)
ind (yo('), (I)0) = ind (yo('), (I),), where t
(I),(z) = (I)(z, 1) = z ( t ) -
p(l~(x(O)),z(O)) - /~(x(O))/f(x(s)) ds. 0
492
(3.18)
Let us now consider the completely continuous family of vector fields
~(~,~) =
~ ( t ) - p(,(~(0))
ff(~(~)) d~
, ~(0)) - s, o ~(~(0))
(z E C[O, 1], 0 < r < 1),
(3.19)
0 where
x(t)
if 0 _ t < r,
s,(,) = { x(r)
(3.20)
if r < t <__ 1.
The function ~(~ T)
f yo(t)
/
y0(r)
if 0 _< t _< ~,
if r < t < l
is a uniformly, with respect to r E [0, 1], isolated zero of the field @(-, r), and therefore ind (z(., 0), @(., 0 ) ) = ind (x(., 1), @(., 1)). But ~(-, 1 ) = r
(3.21)
z(., 1) = yo('), and, consequently, (3.22)
ind (z(., 0), q'(., 0)) = ind (yo('), r Properties 6 and 7 of rotation imply the relation
(3.23)
ind (x(., 0), @(., 0)) = ind (xo, r Relations (3.12), (3.14), (3.17), (3.22), and (3.23)imply (3.8). We have proved the theorem.
Forced
G
4.1.
A linear
link with
a rational
Oscillations
fractional
in C o n t r o l
transfer
Systems
function.
The linear link W with a rational
fractional transfer function is defined by two coprime polynomials L(p) = pt + aapt-1 + . . . + at,
(4.1)
(b0 # 0)
M (p) = bop" + b~p~- ~ + . . . + b~
(4.2)
wit h real coefficients. We assume that 1 > m. A transfer function is the ratio (4.3)
W(p) = M(p)/L(p).
The differential equation d is also related to the link W. In order to describe the link
W~
(4.4)
we construct a system of differential equations dz = Az + 7u(t), dt
(4.5)
z E ~t,
(
"where 0 0
1
0
...
0
0
1
...
0
A __
"~'1 7~ ,
0 --al
0 --al-1
0 --al-2
...
1
...
--al
7 =
71 493
the components of the vector 7 being defined by the relations "/I : "[2 --
= 7t-m-1
7 t - , ~ = bo,
"[i-m-
= O,
1 -- ~l-m
a I
= bl,
7t + 7t- ~a ~ + ... + 7t-,7, am = b,,, . The space R t is the state space of the link W. For the locally integrable input u(t) every initial state z(O) = z0 E Rt defines the law of variation of the state
z(t)
= (z(t), z0).
(4.6)
Finally, z(t) can be used to find the scalar output x(t) by means of the relation
(4.7) in which 1
0 C
-
-
0 0 It should be emphasized that the linear link W is not an operator the output x(t) is defined not only by the input u(t). Equation (4.4) appears when we describe the link W for the following reason. When the inputs u(t) are smooth, relation (4.7) establishes a one-to-one correspondence between the solutions z(t) of the vector equation (4.5) for the dynamics of the states of the link and the solutions z(t) of the scalar equation (4.4) which can be outputs of the link. When u(t) are not smooth, Eq. (4.7) is the definition of the solutions (generalized solutions!) of Eq. (4.4). Note that when we study the link W, for a number of important reasons we must use precisely the initial states in the initial problems. Below we assume that the numbers
2rri 4ri O,:k--~--,-F T '
2kTri "" T " ' "
(4.8/
are not roots of polynomial (4.1). Therefore, every T-periodic locally integrable input u(t) is associated with the only T-periodic output x(t) of the link W. This output is defined by the relation T
9 (t) - fh(t
-
0
The scalar-valued function u(t, T) is defined on the whole axis, is T-periodic with respect to the variable t, and h(t,T) = ( ( I - e A T ) - l e A t T , c ) (0 < t < T). (4.9) Function (4.9) is known as an impulse-frequency characteristic of the linear link W with the rationalfractional transfer function (4.3). 4.2. T h e p r o b l e m on forced oscillations. Let us consider a control system whose dynamics is described by the equation 9
)
(4.101 in which f ( t , z ) satisfies the Lipschitz condition with respect to the variable z and the periodicity condition
f ( t + T,x) = f ( t , x ) 494
with a fixed period T. In systems of this kind the forced oscillations are T-periodic solutions of Eq. (4.10). The forced oscillations of x(t) are defined by Eq. (4.7) from the T-periodic solutions z(t) of the system dz
dt
= Az + "If(t, (c,z))
(4.11)
(z e ~').
One of the important problems of control theory consists in finding the conditions for the existence of forced oscillations and in constructing the algorithms for approximating these oscillations. The forced oscillations coincide with the solutions of the integral equation T
f h ( t - s;T)f(s,z(s))ds,
(4.12)
0
where h(t, T) is the impulse-frequency characteristic (4.9). 4.3. The affinity t h e o r e m . Suppose that Eq. (4.10) has an isolated T-periodic solution x0(t). Then the function xo(t) is an isolated zero of the completely continuous vector field
r
(4.13)
(z e C[0, T]),
in which the completely continuous operator A : C[0, T] ; C[0, T] is defined by the right-hand side of Eq. (4.12). Let ind (z0(.), ~) be the topological index of this zero. The T-periodic solution Zo(t)of Eq. (4.10) corresponds to the T-periodic solution zo(t) of system (4.11). In this case, the point z0 = z0(0) is an isolated zero of the finite-dimensional vector field
=
(z E Rt),
- U(z)
(4.14)
where U is the shift operator at the moment of time T along the trajectories of system (4.11). Consequently, we have defined the topological index ind (z0, qo). T h e o r e m 4.1.
Let L(p) be a Hurwitz polynomial. Then the relation ind (z0(.), r
(4.15)
= ind (Zo, ~)
holds. Proof. In the space C[0, T] of vector-valued functions z(t) = {zx(t),...,zt(t)}, completely continuous on [0, T], we consider the completely continuous family of vector fields t
z(t) - eAtU(z(O)) -- f eA('-~
, (c, z(s))) ds
if0
0
=
(4.16)
z(t) - eaXU(z(O)) -/eA(X-')Tf(s, (c, z(s))) ds ifA < t < T . 0
It is easy to make sure by direct verification that the function za = z(t, A), defined by the relation
{ o(t) = o(a)
if0_
is a uniformly, with respect to A e [0, T], isolated zero of the field r (., A). Therefore, ind (z(., 0), ~,(., 0 ) ) = ind (z(., T), r},(., T)),
(4.17)
where r
o) =
-
U(z(O)),
(4.18) 495
C,(z, T) = z(t) - en'U(z(O)) - - / e a ( ' - ' } v f ( s , (~, ~(~))) &.
(4.19)
0
Properties 6 and 7 of rotation imply the relation ind (z(., 0), r
0)) = ind (Zo, ~o),
(4.20)
and it follows from Eqs. (4.17), (4.20), and z(.,T)= zo(.) that ind (z0,qo) = ind (z0(.), C,(-,T)).
(4.2x)
Let us now consider on C[0, T] the completely continuous family of vector fields T
~' (AU(z(0))+ (1
r
A)(I
-
eaT)-' f ea(T-')vf( ,, (~, z(,))) d,)0
fear
(z e C[0,T], 0 < A < 1)
(4.22)
0
and show that the function z0(.) is a uniformly, with respect to A E [0, 1], isolated zero of the field ~2(', A). If we assume the contrary, we find sequences of numbers A, E [0, 1] and functions z,, E C[0, T], z,(t) z0(t), (n = 1,2,...), for which 2imoo Ilz. - zollct0,rj-- 0, (4.23) r
A,,) = 0.
(4.24)
It follows from Eq. (4.24) that the functions z,(.) are solutions of system (4.11) for which T
z,~C0) = A, uCzn(0)) + (1 - A,)(I - eaT) -1
feA(T-')vfCs, Cc,z.Cs)))&.
(4.25)
0
Since z,(t) is a solution of system (4.11), it follows that
z,,(T)=U(z,.,(O))
(4.26)
and T
z.(T) = ~Arz.(O) + f ~ -
s)Tf(s, (c, z,,(s))) ds.
(4.27)
0
Relations (4.25)-(4.27) imply
( I - A,,eaT)(z,,(O) - z,,(T)) = O,
(4.28)
and since L(p) is a Hurwitz polynomial, the matrices I - A,,e AT are invertible. Therefore, Eq. (28) implies the relations z,,(O) = z,~(T). (4.29) Consequently, z,~(t) are T-periodic solutions of system (4.11). But then Eq. (4.23) contradicts the isolatedness of the solution zo~t). Thus the function zo(t) is a uniformly isolated, with respect to A E [0, 1], zero of the field r in C[0, T]. Therefore, ind (z0(-), r 0)) - ind (z0(.), r 1)). /
But r
1)= r
T), and, consequently. ind (Zo(.), ~2(., 0)) = ind (Zo(.), ~,(., T)).
496
(4.30)
We set
(~(t),...,z,(t)}
~,(t)=~(t), In the coordinates {x(t),y(t)}, the field r
assumes the form
r where
= y(t).
= {r
B(~)},
T
B(z) = p(eAt(I - eAT) -'/eA(T-')Tf(s,X(S))
ds -
O
/
)
eA(t-')Tg(s,x(s)) ds ,
0
and P is an orthoprojector from I~t onto the subspace of vectors with the vanishing first component. Let us consider the completely continuous family of vector fields
+~(z, a) = (r
(0 < ~ _ 1).
u - aB(~)}
(4.31)
For each value of the parameter ~ e [0, 1] the field r ~) has a zero {xo,$B(xo)} uniformly isolated with respect to A. Therefore, (4.32) ind (Zo('), ~3(', 1)) = ind (vo('), r 0)), where Vo(t) = {xo(t), 0},
+~(., ~) = r r But the field rotation,
(I)3(" ,0) is
(4.33)
0),
(4.34)
0) = {r
the direct sum of the field 9 and the unit field. Therefore, by virtue of property 9 of ind (vo(.), r
0)) = ind (z0, r
(4.35)
The chain of relations (4.21), (4.30), (4.32), (4.33), (4.35) implies relation (4 .15). We have proved the theorem.
.
Free Oscillations in Control S y s t e m s
5.1. S t a t e m e n t of the problem. Here we shall study control systems whose dynamics, in contrast to Eqs. (4.10) and (4.11), is described by the differential equation
(5.~) and the system dz
dt
= Az + 7f((c,z))
(z e R').
(5.2)
We use here the notation from Sec. 4.1. In particular, the solutions of Eqs. (5.1) and (5.2) are related as
~(t) = (~,~(t)). Periodic solutions that are different from equilibrium states are called free oscillations. Since the nonlinear equations (5.1) and (5.2) are autonomous (the nonlinear function f(z) does not depend explicitly on time), it follows that the periods of free oscillations are a priori not known. This circumstance essentially complicates "the fundamental problem of studying free oscillations and explains its additional difficulties as compared to problems on forced oscillations. A free oscillation is described by the T-periodic (of a certain period T > 0) solution xo(t)of Eq. (5.1). We can write this solution as xo(t) = (c,zo(t)),where zo(t)is a T-periodic solution of system (5.2) different from the equilibrium state. In turn, the solution z = zo(t) is associated with the cycle F of system (5.2) in the phase space ~t. The same cycle F is associated with all T-periodic solutions zo(t + h). 497
Below we assume that the cycle [' under study is isolated. Therefore, the topological index ind [' of the cycle F and its functional characteristic x(F) are defined. In this case, by virtue of Theorem 3.1, ind F = x(F). 5.2. T h e t o p o l o g i c a l c h a r a c t e r i s t i c . Let us define the topological chat'acteristic of the T-periodic solution zo(t) and establish its relations with the index of the cycle F. The construction of this characteristic is based on the method of functionalization of a parameter. Along with Eq. (5.1) we shall consider the equation ld
ld
(5.3)
0. The finding of periodic solutions of Eq. (5.1) is equivalent to the finding of positive values of the parameter ,~ for which Eq. (5.3) has periodic solutions with period 1 and the subsequent construction of these solutions. If the numbers (4.8) are not zeros of the polynomial L(p) (as we assume in the sequel), then the last problem is equivalent to the finding of the solutions {A, z(t)} of the integral equation
with a parameter
A >
=(t)= f h(t- s,A)f(x(s))ds.
(5.4)
0
Here (cf. Sec. 4.1), ,~ > 0 and the kernel
h(t, ~) is periodic with respect to t with period 1,
h(t,)~) = ((I - eA(~))-leA('~)t, c) 0
0
A(A) =
-at
(O
1
0
...
0
0
1
...
0
al-I
a|-2
~
~
...
1),
(5.5)
-
0 m
0
u
1 C
-
(5.6)
bo
-
0
+
boa l ,~l-1 -
bl,
(5.7) al
+... I
boam ~l--rn
=bm.
Since Eq. (5.1) has a T-periodic solution xo(t), it follows that for ,~ = T the integral equation (5.4) has a 1-periodic solution yo(t) = xo(Tt). This solution is not isolated, i.e., besides yo(t) every one of the functions yo(t + h) (h E ~) is also a 1-periodic solution of Eq. (5.4). The transition to an isolated solution will be carried out by means of the functionalization of the parameter in Eq. (5.4). 9 Let us consider the continuous functional ,~(z) which is defined on a certain neighborhood U C [0, 1] of the point y0 = yo(t) E C[0, 1] and satisfies conditions (3.5), (3.6). A direct verification shows that the function yo(t) is an isolated solution of the equation 1
=(t) = f h(t 0
498
ds
(5.8)
and, consequently, an isolated zero of the completely continuous vector field
r
= ~(t)- f
(5.9)
h(t - s, A(x))f(x(s)) ds
0
defined in C[O, 1]. The index ind (y0, r of this zero does not depend on the choice of the functional 2(x) that satisfies conditions (3.5), (3.6). We call it a topological characteristic of the T-periodic solution zo(t) of Eq. (5.1) and denote it by #(z0). 5.3. T h e affinity t h e o r e m . Theorem
5.1.
Suppose that L(p) is a Hurwitz polynomial. Then the relation (5.10)
ind F = #(x0)
holds. Proof. We divide the proof of Eq. (5.10) into stages, and at every stage establish the equality of the topological indices of the zeros of the auxiliary vector fields. The chain of these relations leads to Eq. (5.10). 1. Suppose that the functional A(z) on the right-hand side of Eq. (5.9) satisfies conditions (3.5), (3.6). We set v(z) = A((c, z(t))) and consider, in the space C[0, 1] of vector-valued functions z(t), the completely continuous vector field t
~(~) =
z(t)-
~(~) -/(A(.(~))~(,) + 7(.(z))f((~. z(~))))d,,
(5.11)
0
where the matrix A(A) and the vectors with 7(A) are defined by Eqs. (5.5)-(5.7). The function wo(t) = zo(Tt), where zo(t) is a T-periodic solution of system (5.2), is an isolated zero of the field @, and its topological index ind (wo('), @) coincides with the functional characteristic x(F) of the cycle F. Using the scheme of the proof of Theorem 3.1, we can establish the equality (5.12)
ind F = A(F). Therefore, in order to prove Theorem 5.1, it is sufficient to establish the equality
(5.13)
ind (wo(.), qJ) = ind (yo('), (I)). 2. Let us consider the completely continuous family of vector fields t
- fe'A(~(~))('-')((1
-- r)A(v(z)))z(s)+
0
7(v(z))f((c, z(s))) ds
(, ~ c[o, ~l, o 5 ~ _ 1).
(5.14)
A direct verification shows that the function wo(t) is a zero of the field ~,(., r) uniformly isolated with "respect to r E [0, 1]. Therefore, ind (wo(.), (I),(., 0)) = ind (wo(.), (I), (., 1)). But r
0)= r
and, consequently,
ind (wo('), ~) = ind (wo(.), r
I)).
(5.15) 499
3. Let us consider the completely continuous family of vector fields l
r
z(t)- eA(~('))t(Tz(1)) + (1
=
-
~)(I - ~,c~c.)))-,/~c~c.))c,-,).rC~(z)) f((c,z(s)))ds0
t
/~,c~(.))c,-.).y(,(~))/((~, z(,))) d,
(~ e c[0, ~], 0 < ~ < ~).
(s.~6)
0
We shall show that the function wo(t) is an isolated zero of the field r respect to r E [0, 1]. Indeed, let the function z~(t) be close to wo(t)in C[O, 1] and
uniformly isolated with
(s.~7)
~=(z,, "r) = O. Then
zl(t)
is a solution of the system dz
dt where vl
=
v(zl).
(5.~8)
= A(v,)z + "7(v,)f((c, z)),
Therefore,
z,(t)
-
~*c~,)'z,(O)+/~c .,)c,-.)~(,,):((~, z,(~))) d~ 0
and, consequently, 1
z,(~) = ~,c~,),z,(O) + f~,c~,)c,-.).y(,,)/((~, z,(~)))d~.
(5.~0)
0
On the other hand, it follows from Eq. (5.17) that
z,(O) = rz,(~) + (1
-
r)(I-
~c~,))-,
:~,c~,)c,-.)~(~,)/((~, z,(~)))d~.
(5.20)
0
Relations (5.19)and (5.20)imply (I - ,~*c~,))(z,(0)
- ~,(~))
- 0.
(5.2~)
Since L(p) is a Hurwitz polynomial, the matrix A and, consequently, the matrix A(ul) are Hurwitz matrices. But then the matrix I - re A(~') is invertible, and it follows from Eq. (5.21) that zl(t) is a 1-periodic solution of Eq. (5.18). Therefore, u, = T and z,(t)= wo(t). Thus the function wo(t) is a uniformly isolated zero of all fields of family (5.16). Consequently, ind (w0('), ~ ( ' , 0)) = ind (w0('), r
But r
1)).
1) = ~1 (', 1), and therefore ind (wo(.), r
= ind (wo(-), r
1)).
4. Let us consider the completely continuous vector field
r
: 0
500
~.c,,c.):)c,-.)..,,(.(z))/((,:, z (~))) d.~-
(5.22)
/ ea("(=))C'-')'t(V(z))f((c, zCs)))ds 0
and set = =(t),
In the coordinates {x(t), y(t)}, the field r r
=
v(t).
O) has the form o) = { r
(5.23)
y - c(=)},
where 1
t
C(x) = p(eA(~('))'(I-eA(~(=)))-~ 0
0
and P is an orthoprojector from ~t onto the subspace of vectors with vanishing first component. Let us finally consider the completely continuous family of vector fields
=
(0_< r_< 1).
(5.24)
For every value of the parameter r 6 [0, 1] the field Ca(', r) has a zero {yo, rC(yo)} uniformly isolated with respect to r. Therefore, ind (Wo(.), Ca(', 1)) = ind (Vo(.), Ca(-, 0)), (5.25) where Vo(t)= {yo(t), 0}. But the field Ca(', 0)is the direct sum of the fields r and I. Therefore, property 9 of rotation implies the relation ind (Vo(.), Ca(', 0)) = ind (yo('), ~). (5.26) Since Ca(', 1) = r
0), it follows from Eqs. (5.25) and (5.26) that ind (wo(.), 02(', 0)) = ind (yo('), r
(5.27)
5. Relations (5.15), (5.22), and (5.27)imply (5.13). This means that relation (5.10) holds. We have proved the theorem.
11
T h e T h e o r e m on a S p l i t t a b l e Field
6.1. D o m a i n s w i t h t h e s a m e core. Suppose that E0 and E1 are Banach spaces, r = I - A0, and Cx = I - A1 are completely continuous vector fields defined in E0 and E1 respectively. We assume that the sets Mo and M: of zeros of the fields r and el are in a one-to-one correspondence by f : Mo , Ml. Two bounded domains flo C Eo and fix C E1 are domains with the same core if the fields CI)o and r are nondegenerate, respectively, on 09~o and 0fix, with the zero zl of the field Cl lying in fll if and only if xl = fzo and Zo 6 f/o. The relations that connect the rotations of the completely continuous vector fields r and ~1 on the boundaries of the domains 120 and 9tl, which have the same core, are known as affinity principles. "6.2. L e m m a s . Let r and r be completely continuous vector fields defined in Eo and E1 respectively. The following lemma follows directly from the properties of rotation. L e m m a 6.1. Let the domain ~o C Eo have the same core as each of the domains ~t C E1 and G1 C El. and 7(#Pl,OG,) of the field ~l on the boundaries of the domains ~ and G~ are Then the rotations 7 ( r the same. 501
Suppose that the Banach space E is the direct sum of the subspaces Eo and El, i.e., (6.1)
E = E o e El.
We denote by Po: E , E0 and ,~ E * E1 the projection operators corresponding to representation (6.1) and by u and v, respectively, the elements of the subspaces Eo and El. Then Pou = u and Pov = O, Plu = 0 and Ply = v. Let us consider the completely continuous vector field ep(z) = z -
Ao(z) - A,(x)
(z e E),
defined in E, where Ao: E * Eo and A l : 2 ~ E1 are completely continuous operators. H: Eo ~ E is a continuous bounded operator such that for every fixed u E Eo the equation x=u+AI(x)
(6.2) Suppose that (6.3)
has a unique solution which is determined by the relation (6.4) The field (6.2) that possesses the indicated property is said to be splittable. Lemma
6.2.
The point x. is a zero of the field (6.2) if and only if x.=H(u.),
(6.5)
where u. is a zero of the completely continuous vector field (~o(U) = u -
Ao o H ( u )
(6.6)
in the subspace Eo.
Proof.
The relation z . = Ao o H ( z . )
and the properties of the operator H imply the chain of equalities g ( u . ) = u. + Al o H ( u . ) = mo o H ( u . ) + A1 o g ( u . ) .
Therefore, the point z.-H(u.)
is a zero of the field (I). Conversely, z . = Ao(z.) + A , ( z . )
implies the equality z. - H o Ao(Z.).
Therefore, A o ( z . ) = Ao o H o Ao(z.)
and, consequently, the point u.-
Ao(x.)
is a zero of the field (I)0. We have proved the lemma. Thus, the domains fi C E and f/o C Eo' have the same core if there are no zeros of the fields (I) and (I)0, respectively, on their boundaries 0fl and 0fl0 and if every zero z. E f / o f the field 4)1 can be represented by relation (6.5), where u. is a zero of the field (bo lying in flo, and every zero u. E f / o f the field (I)o can be represented as u. = Pox.,
where x. is a zero of the field (I)x lying in ft. 6.3. T h e affinity t h e o r e m . 502
Theorem 6.1. Let fi C E and flo C Eo be domains with the same 7(~0, 0flo) of the fields ~p and dpo are related as 7(r 0fl) = ind (v., r162
COre.
Then the rotations 7 ( r Off) and
(6.7)
0flo),
where v. = H(O) E E1 is the only zero of the completely continuous vector field
r (z) = z - A, (z)
Proofi
(z e E).
(6.8)
Let us consider the domain fl(r) in the space E defined by the relation a ( r ) = {x e E :
Pox e 12o, I1~11.< "}
(6.9)
> o).
The boundary OFt(r) of the domain ~(r) is, obviously, the union of two sets M1 and M2 defined by the relations (6.10) M, = {z E E : Pox E 0flo, I1~!1 r}, (6.11) It follows from Eq. (6.9) that the domains flo and [2(r) have the same core if the number r satisfies the inequality IITYI %11 (6.12) r > sup IlntU)ll. uEt%
We assume that inequality (6.12) is valid. By virtue of Lemma 6.2, it is sufficient to prove relation (6.7) for the domain F / = fl(r). Let us consider the completely continuous family of vector fields ~(x, A)
=
x
--
(1
--
A)Ao(z) - AAo o H o Pox - Aa(z)
(x E 0~(r),
0 < A < i).
(6.13)
We shall show that this family is a homotopy between the field r and the field r
= z-
(6.14)
Ao o H o P o x - A , ( z ) .
It is sufficient to verify that the fields of family (6.13) are nondegenerate on 0ff(r). Indeed, let the relation ~(Zo, ~o) = 0 be satisfied for certain Xo E 0ff(r) and $o E [0, 1]. Then
(6.15)
Zo = (1 - Ao)Ao(zo) + AoAo o H o Pozo + Aa(zo). It follows from Eq. (6.15) that
&zo = Al(zo) or
Zo = Pozo + A1 ( Zo ).
But then zo = H o Pozo.
Consequently, Zo = (1 - Ao)Ao(xo)+ AoAo o H o Pozo + A~(zo) - Ao(zo)+ A~(zo), i.e., ~(z0) = 0, end this contradicts the npndegeneracy of the field ~ on Off(r). Let us consider a new completely continuous family of vector fields 9 (x, A) = x - Ao o H o P o x - A ~ ( z - APoz + Av.)
(x E Off(r),
0 < A < 1).
We shall show that for large r the fields of family (6.16) are nondegenerate on Off(r). Indeed, the relation (zo,
= 0
e ofl( ), 0 <
<
(6.17) 503
implies
P,(xo -
AoPozo + Aov.) = A,(zo - AoPozo + AoV.),
whence it follows that Zo - AoPozo + Av. = H o P o ( z o - AoPozo + AoV.) = H((1 - A)Pozo).
(6.18)
Relation (6.18) implies (6.19) II,.,.11 + IIH((1 - ,X)Poxo)ll. Suppose that the domain ao lies in the ball !1=il ~ o. Then inequality (6.19) implies the estimate
I1=oll <_ IIPoxoll
+
(6.20)
I1=oll <_ p + !1~.11+ sup IIn(~)ll. II~ll_
On the other hand, it follows from Eq. (6.17) that PoZo = Ao
o
H
o
(6.21)
Pozo.
Therefore, if
," > p + Iiv.ll + sup IIH(u)ll, Ilull<__p
(6.22)
then relations (6.20), (6.21) show that the point x0 does not belong to any one of the sets M1, M2 of which the boundary Off(r) of the domain fl(r) consists. Thus, if inequality (6.22) is satisfied, then the vector field ~(.,0) or, what is the same, the field ~, is homotopic on 0ff(r) to the field t~,(z) = z - Ao o H o P o x -
Ax(z-
Poz + v.) + v . .
(6.23)
Let us consider the completely continuous vector field 9 o(z) = z - A , ( P l z + v.) + v.
(zEE).
(6.24)
This field has a unique zero which is the zero of the space E. Let ind (0, ~0) be the index of this zero. A direct verification shows that the field ~a is the superposition of the fields ~o and X(z) = z-
(6.25)
Ao o H o Poz.
Therefore, in accordance with property 5 of rotation, we have 7(~,,Ofl(r)) - ind(O, ~ o ) 7 ( X , Ofl(r)).
(6.26)
But the fields 9 and ~, are homotopic on Off(r), and therefore
7(~, ofl(~))= ~(~,, ofl(~)),
(6.27)
and in accordance with property 6 of rotation we have
7(x, oa(~)) - ~(~o, aflo).
(6.28)
Relations (6.26)-(6.28)imply
~(r oa(~))
=
ind (0, ~o)7( ~o, Oflo).
(6.29)
Therefore, in order to prove relation (6.7), it remains to show that (6.30)
ind (0, ~o) = ind (v., r
In order to prove the last relation, we shall consider the family of completely continuous vector fields O(z, A) = z - A , ( z - APoz + Av.) + Av. 504
(O<_A_< 1),
(6.31)
which is a homotopy between the fields (I)l and ~o, and show that the fields of this family are nondegenerate on the spheres Ilzll = n with large radii R. Indeed, if the relation O(zo, Ao) = 0 is valid for certain Zo and Ao E [0, 1], then Poxo
~
0
and, consequently, Therefore, if
n _> I1~.11+ 1, then the fields (I)1 and ~o are homotopic on (6.30). We have proved the theorem.
m
But then properties 1 and 4 of rotation imply relation
S(R).
T h e Affinity T h e o r e m s for B o u n d a r y - V a l u e P r o b l e m s
7.1. T h e g e n e r a l i z e d shift o p e r a t o r . In what follows, we denote by Ct[0, T] the space of vector functions z(t), I times continuously differentiable on the closed interval [0, T], with values in ~ with the norm
I1~11~,- ~
max
O
I~(~)(t)l
and by P: C~-a[O, T] ~ ~ " the operator defined by the relation Px
=
{ Xl ( 0)
( 0)
,...,xm
,...,x
("-')(0) I , . .. , ~ - ' ) ( 0 ) )
.
(7.:)
We shall study the differential equation s
= f(t,
x,
,
x ('-x))
(7.2)
where x = x(t) is a vector-valued function with values in ~'~, s is a linear differential operator defined by the relation s = z (=) + a,(t)x (n-') + . . . + a,~(t)z. (7.3)
ak(t) are square matrices of order m continuously depending on t E [0, T], and the right-hand side f ( t , x l , . . . . . . , z,~) is continuous with respect to the totality of variables. We shall define the solutions of Eq. (7.2) by the boundary conditions z~(x) = o
(k - 1 , 2 , . . . , ran),
(7.4)
where lk: Cn-l[0, T] , ~ are continuous functionals, bounded on every ball, not necessarily linear. It is convenient to write the boundary conditions (7.4) in the form
B(~) =0, .where the operator B: C m-1 [0, T]
(7.5)
~m,,'is defined by the relation
B(z) = { h ( z ) , . . . , / m , ( z ) } .
(7.6)
We assume that the solutions of Eq. (7.2) are uniquely defined by the initial conditions and can be extended to the interval [0, T]. We denote by Q(y) the solution of Eq. (7.2) with the initial condition y E IIUn'~. By a direct verification we can make sure of the following statement. 505
Lemma .7.1. The point y E R m" is an initial condition of the boundarTj-value problem (7.2), (7.5) if and only if it ss a fized point of the operator U(y) = D o Q(y), where D = P -
(7.7)
B.
We call operator (7.7) the generalized shift operator for problem (7.2), (7.5). In the case of periodic boundary conditions, operator (7.7) coincides with the ordinary shift operator. 7.2. Equivalent operator equation. We denote by V: R'~" , C"-l[0, T] the operator which associates the initial condition
(7.8)
Px=y
with the solution of the homogeneous differential equation s
(7.9)
Then the solution of the nonhomogeneous equation = f(t)
(7.10)
can be written in the form z(t) = V o P z +
f K(t, a ) f ( s ) d s .
(7.11)
0
Relation (7.11) implies the following statement.
Lemma 7.2.
The function z E C"[0, T] is a solution of the boundary-value problem (7.2), (7.5) /f and only if it is a fized point of the completely continuous operator A ( z ) = V o 7~(z) +
/
K ( t , s ) f ( s , z(s), . . . , zC"-l)(s)) ds
0
which acts in the space C"-1[0, T].
Thus, the solutions of the boundary-value problem (7.2), (7.5) coincide with the solutions of the operator equation z = A(z).
(7.13)
There are other operator equations equivalent to problem (7.2), (7.5). For instance, if the boundary conditions (7.5) are linear and the Green function G(t,s) exists for the differential operator s under these conditions, then the solutions of the boundary-value problem (7.2), (7.5) coincide with those of the integral equation 1
9 (t)
=
fc(t,
s)f(s, z ( s ) , . . . , z('~-l)(s)) ds.
(7.14)
0
The operator defined by the right-hand side of Eq. (7.14) acts and is completely continuous in the space C"-'[0, T]. 7.3. The affinity theorem. Let us consider the completely continuous vector field t
(~(z) = z -
A(z)
(z e C~-'[0, T]),
(7.x5)
where A is operator (7.12). We shall show that the field ~ is splittable. Indeed, let E0 be a subspace of the space C'"-I[0, T] consisting of the solutions of the homogeneous equation (7.9). Then the operator P0: (5'"-110, T] ~ E0 can be defined by the relation poz= VoPz. 506
The operator P1 = I - Po projects the space C "-l[0, T] onto the subspace E1 of functions which possess zero
initial Cauchy data. Obviously, the operator
Ao(z) = V o D(z) acts from C"-t[0,T] into Eo and the operator t
A,(=)= f K(t,s)f(s , x ( s ) , . . . ,
x(n-l)(s)) d3
0
acts from C"-'[0, T] into Ea. Let us consider the equation
x=u+Al(z)
(ue Eo).
(7.16)
Its solution z is the solution of the differential equation (7.2) which satisfies the initial conditions Px
-- P u .
But then
x
=
H(u),
where
H =QoP. is splittable.
Thus the vector field iemma
7.3.
The topological indez ind (z., '~) of the zero z. = H(O) of the completely continuous vector
field
r
=
~-
Aa(z)
(7.17)
(z e C"-'[0,TI)
is equal to unity. Proof. It is sufficient to show that the rotation of the field r o n the spheres ]]zll = R of a sufficiently large radius is equal to unity. Let us define the family of operators S~ (0 < ,k < T) by the relation
z(t)
S~(z)
if 0 < t < I, if ,~ <_ t <_ T,
V~(z)(t)
where V~(z) is the solution of the homogeneous equation (7.9) satisfying the conditions
(k = 0 , 1 , . . . , , -
=
Suppose that x0
=
1).
xo(t) is the solution of Eq. (7.2) satisfying zero initial conditions. We set max
O<,X
IIS ,(xo)il ',-'tO,Tl + 1
and consider the completely continuous family of vector fields
r
=
z ( t ) - S~ o f K(t,s)f(s,x(s),
9 ..,x(n-1)(s))ds
(i] 3i --'co,rl = R o , 0_< A_< T).
(7.18)
0
We shall show that the fields of this family are nondegenerate on the sphere Indeed, the relation
II llc"-'to.T3
=Ro.
=o
507
means that = = z(t) is a solution of the homogeneous equation (7.9) for ,~ < t < T and |
=(t) = f K(t, s)f(s, x(s) ,...,
=c--,~(,)) a,
(0_
0
But then = = S, (=o)
and
II = lie--, to,~ < Ro. Thus the fields of family (7.18) are nondegenerate on the sphere
Ilzllc.-,to,~
= ~ , and therefore the field
~(., T) = ~x is homotopic to the field ~(., 0) = I and, consequently, the rotation of the field ~ on the sphere II=llc--,t0,~ = ~ is equal to unity. We have proved the theorem. As we have shown, the zeros of the completely continuous vector field I, are in a one-to-one correspondence with the zeros of the finite-dimensional vector field
~(u) = u - u(u)
(u e ~""),
(7.19)
where U is the shift operator defined by Eq. (7.7). The bounded domains f~ C C"-l[0, T] and G C Rm,, have the same core if and only if the set of initial values of the solutions of the boundary-value problem (7.2), (7.5) that lie in 12 coincide with the set of elements of the space ~."" which lie in G and are the initial conditions of the solutions of problem (7.2), (7.5). Moreover, on the boundaries of the domains f~ and G there are no solutions of the boundary-value problem (7.2), (7.5) and initial conditions of these solutions respectively. Theorem
7(~, on)
Let the domains G C ~ ' ~ and f/ C C"-x[0, T] have the same core. Then the rotations ~ d ~(~, Oa) of the fields do and go on the boundaries Of~ and OG of the domains fl and G are equal, 7.1.
i.e.j
(7.20)
7(~, 0a) = 7(~, oe).
Proof. Since the operators V: ~ ' " , C"-a[0,T] and P : C"-x[0,T] , ~ " " are mutually inverse isomorphisms, it follows, by virtue of property 7 of rotation, that the rotation 7(So, aG) of the field qo coincides with the rotation 7 ( ~ , OF) of the field ~(u) = u-
(7.2~)
V o U o Pu
on the boundary OF of the domain F = V(G) C Eo. The field (7.21) coincides with the field (7.6) and the domains F and f2 are, obviously, domains with the same core. In order to complete the proof, it remains to recall Theorem 6.1 and Lemma 7.3. We have proved the theorem.
1
Some Applications
8~176P e r i o d i c s o l u t i o n s of e q u a t i o n s w i t h a r e t a r d e d a r g u m e n t . Let us consider the equation with a retarded argument
dz(t) dt
= f(t,z(t))+ eg(t,z(t),z(t-
h)).
(s.~)
Here e is a small parameter and the vector-valued functions f ( t , z ) and g ( t , z , y ) (t E ~, z , y E Rlv) are continuous with respect to all the variables and T-periodic with respect to t:
f(t + T,z) = f(t,z), g(t+T,z,y)=g(t,z,y). 508
We are interested in T-periodic solutions of Eq. (8.1). Besides Eq. (8.1), we shall consider the unperturbed equation d~
dt
= f(t,z)
(x 6. RN).
(8.2)
We suppose that the solutions of Eq. (8.2) are uniquely defined by the initial values and can be extended to the interval [0, T]. Then the shift operator U along the trajectories of Eq. (8.2) at time T is defined on RN. The following statement is valid.
Theorem 8.1. Suppose that on the boundary OG of a certain domain G C ~ ' the field (8.3)
~o= I - U
is nondegenerate and its rotation 7(~0, OG) is nonzero. Then, for sufficiently small ~, Eq. (8.1) has at least T-periodic solution.
one
Proof.
We introduce into consideration an operator Fin, defined by the relation
IIhz(t) = {
z((t-h)
ifh
z T-h+t)
(8.4)
and use this operator to construct, in the space C[0, T], the completely continuous vector field 6
~#,(x)
x9( T ) -
= x(t)-
/
t
t
f ( s , x ( s ) ) d s - ~ / g(s ,
0
(8.5)
0
A direct verification shows that the zeros of the field 9 (and only they!) are T-periodic solutions of Eq. (8.1). Besides the field ~,, we shall consider the completely continuous vector field r
= z(t) - z(T) -
f
(8.6)
0
Suppose, furthermore, that ~ C C[0, T] is a domain which has the same core as the domain G relative to the fields r and ~. Since 7(r a a ) = 7(~, OG) # 0, by virtue of Theorem 2.1 it follows (for small e) that
Consequently, for small r the field ~ has zeros in ~. We have proved the theorem. The effectiveness of Theorem 8.1 is defined by the possibility of computing or estimating the rotation 7(~0, OG). In a number of cases it is very easy to prove the inequality 3'(~0, OG) r O. Here is an example. The continuously differentiable function V(z) (z E ~N) is said to be guiding for Eq. (8.2) if > 0
(O
> ,).
(8.7)
The indez ind V of the guiding function V(z) is the rotation of the field VV on the spheres S(r) = {z E ~N: [[z[] = r} of large radii r. If V ( z ) i s a guiding function for Eq. (8.2), then the relation
7(~o, S(r)) = ind V
(8.8)
is valid for large r. Therefore, Theorem 8.1 implies the following statement. 509
Theorem 8.2. Suppose that the solutions of Eq. (8.8) can be nonlocally extended and every initial condition defines a unique solution. Suppose, furthermore, that a guiding function with a nonzero index exists for Eq. (8.2). Then Eq. (8.1.) has at least one T-periodic solution for sufficiently small ~ > O. Note that the index of an even guiding function is certainly nonzero since it is an odd number. The requirements of the uniqueness of solutions of the Cauchy problem for Eq. (8.2) and their nonlocal extendability are very restrictive. Therefore, the following theorem essentially complements Theorem 8.2.
Suppose that for Eq. (8.2) there ezists a totality of guiding functions V~(x) , . . . ,
Theorem 8.3. which
fo,
(8.9)
lim ~ IVk(~)l = I=1--.~,,k=1
Let the indez of the function V~(z) be nonzero. Then Eq. (8.1) has at least one T-periodic solution for small We have restricted our considerations to the simplest theorems on the solvability of equations with retarded arguments whose proof is based on the affinity theorems. Analogs of these theorems are also valid for autonomous systems, complicated retardations, various perturbations, etc. 8.2. Tests of convergence for a p p r o x i m a t e procedures. The affinity theorems prove to be of use when we analyze various approximate procedures for solving nonlinear problems (the methods of harmonic balance, of mechanical quadratures, collocation, of finite differences, etc.). Here are some statements referring to the harmonic balance method. Recall the procedure on which this method is based in the problem of approximate construction of T-periodic solutions of Eq. (8.2). Let x.(t) be the required T-periodic solution of Eq. (8.2). We seek the approximation z,,(t) of z.(t) as the trigonometric polynomial
z,,(t) = Ao " ( 2rkt + 2 + y~ Ak sin T
Bk cos
2~rkt T )
(8.10)
k=l
with the unknown vector coefficients " bf
af Ak
(k =
--
0,...,n),
Bk =
b!
(8.11)
(k- 1,...,n).
In order to find the coefficients (8.11), we compose the equality
dz,,(t)dt
P,,f(t
where P,,f(t,x,~(t)) is a part of the expansion of the T-periodic function f(t,x,,(t)) in the Fourier series in which the harmonics up to order n are retained. Equating the coefficients in the same harmonics on the left-hand and right-hand sides of the last relation, we get a system of (2n + 1)N scalar equations ~
}
.
.
.
~
}
.
.
9
}
,;
}
.
-
.
~
~
bl
~
"
"
"
}
}
9
-
.
}
b;
}
.
.
-
}
b;,)
"--"
o
(k
=
+ 1)N)
with (2n + 1)N scalarunknowns. System (8.12) is known as a system of equations of a harmonic balance. Solving this system and substituting the soldtions obtained into Eq. (8.10), we obtain the approximation z~(t) of the periodic solution z.(t) of Eq. (8.2). Suppose that E,, is a subspace of trigonometric polynomials (8.10) of degree not higher than n and P,,: L2[0, T] ~ E,, is an orthogonal projector. Then the approximations z~,(t) of the T-periodic solution z.(t) of Eq. (8.2) found by the harmonic balance method coincide with the solutions of the equation z,, = P,A(z,)
510
E.),
(8.13)
where t
A(z) = z(T) +
f
(8.14)
0
Therefore, we can apply the general theory of projection procedures to the investigation of the harmonic balance method. For instance, the following theorem is valid. T h e o r e m 8.4. Suppose that the vector field gp = I - A domain f / C C[0, T] and
is nondegenerate on the boundary cgfl of the bounded
7(~, c3f~) -~ O.
(8.15)
Then, for sufficiently large n, the sets ~,, of solutions that approximate Eqs. (8.13) have a convergence:
lirn
sup
inf I1 ; -
are.
nonempty and we
.llc[0.r] = 0,
where ~ is a set of T-periodic solutions of Eq. (8.2) lying in f~.
The affinity theorems prove to be useful for verifying conditions (8.15). Here is an example. T h e o r e m 8.5. Suppose that the solutions of Eq. (8.2) are uniquely defined by the initial conditions and can be extended to the closed interval [0, T]. Suppose, furthermore, that for Eq. (8.2) there exists a guiding function of a nonzero index. Then, for sufficiently large n, the sets ~,~ of solutions of Eq. (8.13) are nonempty and convergence (8.16) holds. Proof.
For sufficiently large r, the field qa = I - U is nondegenerate on the spheres S(r) C R N and 7(tp, S(r)) = ind V # O.
But then, for large r, the field r = I Theorem 2.1,
A is nondegenerate on the spheres S(r) C C[0, T] and, by virtue of 7(r S(r)) = 7(~, S(r)).
Consequently, # 0,
and the theorem being proved follows from Theorem 8.4. We have proved the theorem. If the solutions of Eq. (8.2) do not possess the property of nonlocal extendability, then the existence of one guiding function does not guarantee either the existence of periodic solutions of Eq. (8.2) or the convergence of the harmonic balance method. T h e o r e m 8.6. Suppose that the conditions of Theorem 8.3 are satisfied. Then, for sufficiently large n, the sets ~,, of solutions of Eq. (8.13) are nonempty and we have convergence (8.16). Suppose that it is known a priori that the required T-periodic solution is asymptotically stable. Then -the method of harmonic balance converges without any additional assumptions. T h e o r e m 8.7. Let the periodic solution z. = z.(t) of Eq. (8.2) be asymptotically stable. Then, for sufficiently large n, Eqs. (8.13) have solutions z~, = z~,(t) that form a sequence converging to z.: I
lim IIx,, I'l---* O 0
O.
Sll
Proof. Let us consider a ball B ( z . , r ) = {z e RN: < o f ~ sufficiently small radius r > O. Since the solution z . ( t ) is asymptotically stable, there exists no such that c
for n _> no. Here U is a shift operator at time T along the trajectories of Eq. (8.2). Then, by virtue of the Browder theorem, ind (z., ~o) = 1, where z. = x.(0), ~o = I -
U. By virtue of Corollary 2.1, ind (z.(.), (I)) = ind (z., qo) = 1.
It remains to recall Theorem 8.4. The proof of the theorem is complete. We have illustrated the application of affinity theorems to the justification of the convergence of approximate procedures in the problem on periodic solutions of the elementary equation (8.2). The affinity theorems can also be used in the investigation of problems on free oscillations, on the solution of boundary-value problems, etc.
9.
Additional Remarks
9.1. G e n e r a l t h e o r e m on s p l i t t a b l e fields. Here is one of the possible generalizations of the theorem on a splittable field presented in Sec. 6. Suppose that the Banach space E is the direct sum of the subspaces E0 and El, i.e., E= We denote by P0: E
(9.1)
Eo@E1.
* E0 and P I : E ~ E1 the projection operators corresponding to decomposition (9.1).
Let us consider the equation z
with a completely continuous operator A: E
:
(9.2)
A(z)
~ E. Equation (9.2) is equivalent to the system of two equations u = PoA(u + v),
(9.3)
v = P1A(u + v),
(9.4)
where u E Eo, v E El.
We suppose that for every fixed u E Eo Eq. (9.4) has the unique solution
(9.5)
=
where R: Eo
, E1 is a certain continuous and bounded operator. Then Eq. (9.1)is equivalent to the equation u = PoA(u + R(u)),
(9.6)
and the relations u. = Pox.,
9.==.
+ R(u.)
establish a one-to-one correspondence between the solutions z. and u. of Eqs. (9.1) and (9.6), respectively. Suppose that fl C E and 9t0 C E0 are bounded domains in the spaces E and E0 and the domain flo is connected. Let us consider the completely continuous vector fields r
A(x)
(9.7)
PoA(u + n ( u ) )
(9.8)
= x-
and (~o(U) = u 512
on the closures ~ and ~o of these domains. We say that the domains ft and rio have the same core if the fields r and @o are nondegenerate, respectively, on Ofl and oqf/o and if the set ~ o of zeros of the fields @o and f/o coincide with the projection Po~ of the set of zeros ~ of the field @ lying in ft. Let us consider the completely continuous vector field t
@,(v) = v - P1A(uo + v). For every fixed uo E flo, this field has the unique zero vo = R(uo) whose index ind (Vo, @I) does not depend on uo. We denote this general value of the index by po. Theorem
9.1.
Let the domains f~ and flo have the same core. Then 7(r on)
o7( 0, o 0).
=
(9.9)
9.2. T h e affinity p r i n c i p l e for e v o l u t i o n a r y e q u a t i o n s . Let C"-I[a,B] be a space of m-dimensional vector-valued functions with n - 1 continuous bounded derivatives on [a, ~] (the closed interval [a, ~] can be unbounded). Suppose, furthermore, that -oo
< a < b<
(9.10)
cr
We denote by T[a, al= the restriction of the function z(t) E C"-'{a,b] to the closed interval [a,a] and by Tin, b]z the restriction of the function z(t) E C"-'[o, b] to In, b]. We say that the continuous operator F : C=-~[a, b] ; C[a, b] is a Voltewa operator if the relation
a
(a
F ( x ) ( t ) - F(y)(t)
(a <_ t < ,~).
The equation
(9.11)
z:x = F ( x ) ,
where/: is the differential operator (6.3) and F is a Volterra operator, is said to be evolutionary. By a solution of Eq. (9.11) we mean a function z(t) E C"-'[a,b] which satisfies this equation on the closed interval [a,b]. The solutions of Eq. (9.11) are defined by the initial conditions -
(9.12)
y,
where y is a fixed function from C"-~[a,a]. Condition (9.11) is the initial Cauchy condition. The analog of the boundary conditions that appear in the theory of boundary-value problems for ordinary differential equations is the boundary condition T[a, a]z = B,(z) + B~ o T[a, b]z,
(9.13)
where B1 : C "-1 [a, b] C"-l[a,a] is a completely continuous operator and B2: C"-l[a, b] ; Cn-l[a, a] is a continuous operator. Let V be a linear operator that associates the function y E C"-l[a, a] with the solution of the homogeneous equation Z:z = 0
(9.14)
satisfying the conditions 9
=
..,
y
(9.15) 513
Lemma
9.1.
For the function z = x(t) to be a solution of the boundary-value problem (9.11), (9.13), it is necessary and sufficient that z be a zero of the completely continuous, in C"-t[a, b], vector field
4~(z) = z - A(z),
(9.16)
where
B,(=) +
o (v o Tt ,, o1= + / K ( t ,
A(z) =
if t e [a,a],
s)F(x)(s)ds)
r
B.<=)+ , o =.o ( , o
Io.oj=+
+
,:, (9.17/
and K(t, s) is a Cauchy matriz-valued function.
A different approach to the problem of finding the solutions of the boundary-value problem (9.11), (9.13), which is based on the concept of the shift operator, is also possible. Let [a,A] C [a,b]. We say that the function z E C"-~(a,A)is a solution of Eq. (9.11) if the relation
L=(t) = F(=)(t)
(~ < t <_ A)
is valid. Here F(x) is the value of the operator F on any (n - 1)-times continuously differentiable extension of the function z(t) to [a, hi. We assume that all solutions of Eq. (9.11) are uniquely defined by the initial Cauchy conditions (9.12) and can be extended to the interval [a, b]. We also assume that the set of solutions of this equation that correspond to the set of initial conditions bounded in C "-I [a, a] is bounded in C "-1 [a, b]. Under these assumptions there exists an operator Q that associates every function y E C"-l[a, a] with the solution Q(y) of the evolutionary equation (9.11) satisfying condition (9.12). This operator acts from C"-X[a, a] into C"-~[a, b] and is continuous and bounded. We set U(y) = B, o Q(y) + B2 o T[a, b] o Q(y). (9.18) The operator U is a shift operator along the trajectories of Eq. (9.11). The shift operator U is completely continuous in the space C"-X[a, a]. L e m m a 9.2. The function y E C"-~[a,a] is an initial value of the solution of the boundary-value problem (9.11), (9.13) if and only if y is a zero of the completely continuous vector field
v(v)=y-u(y).
(9.19)
It follows from Lemmas 9.1 and 9.2 that the zeros of the completely continuous vector field (I) are in a one-to-one correspondence with the zeros of the completely continuous vector field ~o. In this case, the bounded domains fl C C"-l[a, b] and G C C"-~[a, a] have the same core if and only if the fields (I) and ~o are nondegenerate on Of2 and OG, respectively, and the restriction of the set of solutions of problem (9.11), (9.13), that lie in fi, to [a, a] coincides with the set of initial conditions of the solutions of this problem lying in G. Theorem
9.2.
Let the domains fl C C"-~[a,b] and G C C"-~[a,a] have the same core. Then
7(+, 0s)= 9.3. T h e affinity principles for elliptic equations. boundary. We shall consider the Dirichlet problem
0a).
Let fl be a bounded domain in RN with a smooth
Lu(x)-v(x), 514
(9.20)
(9.21)
I
(9.22)
= w(,).
o.
Here L is a uniformly elliptic differential operator N
Lu
=
N
,71"au(z)OziOzj + ~ai(z) .:
Ou( )
(9.2:3 /
"~z; F aCxluCz)
i=1
of the second order with smooth coefficients aij(z), ai(z), a(z) (z e fl) such that the estimate N
N
i=1
i,j=l
is valid for a certain v > 0. Suppose that for v(x) = 0, w(x) = 0 the Dirichlet problem (9.21), (9.22) has only a zero solution. We say that the triple of functional Banach spaces {E, E0, E~ } that consist of the functions v(x) (z E ~), w(z) (z E OFt), u(z) (z E -~), respectively, is compatible with problem (9.21), (9.22)if E~ is compactly embedded in E and, for every fixed v E E and w E E0 , the Dirichlet problem (9.21), (9.22) has a unique solution u E E, representable in the form (9.24) u = Aow + Alv, 9
where A0: Eo * El, A1 : E ~ E1 are linear bounded operators. In problems concerned with classical solutions the natural triples of functional spaces compatible with problem (9.21), (9.22) are, for instance, the triples {C(fl), C(Of~), C'(~)}, {C~(~), C"(Ofi), C2+"(~)}, where C~(fl), C"(Ofl), C2+~('fl) are HSlder spaces. In the problem on generalized solutions, the triple {L~(s
B~/2(Of~), W~(fi)}
can serve as an example
of a compatible triple. Here B~/2(Ofi) is a Besov space and W~(ft) is a Sobolev space. Note that the operators A0 and A1 in Eq. (9.24) admit the representation
Aow = f a, ( z, ~)w(E,) d~,
(9.25)
8n
A, v =
f ao(=,y),(y)
(9.26)
where G0(x, y), GI(X, ~) are the Green functions corresponding to the boundary-value problems
Lu(z) = 0 ,
(9.27) (9.28)
and
Lu(x)=v(z),
(9.29)
( )loa = O.
(9.30)
We denote by P the restriction (or trace) operator of the function u(z) (z E fi) to Of/. In addition to the problem (9.21), (9.22), we shall consic]er the boundary-value problem
Lu(x) = f(x, u(x)), =
(9.31) (9.32)
where the operator/3: E ~ E0 is continuous and bounded on bounded sets and the superposition operator
f(u) = f(x, u(x)) 515
acts from E into E, is continuous, and bounded on bounded sets. Then problem (9.31), (9.32) is equivalent to the operator equation u = A(u) (9.33) with a completely continuous operator A: E
E defined by the relation (9.34)
A(u) = Ao o B(u) + Ax o f ( u ) .
Let us consider a different approach to problem (9.31), (9.32). Suppose that for every fixed function w E Eo the Dirichlet problem Lu(z)
=
(9.35)
f ( z , u(z)), =
(9.36)
has a unique solution u = Q(w), where Q: E0 } E~ is a continuous operator bounded on bounded sets. Let us consider, in the space E0, the operator equation (9.37) with a completely continuous operator B o Q : E0
; Eo. If w0 is a solution of Eq. (9.37), then
L o Q(wo) -- f o Q(wo)
and P o Q(wo) = Wo = B o Q(wo).
Therefore, the function Uo = Q(wo)is a solution of problem (9.31), (9.32). Conversely, if the function u0 E E, is a solution of problem (9.31), (9.32), then uo is a solution of the equation = Q o B(u)
in the space E, and therefore the function Wo = B(uo) or, what is the same, the function Wo solution of Eq. (9.37). Thus the relations ,,
=
-
Puo
is a
w=Pu
establish a one-to-one correspondence between the solutions of Eqs. (9.33), (9.37)or, what is the same,
between the zeros of the completely continuous vector fields ~(u) = u - A(u)
(u E E)
(9.38)
and ~(w) = w - B o Q(w)
(w e eo).
(9.39)
The bounded domains 9/ C E and f20 C E0 are domains with the same core if the fields 9 and ~ are nondegenerate on 0f~ and 0f/0 respectively, if it follows for every zero uo E 12 of the field ~ that Puo E 12o, and the inclusion Puo E f2o implies that uo E f'/. Let us consider in E the completely continuous vector field X(U) = u - Ax o f ( u + Q(0)) - Q(0).
It is obvious that the zero function is the only zero of the field X. We denote the topological index of this zero by ind (0, X)T h e o r e m 9.3. Let the triple of spaces {E, E0, El} be compatible with the boundary-value problem (9.21), (9.22). Suppose that the domains f~ C E and f~o C Eo have the same core. Then the equality 7((}, Off) = ind (0, X)7(qo, Of2o) holds. 516
(9.40)
9.4. Fields w i t h s u p e r p o s i t i o n s of o p e r a t o r s . Suppose that Eo and Et are Banach spaces and Ao: Eo El, AI: E1 * Eo are completely continuous operators. In the spaces Eo and El we consider the completely continuous vector fields Co(Z) = z - A, o Ao(z) (z e Eo), (9.41) and r
= y - Ao o A , ( y )
(y e E , ) .
(9.42)
The zeros of the fields (9.41), (9.42) are in a one-to-one correspondence, i.e., if xo is a zero of the field r then !he point yo = Ao(zo) is a zero of the field r and, conversely, if yo is a zero of the field r then Xo = A,(yo) is a zero of the field ~o. Suppose that 12o and I21 are bounded domains in the spaces Eo and El, respectively, and ~to and ~1 are the sets of zeros of the fields r and ~x lying in ~o and f/x respectively. We say that the domains flo and fix have the s a m e core relative to the fields r and r if
Ao(m o) c
(9.43)
A1(~1) C rio.
(9.44)
and It follows from Eqs. (9.43) and (9.44) that the fields r
Theorem Then
9.4.
and r
are nondegenerate ou 012o and Off1 respectively.
Let the domains f~o C Eo and f~l C E1 have the same core relative to the fields r
7(r
O o) = 7(r
oa,)
and r
(9.45)
Let us now discard the assumption concerning the complete continuity of every one of the operators A0 and A1. T h e o r e m 9.5. Suppose that the operators Ao: Eo ~ E1 and AI : E1 ~ Eo are continuous and the operators A1 o Ao: ~io ; Eo and Ao o AI : fll ~ Ea are completely continuous. Suppose, furthermore, that there ezist convex compact sets Ko C Eo and Ka C Ex for which A, o Ao(~o) C Ko, Ao o
AI(~I)
C
K1,
A I(~, fl K1) C Ko,
(9.46)
Ao(flo N K0) C K1.
(9.47)
Suppose, finally, that the domains flo and 121 have the same core relative to the fields ~o and (~1" Then
7(~o, Oflo) = 7(~,, Off,).
(9.48)
9.5. I n v a r i a n c e of r o t a t i o n . In many situations it is convenient to study the same vector field 9 = I - A in different spaces. The relations between the rotations of the field ~ on the boundaries of the corresponding domains in these spaces are of importance. Under natural assumptions, these rotations prove to be similar. The theorems that establish the equality of rotations are known as the principles of invariance of rotation. In their character, principles of invariance of rotation are close to the affinity theorems. Here we give one of the principles of invariance of rotation. Suppose that two Banach spaces Eo and E1 lie in a certain linear space E. Suppose, furthermore, that fl0 C E0 and 9/1 C Ex are bounded domains in E0 and El, respectively, and the operator A is defined on the set f~'oU f~l, with A(~o) C E0 and A(~I) C Ea. We say that the domains f/o and f~l have the same core relative to the field r = I - A if it is nondegenerate on 0120 tO 0ill and if the set of its zeros lying in flo coincides with the set of zeros lying in fix, T h e o r e m 9.6. Suppose that the bounded domains rio C Eo and fll C Ex have the same core relative to the field ~ = I - A and the operator A on -flo is completely continuous as an operator in Eo and continuous as an operator acting from Eo into El. Suppose, furthermore, that the operator A on -~1 is completely continuous as an operator into E1 and continuous as an operator acting from E1 into Eo. Then
-y(r aflo) = -y(r
). 517
10.
Bibliographical Comments
1. For a finite-dimensional case, the concept of rotation in the equivalent terms of the degree of mapping of the domain f/with respect to the zero point (or the degree of a normed mapping of the boundary c0il of the domain f/onto a unit sphere) goes back to Brouwer and Poincar~. Directly in terms of rotation the theory was constructed in Aleksandrov's monograph [1], for instance. The transition to infinite-dimensional analogs was first suggested for Fredholm maps, where the degree of mapping modulo 2 was used. Leray and Schauder suggested the generalization of the Brouwer-Poincar~ general theory to maps in infinite-dimensional spaces in connection with the famous Leray-Schauder principle of a priori estimates of solutions in problems on the solvability of nonlinear equations. In terms of rotation of the fields (I) = I - A with the completely continuous operators A the theory was constructed in [13]. The reader can find the main facts of the theory of completely continuous vector fields in many books (see [18], for instance). 2. The first affinity theorems were formulated by Krasnosel'sky in 1962-63 in seminars at Voronezh University as a method for investigating problems on forced oscillations in systems with a retarded argument. Theorem 2.1, its first modifications, and applications were indicated by Krasnosel'sky and Strygin in [14, 23, 32, 33]. 3. The index of a cycle and the functional characteristic of a periodic solution as well as the affinity theorems connecting them were suggested by Bobylev and Krasnosel'sky in [5]. For further developments see, for instance, [2-4, 6-8, 34]. The method of functionalization of a parameter is convenient in problems with a parameter in which, for its certain unknown value, connected continua of solutions arise. The replacement of the parameter by a linear or nonlinear functional reduces the problem to a search for isolated solutions. The simplest example is the problem of finding the eigenvalues and eigenvectors of linear operators. Already in this example (see [12]) is the method of functionalization of a parameter very effective. The method of functionalization of a parameter, used as an instrument for investigation of the periodic solutions of autonomous systems of differential equations, was developed in the articles by Bobylev and Krasnosel'sky mentioned above. This method is convenient for the investigation of phenomena of the type of Hopf bifurcation (see [16, 26]). 4. The main facts of the method of state spaces in the investigation of linear systems with rational fractional transfer functibns can be found in [35, 37], for instance. The concept of the pulse-frequency characteristic and its first systematic use in problems of control theory belongs to R,ozenwasser. The affinity Theorem 4.1 for problems on forced oscillations in control systems is indicated in [4]. 5. The affinity Theorem ,5.1 for problems on free oscillations in autonomous control systems is proved and used in [7, 8]. In this article we suggest a new simple proof. 6. Theorem 6.1 belongs to Lifschitz [28, 29]. The methods given in [29] played an essential part in the further development of the affinity theory and its applications. 7. The affinity Theorem 7.1 for boundary-value problems was suggested by Krasnosel'sky and Lifschitz [19, 291 (see also [15, 18]). 8. The schemes of application of the affinity theorems in problems on forced oscillations in distributed argument systems are given in [14] (see also [18]). The method of guiding functions was suggested by Kra nosel'sky and Perov [22]. Later this method was further developed and had various applications in works by different authors. Voluminous journal and monographic literature is devoted to the general theory of projection methods (the harmonic balance method is one of them). The proof of Theorem 8.4 can be found, for instance, in [17]. The remarkable theorem of Browder used in the proof of Theorem 8.7 can be found in [18], for instance. 9. In the concluding part of the article, a number of theorems are given without proof. For the proof of Theorems 9.1, 9.3-9.5, and some other affinity theorems, see [18]. Theorem 9.2 was proved by Lifschitz [28, 29]. Theorem 9.3 belongs to Krasnosel'sky and Lifschitz. Theorem 9.4 was proved by 518
Zabreiko and Krasnosel'sky [9]. The invariance principle was first established under additional assumptions by Krasnosel'sky and Strygin [24, 25]. These assumptions were removed by Sapronov [31]. A new general approach was suggested in [10]. We presented the formulation of the invariance principle since it must be regarded as one of the most important affinity theorems. This work was performed within the framework of the EU-Russian Colloboration Project ESPRIT P9282 ACTCS.
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0
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