Stability Domains

NONLINEAR SYSTEMS IN AVIATION AEROSPACE AERONAUTICS ASTRONAUTICS

STABILITY DOMAINS

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Nonlinear Systems In Aviation Aerospace Aeronautics Astronautics

A series edited by: S. Sivasundaram Embry-Riddle Aeronautical University, Daytona Beach, USA

Volume 1 Stability Domains L.Gruyitch, J-P. Richard, P. Borne and J-C.Gentina Volume 2 Advances in Dynamics and Control S. Sivasundaram Volume 3 Optimal Control of Turbulence S.S. Sritharan

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NONLINEAR SYSTEMS IN AVIATION AEROSPACE AERONAUTICS ASTRONAUTICS

STABILITY DOMAINS

L. GRUYITCH, J-P. RICHARD, P. BORNE AND J-C. GENTINA

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. © 2004 by Chapman & Hall/CRC

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Contents Preface Notations General introduction 1 Introductory comments on stability concepts

1.1 Comments on Lyapunov's stability concept 1.1.1 Lyapunov's denition of stability and the denitions of stability in the Lyapunov sense 1.1.2 Denitions of attraction 1.1.3 Denitions of asymptotic stability 1.1.4 Denitions of exponential stability 1.1.5 Denitions of absolute stability on Ni (L) 1.1.6 Denitions of attraction with nite attraction time . 1.1.7 Denitions of stability with nite attraction time . 1.1.8 Denitions of absolute stability with nite attraction time 1.2 Comments on the practical stability concept 1.2.1 Introductory comments 1.2.2 Denition of practical stability 1.2.3 Denition of practical contraction with settling time . . 1.2.4 Denition of practical stability with settling time . .

2 Stability domain concepts

2.1 Introductory comments 2.2 Domains of Lyapunov stability properties 2.2.1 The notion of domain 2.2.2 Denitions of stability domains . . 2.2.3 Denitions of attraction domains . . 2.2.4 Denitions of asymptotic stability domain 2.2.5 Denitions of exponential stability domains . . 2.2.6 Denitions of asymptotic stability domains on N( ) ( ) 2.3 Domains of practical stability properties 2.3.1 Denitions of domains of practical stability

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2.3.2 Denitions of domains of practical contraction with settling time 2.3.3 Denitions of domains of practical stability with settling time

3 Qualitative features of stability domains properties 3.1 Introductory comments 3.1.1 Denition of a motion . 3.1.2 Existence of motions . 3.1.3 Existence and uniqueness of motions . 3.1.4 Continuity of motions in initial conditions . 3.1.5 Dierentiability of motions in initial conditions . 3.2 Generalised motions 3.2.1 Motivation 3.2.2 Dini derivatives 3.2.3 Generalised motions . . . 3.2.4 Limit points and limit s 3.2.5 Limit sets, Lagrange stability, precompactness and stability domains 3.2.6 Invariance properties of sets 3.2.7 Invariance properties of limit sets . 3.3 System regimes . . . 3.3.1 Forced regimes and the free regime . . . 3.3.2 Periodic regimes 3.3.3 Stationary regimes and stationary points 3.3.4 Equilibrium regimes and equilibrium points 3.4 Invariance properties of sets of equilibrium states . 3.5 Dynamical and generalised dynamical systems . . . . . 3.5.1 Denition and properties of dynamical systems . . . 3.5.2 Denition and properties of generalised dynamical systems . . 3.6 Stability properties and invariance properties of sets 3.7 Invariance features of stability domains properties 3.8 Features of equilibrium states on boundaries of domains of stability properties

4 Foundations of the Lyapunov method

4.1 Introductory comment 4.2 Sign denite functions 4.2.1 Sign semi-denite functions . 4.2.2 Sign denite functions 4.2.3 Comparison functions 4.2.4 Positive denite functions and comparison functions . 4.2.5 Radially unbounded and radially increasing positive denite functions 4.3 Uniquely bounded sets

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4.4

4.5 4.6

4.7 4.8 4.9

4.10 4.11

4.3.1 Denition of uniquely bounded sets 4.3.2 Properties of uniquely bounded sets 4.3.3 O-uniquely bounded sets and positive denite functions 4.3.4 Denition of uniquely bounded neighbourhoods of sets 4.3.5 Properties of uniquely bounded neighbourhoods of sets Dini derivatives and the Lyapunov method 4.4.1 Fundamental lemmae on Dini derivatives 4.4.2 LaSalle principle 4.4.3 Dini derivatives, positive deniteness, positive invariance and precompactness . Stability theorems 4.5.1 Stability of a set 4.5.2 Stability of X = 0 4.5.3 Comment Asymptotic stability theorems . 4.6.1 Asymptotic stability of a set 4.6.2 Complete global asymptotic stability of sets 4.6.3 Asymptotic stability of X = 0 4.6.4 Complete global asymptotic stability of X = 0 4.6.5 Absolute stability of X = 0 of Lurie systems Exponential stability of X = 0 4.7.1 Krasovskii criterion 4.7.2 Yoshizawa criterion Stability domain estimates . . . 4.8.1 Denitions of stability domain estimates . . . 4.8.2 Estimates of the stability domain of a set 4.8.3 Estimates of the stability domain of X = 0 Asymptotic stability domain and estimates 4.9.1 Classical approach 4.9.2 Denition of asymptotic stability domain estimate . . . 4.9.3 Estimates of the asymptotic stability domain of a set . 4.9.4 Estimates of the asymptotic stability domain of X = 0 Exponential stability domain estimate . . . . 4.10.1 Denition of exponential stability domain estimate . . . 4.10.2 Estimates of the exponential stability domain of a set 4.10.3 Estimates of the exponential stability domain of X = 0 Asymptotic stability domains on N( ) ( ) 4.11.1 Denition of estimate of the asymptotic stability domain on N( ) ( ) 4.11.2 Algebraic approach 4.11.3 Frequency domain approach

5 Novel development of the Lyapunov method 5.1 Introductory comment 5.2 Systems with dierentiable motions . 5.2.1 Smoothness property 5.2.2 Two-stage approach

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5.2.3 Approach via O-uniquely bounded sets 5.2.4 General one-shot approach 5.2.5 Exponential stability 5.3 Systems with continuous motions (generalised motions) 5.3.1 Smoothness property 5.3.2 Approach via O-uniquely bounded sets 5.3.3 General one-shot approach 5.4 Conclusion

6 Foundations of practical stability domains

6.1 Introductory comment 6.2 System aggregation function and sets . . 6.2.1 System description and sets . 6.2.2 Denition of estimates of practical stability domains of systems . . 6.2.3 System aggregation function extrema and sets . . . 6.3 Estimate of the system practical stability domain . 6.4 Estimate of the domain of practical stability with settling time s 6.5 Conclusion

7 Comparison systems and vector norm-based Lyapunov functions 7.1 Introductory comments and denitions 7.1.1 Presentation 7.1.2 Comparison systems . . . 7.1.3 Dierential inequalities, overvaluing systems . 7.1.4 ;M matrices . . 7.2 Vector norm-based comparison systems 7.2.1 Denition and aim of vector norms . 7.2.2 A rst statement 7.2.3 Computation of overvaluing systems . 7.2.4 Overvaluation lemma 7.3 Vector norms and Lyapunov stability criteria 7.3.1 Stability of equilibrium points 7.3.2 Stability of sets 7.3.3 Examples . 7.4 Vector norms and practical stability criteria with domains estimation . . . 7.5 Conclusions

References

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Preface \At the same time it becomes clear why some stability investigations should not be taken too seriously. One needs to know the size of the region of asymptotic stability." J.P. LaSalle and S. Lefschetz1 The kind of persistency of dynamical behaviour is important not only for various sciences like mathematics, mechanics, uid mechanics, thermodynamics, electricity, electronics, chemistry, control, econometrics, biology, medicine, and not only for engineering and science, but also for individuals, social life and state development. An appropriate persistency is the essential sense of the corresponding stability property. Relativeness of the notion of persistence resulted in a number of stability concepts among which was that of practical stability probably initiated by Chetaev in the thirties of the twentieth century. The above cited comment by LaSalle and Lefschetz, who were probably those who attracted the attention and interest of English speaking scientists to the Lyapunov stability theory, explains well the need to study stability domains in general and stability regions in particular. This book is aimed at what we consider basic from the theory of stability domains for various direct eective applications and/or for further research. In doing so we express freely and openly our views on stability theory in the framework of the Lyapunov stability concept and practical stability concept. Authors

1 Stability

by Lyapunov's Direct Method

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, Academic Press, New York, 1961, p. 121.

Notations Capital script Roman letters denote sets and spaces, capital block Roman letters designate matrices, lower case script Roman letters are used for scalars, lower case block Roman letters represent vectors, most Greek letters denote scalars. Latin

A A A A a

Ba Ba

+

B B b b

C C (k)(S ) C0(I0  X0 ) Ct(I0  X0 ) C c

D D(A) Da

a nonempty subset of
© 2004 by Chapman & Hall/CRC

Da (A) Dp Dp (A) Dpc Dpc (A) Dps Dps (A) Ds Ds (A) Ds (") Ds (" A) D () D () D1 () 1 2

d

E () F F ()

0

f fX g H h

I I

0

the attraction domain of a set A, the domain of practical stability with the settling time s with respect to f
XA
XF
Ig: Dp = Dp (
s
XA
XF
I ), the domain of practical stability with the settling time s of a set A with respect to f
XA
XF
Ig: Dp (A) = Dp (
s
XA
XF
I  A), the domain of practical contraction with the settling time s with respect to f
XF
Ig: Dpc = Dpc (
s
XF
I ), the domain of practical contraction with the settling time s of a set A with respect to f
XF
Ig: Dpc (A) = Dpc (
s
XF
I  A), the domain of practical stability with respect to f
XA
Ig: Dps = Dps (
XA
I ), the domain of practical stability of a set A with respect to f
XA
Ig: Dps (A) = Dps (
XA
I  A), the stability domain of X = 0, the stability domain of a set A, the largest neighbourhood of X = 0 such that X0 2 Ds(") implies kX(t X0 )k < " for all t 2 <+ , the largest neighbourhood of a set A such that X0 2 Ds (" A) guarantees X(t X0 ), A] < " for all t 2 <+ , rhomboid closed set fX 2
0

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0

0

i I

an integer, the identity matrix of the appropriate order, i an input vector function: i : < ! qg, M 2 M, q  0, k an integer, K a diagonal matrix: K 2 <m
m , or a possibly non-diagonal matrix K 2 0, the family of all the Lurie functions f in a matrix sector L N0 (L S ) over a set S , N1 (L M S ) the family of all the Lurie functions f in a matrix sector L over a set S , which are dierentiable with Jacobian fw in a matrix set M over S , n an integer, O the singleton: O = fX : X = 0g = f0g 
<; <; < < <

0

+ +

<
<
s r

r

S S S @S Sd Se

s

sat( ) sgn( ) sign( )

T + Tt ( ) ; 

T
s

t t0

U

u u(M) u

V ; ( ) V (A) 1

V

= ft : t 2 <
;1 < t  0g = ] ; 1
0], the interior of <; : <; = ] ; 1
0 = ft : t 2 <;
t < 0g = <; , a time interval: <0 = t0
+1, i t0 = 0 then <0 = <+ , = ft : t 2 <
0  t < +1g = 0
+1, <+ = 0
+1], the interior of <+ : <+ = ]0
+1 = ft : t 2 <+
t > 0g = <+ , a time interval: <
= 0
 = ft : 0  t < g, a time interval: <
s = 0
s = ft : 0  t < s g, a real number, a vector function, r : <  t (tk < t )
k = 0
1
: : :
tk ! t as k ! +1g, a time interval: T
s =  s
 = ft : s  t < g, an independent scalar variable denoting time, an initial moment, t0 2 <, a uniquely bounded set, a generating function of a uniquely bounded set, an importance vector of a ;M-matrix M, u(M) 0, either a control vector (function), u = (u1 u2 : : : ur )T , with the elements ui whose values are measured with respect to their total zero values, or an elementwise positive vector of size k or n, = fX : X 2
© 2004 by Chapman & Hall/CRC

v w W W

X XA XF X

0

XA XF 0 0

x X

X0 X

X(t X0 i) X(t X0) X(T  X0)

a tentative scalar Lyapunov function, in general v : < 
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the real output vector (function) with the elements whose values are measured with respect to their total zero values, or a vector y 2 <m , a desired output vector (function): a desired (prespecied, demanded, reference) output vector (function) y, the class of square matrices with non positive o-diagonal elements, at most nite set, a vector, z 2
y yd

Z Z z

Greek   ij



(A)

(M) (A)  a s

  !



a scalar, a scalar, Kronecker delta: ij = 1 for i = j, ij = 0 for i 6= j, a scalar, the empty set, the minimal eigenvalue of the matrix A = AT , the importance (eigen) value of a ;M]-matrix M,

(M) < 0, the maximal eigenvalue of the matrix A = AT , a distance function, (X
A) is the distance from X to A: (X
A) = inf(kX ; yk : y 2 A), a nal time, 2 <+ or = +1, an attraction time: a 2 <+ , a settling time: s 2 ]0
], a transformed time variable: d = dt(1 + kf(X)k), a scalar function associated with the derivative of v,  : < !(x
x) _ = > 0
(x = 0
x_ = 0)
: +1
(x 2 <+
x_ 2 <) or (x = 0
x_ 2 <+ )
_ = a matrix function,  :
2 26 ) , ! 9 9!

belongs to, is member of, does not belong to, is not member of, implies, if and only if, necessary and su cient converges to, tends to, exists, exists exactly one,

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A;B AB A\B AB A B 8 8a ( ) sat x sgn x sgn X sign x sign X ]
 
] (
)

rrr _

1 jX j jX j  jS j jX j < jS j k k Xk

set dierence: A ; B = f :  2 A
 62 Bg set union: A  B = f :  2 A or  2 B or bothg set intersection: A \ B = f :  2 A and  2 Bg there is  2 A,  2= B, permits both A  B and A = B, for every, for every ( ) almost everywhere, almost for every ( ), ( x
jxj  1
= sign x
jxj 2 <+
8 = ;1
x < 0
> < 2 ;1
1]
x = 0
> : =1
x>0 = (sgn x1 sgn x2 : : : sgn xn)T , ( sgn x
x 6= 0 = 0
x = 0
= (sign x1 sign x2 : : : sign xn)T , an open interval, ]
 = ft : t 2 <
 < t <  g, a closed interval, 
] = ft : t 2 <
  t   g, a general interval that is not completely specied, it can be ]
] or 
 or 
], or ]
, end of a theorem, (inclusive) logical \OR", a _ b means either \a" or \b" or both \a" and \b", = (1 1 : : : 1)T 2
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Chapter 1

Introductory comments on stability concepts 1.1 Comments on Lyapunov's stability concept 1.1.1 Lyapunov's de nition of stability and the de nitions of stability in the Lyapunov sense

Lyapunov 127] used q = q1 q2 : : : qk q_1 q_2 : : : q_k ]T as the state (vector) of a material system with k degrees of freedom. It satises a vector dierential equation of the rst order and dimension equal to 2k (or to n if we replace 2k by n : n = 2k), which is of the form (1.1): dq = r(t
q)
r : < 
(1.3)

The perturbed motions essentially depend on X0 . Lyapunov 127] used scalar real-valued continuous functions Q1
Q2
: : :
Qm dependent on q and denoted their values taken along the unperturbed motion qu

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at an instant t 2 < by F1 (t)
F2(t)
: : :
Fm (t). This means that Fi(t) = Qi qu(t)]
8i = 1
2
: : :
n. Lyapunov's definition of stability of motion 127]: \Let L1
L2
: : :
Lm be arbitrarily given positive numbers. If for all Ls , irrespective of how small they are, it is possible to select positive numbers E1
E2
: : :
Ek
E10
E20
: : :
Ek0 so that for all real "j
"0j , which satisfy the conditions:

j"j j  Ej
j"0j j  Ej0 (j = 1
2
: : :
k)


and for every t, greater than t0 , the inequalities:

jQ ; F j < L
jQ ; F j < L
: : :
jQm ; Fm j < Lm
1

1

1

2

2

2

are satised, then the unperturbed motion is stable with respect to quantities Q1
Q2
: : :
Qm  otherwise { it is unstable with respect to the same quantities."

Comment 1.1 Lyapunov's denition is related to cases for which the initial instant

does not inuence the stability property 127]. In such cases we may accept t0 = 0, which holds for time-invariant (also called stationary) systems.

Comment 1.2 Lyapunov's denition demands that inequalities jQi ; Fi j < Li
8i = 1
2
: : :
m
are satised for all t 2 ]t
+1] rather than for all t 2 t
+1. However, it has been commonly accepted to require that for all t 2 < , < = t
+1 10], 0

0

0

94]{121], 218].

0

0

Comment 1.3 Lyapunov's denition introduces stability with respect to some con-

tinuous functions Qi dependent on q. In a special case Qi = qi and Qk+i = q_i, i = 1
2
: : :
k. It is interesting to note that Lyapunov himself studied only this special case. A justication for that is claried by the next theorem, which is a generalised version of theorem 1 by Grujic in 88] (page 7).

Theorem 1.1 Let the unperturbed motion qu be a dierentiable vector function. T Let Q = ( Q1 Q2 : : : Qn ) , n = 2k:

a)

X(t) = q(t) ; qu (t)


(1.4)

b)

f(t
X) = rt
qu(t) + X] ; rt
qu(t)]


(1.5)

c)

dX = f(t
X): dt

(1.6)

and, in view of (1), Then, stability of X = 0 of the system (1.4){(1.6) with respect to Q(t
X)  X is necessary and sucient for stability of the unperturbed motion qu(:) of the system (1.1) with respect to any vector function Q, Q : < 
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Proof. Necessity. Let Q(t
q)  q be chosen for system (1.1) and let Q(t
X)  X

be selected for the system (1.6). Such choices are permissible. The function Q is uniformly continuous. Let the unperturbed motion qu of the system (1.1) be stable with respect to any function Q that is uniformly continuous in (t
q) 2 < 
Su ciency. Let Q be arbitrary uniformly continuous function in (t
q) 2 < 
which implies the following: 8t 2 <
8q 2
imply:

jxj (t)j < lj and jxk j (t)j < lk j
8t 2 ]t
+1
8j = 1
2
: : :
k: +

+

(1.9)

0

Now, (1.5), (1.6), (1.7) and (1.9) prove that for the unperturbed motion qu of the system (1.1) the following holds:

8t 2 <
8li 2 ]0
+1
8i = 1
2
: : :
n
9(Ej
Ej0 ) 2 <  <
8j = 1
2
: : :
k
0

+

(1.10)

+

such that

jqj (t ) ; qju(t )j < Ej and jq_j (t ) ; q_ju(t )j < Ej0
8j = 1
2
: : :
k
0

guarantee

0

0

0

jqj (t) ; qju(t)j < lj and jq_j (t) ; q_ju(t)j < lk j
8t 2 ]t
+1
8j = 1
2
: : :
k: +

(1.11)

0

Since all li 2 <+ can be arbitrarily chosen in (1.11) then let them be accepted to obey (1.7). Now, (1.4) with q = qu in it, and (1.11) yield 8t0 2 <, 8Li 2 <+ , 8i = 1
2
: : :
n, 9(Ej
Ej0 ) 2 <+  <+ , 8j = 1
2
: : :
k, such that

j"j j = jqj (t ) ; qju(t )j < Ej 0

and

0

j"_j j = jq_j (t ) ; q_ju(t )j < Ej0
8j = 1
2
: : :
k
0

ensure

0

jQit
q(t)] ; Qi t
qu(t)]j = jQit
q(t)] ; Fi(t)j < Li
8t 2 ]t
+1
8i = 1
2
: : :
n
0

which proves stability of the unperturbed motion qu(:) of the system (1.1) with respect to the vector function Q. Theorem 1.1 reduces the stability analysis of the unperturbed motion qu of the system (1.1) to the stability analysis of the zero state X = 0 of the system (1.4){(1.6). Lyapunov's original denition, Comments 1{3 and Theorem 1.1 explain why the following denition is commonly used as the denition of stability in the Lyapunov sense in case the initial instant t0 is xed, e.g. t0 = 0. Let X(t t0
X0 ) be a motion of the system (1.6) at time t, which passes through X0 at time t = t0, X(t0 t0
X0 )  X0 .

Denition 1.1 a) A point X  2
+

0

0

0

0

0

b) The stability is global (in the whole, in the large) if and only if the maximal (" t0 X  ) denoted by M (" t0 X  ) obeying a) tends to +1 as " ! +1.

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c) If and only if X  = 0 then (" t0  0) and M (" t0
0) are denoted by (" t0 ) and M (" t0) respectively. Comment 1.4 a) The Lyapunov stability concept was originally concerned with stability of a motion and of the origin X = 0, which was later broadened to stability of a set 10], 218]. b) The distance was originally dened by Lyapunov 127] via the absolute values jQi(t) ; Fi (t)j. It is now commonly dened via the Euclidean norm in the literature, e.g. 10], 94], 116], 121], 218]. c) The closeness in the Lyapunov sense means that the distance is less than " for any given " 2 <+ , 94], 218]. d) The closeness in the Lyapunov sense is requested over the innite time interval <0 = t0
+1 <, 10], 94], 116], 121], 218]. This diers from Lyapunov's original requirement for the closeness over the time interval ]t0
+1]. e) The Lyapunov closeness is demanded provided external inuences to the system (1.1) are nominal over the time interval <0. This means the Lyapunov closeness is required for the system (1.6) in its free regime (i.e. under no external disturbance action over the time interval <0 ). f) The Lyapunov closeness is demanded for all the initial states X0 = X(t0 ) whose distance from X = 0 is less than some = (t0
") 2 <+ . g) Denition 1.1 broadens the concept of continuity of a function at a point to motions. For a deep analysis of it see Rouche and Mawhin 178, pp. 1{3]. Comment 1.5 Denition 1.1 can be satised by arbitrarily small 2 <+ even if the corresponding " 2 <+ has been chosen very large. It requires only the existence of 2 <+ obeying its condition. This is a conceptual drawback of the Lyapunov stability concept from the point of view of engineering, technical applications and needs.

Example 1.1 70] Let n = 2, X = x x ]T ,  2 < and the system (1.6) take

the following specic form:

1

2

+

dX = (; + jx j + jx j)X: 1 2 dt The system state portrait is given in Fig. 1.1. The state X = 0 is stable. However, if  is small (e.g.  = 10;3) or very small (e.g.  = 10;10), then the dynamic behaviour of the system is unsatisfactory in the engineering sense even for small initial states (jx10j + jx20j > 10;3) or (jx10j + jx20j > 10;10), respectively. Comment 1.6 Example 1.1 shows that existence of positive obeying Denition 1.1

is not adequate engineering information about the qualitative dynamic properties of the system. Useful information is that about the largest neighbourhood Ds (" t0) of X = 0 such that kX(t t0
X0 )k < " is satised for all t 2 <0 i X0 2 Ds(" t0 ) for any " 2 <+ . Moreover, we need information about the largest neighbourhood Ds (t0 ) of X = 0 containing all Ds (" t0 ), that is that Ds (t0) = Ds(" t0) : " 2 <+ ].

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x2 α

−α

0

α

x1

−α

Figure 1.1: The state portrait of the system of Example 1.1.

Note 1.1 For the sake of simplicity of introducing new concepts this book will

deal only with time-invariant continuous-time systems. In the preceding analysis time-varying systems were treated due to the reference to Lyapunov's original considerations and denition. Motions of time-invariant systems, as well as their properties, do not depend on the initial moment t0 . Hence, it is accepted equal to zero: t0 = 0: This enables the use of the simpler, more concise notation for a system motion, X(: X0): X(t t0
X0 )  X(t 0
X0)  X(t
X0 )  X(t): In case the system (1.6) is time-invariant then it will be described by (1.12): dX = f(X)
f :
X 0 ∈B δ

τ

t

δ ε

X (τ;X 0 )∈B ε Bδ Bε

8" 2 < , 9 2 < ! X 2 B ) X(t X ) 2 B" , 8t 2 < . +

+

0

0

+

Figure 1.2: Stable X = 0.

X0 0X

t

A N(δ,A)

N(ε,A)

N(ε,A)

8" 2 < , 9 2 < ! X 2 N (
A) ) X(t X ) 2 N ("
A), 8t 2 < . +

+

0

0

+

Figure 1.3: Stable set A.

Ds (" t ) reduces to Ds ("), Ds (" t )  Ds("), for time-invariant systems. Comments 1.5 and 1.6 are valid also for stability of a set A dened as follows. The distance function (:) :
0

+

+

0

+

0

+

Comment 1.7 We wish to know more than just about stability of a set A. We want to get information about what is, for any " 2 <+ , the largest neighbourhood Ds (" A) of the set A such that X(t X0 )
A] < " for all t 2 <+ holds i X0 2 Ds (" A), and in addition, to know Ds(A) = Ds(" A) : " 2 <+ ]. © 2004 by Chapman & Hall/CRC

The preceding denitions are illustrated by Fig. 1.2 and Fig. 1.3. From the engineering point of view " can be used to measure accuracy the lower ", the higher the accuracy. Stability can be interpreted as a system's ability to maintain as high nite accuracy of its motions as desired (i.e. arbitrarily small " 2 <+ ) provided initial state accuracy is su ciently high (i.e. is su ciently small, 2 <+ ), respectively.

1.1.2 De nitions of attraction

Although Lyapunov did not dene the notion of attraction, he was the rst to state general qualitative conditions for convergence of system motions to the zero state (Lyapunov's remark 2 in 127], page 61), which led to denitions of attraction and asymptotic stability of X = 0 of the system (1.12). The former is commonly dened as follows (e.g. 10], 94], 218]): Denition 1.4 a) The state X = 0 of the system (1.12) is attractive if and only if there is $ > 0 and for every > 0 there is 2 <+ , = (X0  ), such that kX0 k < $ implies lim kX(t X0)k : t ! +1] = 0. b) The state X = 0 of the system (1.7) is globally attractive (attractive in the whole, in the large) if and only if a) holds for $ = +1. Comment 1.8 The denition of attraction requires only the existence of $ > 0 obeying its condition irrespective of whether $ is large or small, even very small. In the case of the system (1.12) in the form given in Example 1.1 the state X = 0 is attractive. However, this property becomes useless from the engineering point of view as soon as  is small (e.g.  = 10;3) or very small (e.g.  = 10;10).

Comment 1.9 For engineering purposes it is important to derive, or at least to estimate well, the largest neighbourhood Da of X = 0 such that lim kX(t X )k : t ! +1] = 0 holds i X 2 Da . Denition 1.5 a) A set A is attractive with respect to motions of the system (1.12), for short, attractive, if and only if there is $ > 0 and for every > 0 there is 2 < , = (X
 A), such that (X
A) < $ implies X(t X )
A] <

for all t 2 ]
+1, and lim X(t X ) : t ! +1] = X 2 A. b) The set A is globally attractive (attractive in the whole, in the large) if and only if a) holds for $ = +1. 0

0

+

0

0

0

0

The preceding denitions are illustrated by Fig. 1.4 and Fig. 1.5.

Comment 1.10 We wish to know more than only about the attraction of a set A. We want to get information also about the largest neighbourhood Da (A) of the set A such that for every > 0 there is 2 < obeying that X(t X )
A] < for all t 2 ]
+1 i X 2 Da (A). Note 1.2 Notice that the condition \for every > 0 there is 2 < such that (X
A) < $ implies X(t X )
A] < for all t 2 ]
+1" is equivalent to \limfX(t X )
A] : t ! +1g = 0". +

0

0

+

0

0

0

© 2004 by Chapman & Hall/CRC

X0 t →+∞

∆ B∆

9$ 2 < ! X 2 B ) limfX(t
X )
O] : t ! +1g = 0, O = fOX g O = fOX g +

0



0

Figure 1.4: Attractive X = 0.

X0 0X t→+∞ A A N(∆,A)

9$ 2 < ! X 2 N ($
A) ) limfX(t X )
A] : t ! +1g = 0 +

0

0

Figure 1.5: Attractive set A.

1.1.3 De nitions of asymptotic stability

Stability and attraction are in general mutually independent system dynamic properties. It was well illustrated by Vinograd (Hahn 1967 94], pp. 191{194). Both are often desired, which led to the concept of asymptotic stability. Denition 1.6 a) The state X = 0 of the system (1.12) is asymptotically stable if and only if it is both stable and attractive. b) The state X = 0 of the system (1.12) is globally asymptotically stable (asymptotically stable in the whole, in the large) if and only if it is both stable and globally attractive. c) The state X = 0 of the system (1.12) is completely globally asymptotically stable if and only if it is both globally stable and globally attractive.

© 2004 by Chapman & Hall/CRC

Comment 1.11 The notion of asymptotic stability has the same drawback as sta-

bility and attraction. The state X = 0 of the system considered in Example 1.1 is asymptotically stable. Unfortunately, its asymptotic stability can be useless for engineering purposes in the case where  is small (e.g.  = 10;3) or very small (e.g.  = 10;10). Engineering applications of the stability analysis need more information about qualitative dynamic properties of the system. They often need data on the largest neighbourhood D of X = 0 which is contained in both Ds and Da , that is that on D = Ds \ Da . Asymptotic stability of X = 0 is meaningful for engineering purposes provided only that D is suciently large from the engineering point of view.

Denition 1.7 a) A set A is asymptotically stable with respect to motions of the system (1.12), in short, asymptotically stable if and only if it is both stable and attractive.

b) The set A is globally asymptotically stable (asymptotically stable in the whole, in the large) if and only if it is both stable and globally attractive. c) The set A is completely globally asymptotically stable if and only if it is both globally stable and globally attractive.

Comment 1.12 Besides asymptotic stability of a set A we often wish to know the set D(A) = Ds(A) \ Da (A).

1.1.4 De nitions of exponential stability

Asymptotic stability does not provide any information on the rate of system motion convergence to X = 0, or to the set A. Engineering requests for a higher quality of the system dynamic behaviour demand su cient rate of the system motion convergence. For this purpose the concept of exponential stability introduced by Krasovskii 116] is important. Denition 1.8 a) The state X = 0 of the system (1.12) is exponentially stable if and only if there are $ > 0 and positive numbers  and ,  1, such that kX0 k < $ implies kX(t X0 )k   kX0 k exp (; t) for all t 2 <+ . b) The state X = 0 of the system (1.12) is globally exponentially stable (exponentially stable in the whole, in the large) if and only if a) holds for $ = +1. Comment 1.13 Existence of $
 and is not sucient information for engineer-

ing applications. In addition to that, we need the knowledge of the smallest possible value of  , the largest possible value of and of the largest neighbourhood De of X = 0 such that kX(t X0)k   kX0 k exp (; t) for all t 2 <+ holds i X0 2 De . There is certainly a trade-o among  , and De in general, which is illustrated as follows. For the system of Example 1.1 the set De depends on  and via 1, De (1) = fX : jx1j + jx2j  1 g, where 1 =  ; should be positive. For given  and ,  > > 0, the set De(1 ) is the largest neighbourhood of X = 0 such that p X0 2 De (1) implies kX(t X0 )k   kX0 k exp (; t) for all t 2 <+ , where  = 2.

© 2004 by Chapman & Hall/CRC

β∆ e X0

xp(-

γ t) t

0X

∆

p(-γ t

)

β∆ ex

β∆ Bβ∆

B∆

9$ > 0
9(
) 2 <  <
 1 ! X 2 B ) X(t X )
O]  (X
O) exp (; t)
8t 2 < . +

0

+

0

0



+

Figure 1.6: Exponentially stable X = 0.

Denition 1.9 a) A set A is exponentially stable with respect to motions of the system (1.12), for short: exponentially stable, if and only if there are $ > 0 and positive numbers  and ,  1, such that (X
A) < $ implies X(t X )
A]  (X
A) exp (; t) for all t 2 < . b) The set A is globally exponentially stable (exponentially stable in the whole, in the large) if and only if a) holds for $ = +1. Comment 1.14 In addition to information on exponential stability of a set A we need information about the largest neighbourhood De (A) of the set A such that X(t X )
A]  (X
A) exp (; t) for all t 2 < holds i X 2 De(A). Note 1.3 In the sequel the expression \globally" can be everywhere replaced by \in 0

0

0

0

0

+

+

0

the whole", or by \in the large" as pointed out in the preceding denitions, which are illustrated by Fig. 1.6 and Fig. 1.7.

1.1.5 De nitions of absolute stability on

Ni (L)

The notion of absolute stability is due to Lurie and Postnykov 125]. It is related to nonlinear systems called after them Lurie{Postnykov systems, for short, Lurie systems, which are described by (1.13) and (1.14): a) dX (1.13) dt = AX + Bf(w)
b) w = CX + Df(w)
(1.14) where X 2
A

A

X0

∆ β∆

t

0X

N(∆,A)

N(β∆,A)

9$ > 0
9(
) 2 <  <
 1 ! X 2 N ($
A) ) X(t X )
A]  (X
A) exp (; t)
8t 2 < . +

0

+

0

0

+

Figure 1.7: Exponentially stable set A.

(matrix) sector L, L = (K1
K2), Ki 2 <m
m , i = 1
2. The notation (K1
K2 ) has a general meaning and can be replaced by any of the following: K1
K2]
]K1
K2]
K1
K2 
or ]K1
K2: The matrices K1 and K2 are diagonal, 0  K1  K2 . The diagonal entries k2j of K2 can be positive numbers or equal to +1, Ki = diag fki1 ki2 : : : kim g, i = 1
2. The vector nonlinearity f is in the class (family) N0(L S ) i the system (1.13), (1.14) has a unique solution through X = 0 and: a) f(CX) 2 C(S )
b) c)

f(0) = 0


> fj (W ) > > wj >W

8 2 (k
k )
8X 2 S
w 6= 0
< j j j : = 0
8X 2 S
wj = 0
j = 1
2
: : :
m: 1

=

W (X )

(1.15)

2

With Nd (L S ) we denote all odd f 2 N0 (L S ), and N1 (L
M S ) designates the family of all f 2 N0 (L S ) obeying also (1.16) and (1.17): a) fW (X)] 2 C (1)(S )
(1.16) b) where

>

fW (W)> >W =W (X) 2 M
8X 2 S






(1.17)

@fi : <m ! <m
m : fW (W ) = @w j fW is the Jacobian (matrix function) of the vector nonlinearity f, and M = (M1
M2) <% m
m . In case S =
Denition 1.10 The state X = 0 of the system (1.14) is absolutely stable on N : (L) on N (L M)], if and only if it is globally asymptotically stable for every f 2 N : (L) for every f 2 N (L
M)], respectively. 1

()

1

()

Comment 1.15 The requirement for the global asymptotic stability is often unnecessarily stringent from the engineering point of view. There is not a technical system whose initial state can be arbitrary far from the origin. Besides, for many nonlinearities the condition (1.15c) appears too restrictive in the case S =
()

1

()

respectively.

1.1.6 De nitions of attraction with nite attraction time

The attraction property claries behaviour of motions as t ! +1. For engineering needs it is more important to ensure attraction as time t converges to a nite time a 2 <+ .

Denition 1.12 a) The state X = 0 of the system (1.12) is attractive with the nite attraction time a (X ) if and only if there is $ > 0 and for every X 2
0

0

+

0

0

0

the chosen X0 .

0

b) The state X = 0 of the system (1.12) is globally attractive with the nite attraction time a (X0 ) if and only if the preceding conditions under a) hold for $ = +1.

The dierence between attraction and attraction with nite attraction time is illustrated by Fig. 1.8. Denition 1.12 can be generalised to sets as follows.

Denition 1.13 a) A set A is attractive with the nite attraction time a , a = a (X  A) 2 < , if and only if there is $ > 0 and for every X 2
+

0

0

+

0

0

0

0

0

b) The set A is globally attractive with the nite attraction time a if and only if the conditions under a) hold for $ = +1.

The nite attraction time can be also called the nite reachability time.

© 2004 by Chapman & Hall/CRC

X0

X0

∆ 0X

t→+∞

∆ 0X

τ a (X 0 )

t

B∆

B∆

8

a) τa(X0)=+

b) τa(X0)∈R+

Figure 1.8: a) attraction of X = 0, b) attraction of X = 0 with the nite attraction time a (X0 ) 2 <+ .

1.1.7 De nitions of stability with nite attraction time

Denitions 1.6 and 1.12 lead to the following. Denition 1.14 a) The state X = 0 of the system (1.12) is stable with the nite attraction time a if and only if it is both stable and attractive with the nite attraction time a . b) The state X = 0 of the system (1.12) is globally stable with the nite attraction time a if and only if it is both stable and globally attractive with the nite attraction time a . c) The state X = 0 of the system (1.12) is completely globally stable with the nite attraction time a if and only if it is both globally stable and globally attractive with the nite attraction time a .

Similarly, Denitions 1.7 and 1.13 yield the following notions.

Denition 1.15 a) A set A is stable with the nite attraction time

a if and only if it is both stable and attractive with the nite attraction time a . b) The set A is globally stable with the nite attraction time a if and only if it is both stable and globally attractive with the nite attraction time a . c) The set A is completely globally stable with the nite attraction time a if and only if it is both globally stable and globally attractive with the nite attraction time a .

The preceding denitions are illustrated by Fig. 1.9.

1.1.8 De nitions of absolute stability with nite attraction time In the framework of the Lurie systems (1.13) and (1.14) we can now introduce the following notions.

© 2004 by Chapman & Hall/CRC

τ a (X 0 ;A)

τ a (X 0 )

∆ ||X 0 ||

0

∆ ρ(X 0 ,A)

0

a) τ a (X 0 )∈R + , ∀X 0 ∈B ∆

b) τ a (X 0 ;A)∈R + , ∀X 0 ∈N(∆,A)

Figure 1.9: Dependence of a on X0 . a) X = 0 is attractive with the nite attraction time a , b) the set A is attractive with the nite attraction time a . Although a is nite for every X0 2 B in case a) and for every X0 2 N (
A) in case b), in both cases sup  a(X0 ) : X0 2 B ] = +1 and sup  a (X0  A) : X0 2 N (
A)] = +1.

Denition 1.16 a) The state X = 0 of the system (1.14) is absolutely stable on N : (L) with the nite attraction time a if and only if it is globally stable with the nite attraction time a , a = a (X  f), for every f 2 N : (L). ()

0

()

b) The state X = 0 of the system (1.14) is absolutely stable with the nite attraction time a on N(:)(L) if and only if a) holds and sup  a (X0  f) : f 2 N(:) (L)] < +1 for every X0 2
Denition 1.17 a) A set A is absolutely stable on N : (L) with the nite attraction time a if and only if it is globally stable with the nite attraction time a , a = a (X  f), for every f 2 N (L). b) A set A is absolutely stable with the nite attraction time a on N : (L) if and only if a) holds and sup  a (X  f) : f 2 N (L)] < +1 for every X 2
0

()

()

0

()

0

1.2 Comments on the practical stability concept 1.2.1 Introductory comments

The practical stability concept aims at determining closeness in a practical, engineering sense, at clarifying the system behaviour either in the free or in a forced regime over a time interval that can be bounded and for initial states of practical, engineering signicance. The closeness can be dened via a suitably accepted distance function and/or via sets. The latter appears more general and will be utilised

© 2004 by Chapman & Hall/CRC

x2 4α

2α α −4α

−2α

−α

X(τ;X0) X0

0

α

2α

4α

x1

−α −2α xA x0A

−4α

Figure 1.10: The state portrait of the system of Example 1.1 and the sets X0A and XA .

in the sequel for the system (1.18): dX = f(X
i)
f :
1.2.2 De nition of practical stability

By referring to LaSalle and Lefschetz 121], Weiss and Infante 208], 209], Michel 144]{146], Gruji&c 60]{63] and Martynyuk 133] we shall accept the following practical stability denition in which <
= 0
 and I is the family of inputs of our interest. Besides a set X( ) is assumed to have a non-empty interior X( ).

Denition 1.18 The system (1.18) is practically stable with respect to f
X A, XA
Ig if and only if its motions obey X(t X  i) 2 XA for every (t
X
i) 2 <
 X A  I. Comment 1.16 Let the system of Example 1.1 (Section 1.1.1) be reconsidered: 0

0

0

dX = (; + jx j + jx j)X: 1 2 dt © 2004 by Chapman & Hall/CRC

0

x2 4α

2α α −4α

−2α

−α

2α

α

0

4α

x1

−α −2α

x0A

xA −4α

Figure 1.11: The state portrait of the system of Comment 1.17 and the sets X0A and XA .

If

X A = fX : X 2 <
jx j + jx j  2g
XA = fX : X 2 <
jx j + jx j  4g
2

0

2

and

1

1

2

(1.19)

2

I = f0g


then for the system to be practically stable with respect to f
X0A
XA
Ig it is both necessary and sucient that is suciently small, 2 <+ . If it is not satised, i.e. if  (Fig. 1.10) then the system is not practically stable with respect to f
X0A
XA
Ig. Notice again that X = 0 is asymptotically stable.

Comment 1.17 Let X 2 < and let the system (1.18) be specied in the following 2

form:

Evidently, I = f0g. If

dX = ( ; jx j ; jx j)X
 2 <+ : 1 2 dt

X A = fX : X 2 <
jx j + jx j  2g
2

0

and

1

2

XA = fX : X 2 <
jx j + jx j  4g
then the system is practically stable with respect to f
X A
XA
I g for any 2 < despite X = 0 is unstable, Fig. 1.11. The same holds if we redene XA and X A so that XA = X A = fX : X 2 <
jx j + jx j   g for any  2 ]
+1. 2

1

2

+

0

0

© 2004 by Chapman & Hall/CRC

2

0

1

2

X0

x0A xA

xA t0=0

τ

Rτ

Figure 1.12: The system is practically stable with respect to f
X0A
XA
Ig.

X0

Rτs

x0F

xF

xF

t t0=0

τs

ττs

τ

Figure 1.13: The system is practically contractive with the settling time s with respect to f
X0F
XF
Ig.

1.2.3 De nition of practical contraction with settling time The notion of practical stability does not reect any contraction property, which is expressed by the following in which T
s =  s
.

Denition 1.19 Motions of the system (1.18) are practically contractive (or, for short, the system (1.18) is practically contractive) with settling time s with respect to f
X0F
XF
Ig if and only if X(t X0 i) 2 XF for every (t
X0
i) 2 T
s X0F I . The dierence between practical stability and practical contraction with settling time s is illustrated by Fig. 1.12 and Fig. 1.13.

© 2004 by Chapman & Hall/CRC

X0

xA

x0

xA xF

xF t

t0=0

τs

τ

Figure 1.14: The system is practically stable with the settling time s with respect to f
X0
XA
XF
Ig.

1.2.4 De nition of practical stability with settling time

Denitions 1.18 and 1.19 combined imply the following notion.

Denition 1.20 The system (1.18) is practically contractively stable (for short, practically stable) with the settling time s with respect to f ,X ,XA,XF
Ig, XF  XA, if and only if both a) X(t X  i) 2 XA for every t 2 <

0

0

and b)

X(t X0  i) 2 XF for every t 2 T
s


hold for every (X0
i) 2 X0  I .

This denition is illustrated by Fig. 1.14.

© 2004 by Chapman & Hall/CRC

Chapter 2

Stability domain concepts 2.1 Introductory comments In order to get complete information about the causality between initial states and systems motions, concepts of domains of various stability properties were introduced. In the framework of the Lyapunov stability the notion of attraction domain of the origin was dened by Zubov 218] and Hahn 94], and notions of the stability domain and the asymptotic stability domain were dened by Gruji&c 64], 65], 68]{70], 72], and used by Gruji&c et al. 88]{90]. The concept of practical stability domains was introduced by Gruji&c 70]. In the literature (e.g. LaSalle and Lefschetz 121] and Zubov 218]) the notion of \region of asymptotic stability" has been used in the sense of the attraction domain. In what follows the dierence between them will be claried.

2.2 Domains of Lyapunov stability properties 2.2.1 The notion of domain

The term \domain" denotes a set that can be, but need not be, open or closed. Domains of Lyapunov stability properties will be called for short \Lyapunov stability domains" in a general sense incorporating domains of stability, of attraction and of asymptotic stability. In the closer sense the notion \Lyapunov stability domain" will be used for the domain of stability (for short, the stability domain). Lyapunov stability domains will be studied herein in the framework of timeinvariant continuous-time nonlinear systems governed by dX = f(X)
(2.1) dt with possibly certain specic features that will be described when they are needed. In the literature (e.g. LaSalle and Lefschetz 121]) the notion \region" has been used for an open connected set. We do not wish a priori to impose such a restriction

© 2004 by Chapman & Hall/CRC

on the largest set of initial states appropriate for a corresponding stability property. Therefore, we shall use the term \domain" in general rather than \region". In case a domain is open and connected then we can also call it a \region".

2.2.2 De nitions of stability domains

The denitions of stability domains were introduced to comply with the denitions of stability of a state and of a set 64], 65], 68], 69], 70], 72].

Denition 2.1 (a) The state X = 0 of the system (2.1) has the stability domain denoted by Ds if and only if both (i) for every " 2 < there is a neighbourhood Ds (") of X = 0 such that kX(t X )k < " for all t 2 < holds provided only that X 2 Ds ("), +

0

+

0

and (ii) Ds =  Ds (") : " 2 <+ ]. (b) The state X = 0 of the system (2.1) has the strict domain of stability (the strict stability domain) Dsc if and only if (i) it has the domain Ds of stability, (ii) Dsc (") is the largest connected neighbourhood of X = 0 which is subset of Ds ("), Dsc(") Ds ("), for every " 2 <+ , (iii) Dsc =  Dsc (") : " 2 <+ ].

Comment 2.1 The state X = 0 of the system (2.1) has the stability domain i it is stable due to Denition 2.1 and Denition 1.2. (Section 1.1.1). Moreover, it is globally stable i its strict stability domain Dsc is its whole state space
Example 2.1 Let a second order system (2.1) be in the following form: dX = (; + jx j + jx j)X
 2 <+ : 1 2 dt It has the innite set Se of equilibrium states,

Se = X : X 2 <
(X = 0) or (jx j + jx j = ) : 2

© 2004 by Chapman & Hall/CRC

1

2

p

We are now interested only in stability of X = 0. Let " =  2=2. For any " 2 ]0
"], the maximal (") denoted by M (") (Denition 1.2, Section 1.1.1) obeys M (") = ". For any " 2 "
+1 the maximal M (") = " . However 8 > B"
" 2 ]0
"]
> < kX(t
X0 )k < "
8t 2 <+
i X0 2 > B" \ S
" 2 ]"
]
> : S
" 2 ]
+1
where





S = X : X 2 <
jx j + jx j  
which is illustrated by Fig. 2.1. Evidently, @ S p= Se ; O. 2

1

2

It is now obvious (Fig. 2.1) that for " =  2=2

8 > B
" 2 ]0
"]
> < " Ds (") = > B" \ S
" 2 ]"
]
> : S
" 2 ]
+1


so that

Ds =  Ds (") : " 2 < = S : In this example the stability domain is the closed set S , i.e. Ds = X : X 2 <
jx j + jx j  
which is showed in Fig. 2.1f. The strict stability domain Dsc equals Ds , Dsc = Ds . +

2

1

2

Denition 2.2 (a) A set A 
0

+

0

and (ii) Ds (A) =  Ds (" A) : " 2 <+ ]. (b) A set A 
© 2004 by Chapman & Hall/CRC

x2

x2

α

α D s (ε) = B ε

ε

−α

−ε

α

ε

x1

−α

α

ε∗

Se−O

−ε

Sα

ε∗

x1

D s (ε ∗ ) = B ε ∗ −α

−α a)

" 2 ]0 " ) Ds(") = B"

"="





=

x2

p2 2

b)

) Ds ("



)=

B"



x2

α

ε=α B ε =B α

Bε

−α

x1

α

−α

α

D s (ε)

x1

D s (ε) −α

−α c)

d)

" 2 ]"  ] ) Ds (") = B" \ S


" = ) Ds (



x2

)=

B" \ S


x2 α

α

Bε

−α

x1

α

−α

α

D s (ε)=D s

D s =S α

−α

−α

e)

" 2 ]  +1 ) Ds (") = S


f)

Ds =

Ds (") : " 2 <+ ] = S


Figure 2.1: Dependence of Ds (") on ".

© 2004 by Chapman & Hall/CRC

x1

x2 2 1 -1

-4

4

1

x1

-1

A

-2

Figure 2.2: The state portrait of the system X_ = (1 ;jx1 j;jx2 j) (4 ;jx1 j; 2jx2 j)X . The shaded area together with its boundary represents the set A.

Example 2.2 Let the second order system (2.1) be specied by

dX = (1 ; jx j ; jx j)(4 ; jx j ; 2jx j)X: 1 2 1 2 dt We are interested in stability of the set A,

A = X : X 2 <2
jx1j + jx2j  1 : The state portrait of the system is given in Fig. 2.2. The set Se of the equilibrium states of the system is found as

Se = X : X 2 <2
(X = 0) or (jx1j + jx2j = 1) or (jx1j + 2jx2j = 4) : The graphical analysis of dependence of Ds (" A) on " is presented in Fig. 2.3. In this example (Denition 1.3, Section 1.1.1)

# p# 8 > 2 5 > < "
" 2 0
5
"p " M (") = > p 2 5 5 2 > : 5
" 2 5
+1 :

However,

# p# 8 > > N ("
A)
" 2 0
2 5 5
> > < # # kX(t
X )k < "
8t 2 <
i X 2 > N ("
A) \ S
" 2 2p5
3
> 5 > > : S
" 2 ]3
+1
0

+

0

4

4

© 2004 by Chapman & Hall/CRC

x2

x2 A

2

2 1

1 -4

-1

0

ε∈ ] 0, 2√5 ] 5 a)

-1

x2

x2 N(ε,A)∩S 4 =D s (ε,A)

2

2

-1

1

4 x1

1

-4

-1

1

0

-1

ε

0

-1

ε

N(ε,A)

1 -4

4 x1

1

0

-2 N( 2√5 ,A)=D s ( 2√5 ,A) 5 5 ε= 2√5 5 b)

-2

N(ε,A)=D s (ε,A)

-1

ε

-1

-4

4 x1

1

= 2 ε= √ 5 5

S4

-2

4

x1

-2 S 4 =D s (ε,A)=D s (A)

N(ε,A) c)

d)

Figure 2.3: The dependence of the subset Ds(
A) of Ds (A) on  2 ]0p
+1 (Example 2.2): a) Ds(
pA) = N (
A)p= N (
A) \ Sp4
 2 ]0
2 5 5 ]: p b) Ds( 2 5 5
A) = N ( 2 5 5
A) = N ( 2p5 5
A) \ S4
 = 2 5 5 : c) Ds(
A) = N (
A) \ S4
 2  2 5 5
3]: d) Ds(
A) = S4
 2 3
+1: Hence Ds(A) =  Ds(
A) :  2 ]0
+1 ] = S4 :

where





S = X : X 2 <
jx j + 2jx j  4 : 4

2

1

This shows, which is illustrated by Fig. 2.3, that

2

# p# 8 > > N ("
A)
" 2 0
2 5 5
> > < # # Ds (" A) = > N ("
A) \ S
" 2 2p5
3
> 5 > > : S
" 2 ]3
+1
4

4

© 2004 by Chapman & Hall/CRC

x2

3 2 1 -3 -2

-1

0 1

2

x1

3

-1 -2 -3

Figure 2.4: The state portrait of the system (Example 2.3) described via the polar coordinates  = kX k,  = arctan xx2 by _ = ;(1 ; 2 )(4 ; 2 )(9 ; 2 ), _ = ;1. 1

so that the stability domain Ds(A) of the set A is found as Ds (A) = S4 ,





Ds (A) = X : X 2 <
jx j + 2jx j  4 : 2

1

2

In this example Dsc = Ds .

Note 2.1 In case A = O, O = f0g 
Note 2.2 A nonlinear dynamical system can have both a stable limit cycle and the domain of stability of the set O (of X = 0). The next example illustrates this assertion.

Example 2.3 A second order nonlinear system described by

2 ;(1;x ;x )(4;x ;x )(9;x ;x ) dX = 6 dt 4 2 1

2 2

2 1

;1

2 2

2 1

1

2 2

;(1 ; x ; x )(4 ; x ; x )(9 ; x ; x ) 2 1

2 2

2 1

2 2

2 1

2 2

3 75X

has three limit cycles: x21 + x22 = 1, x21 + x22 = 4 and x21 + x22 = 9. The zero state is the system unique equilibrium state. It is also stable. This is illustrated by Fig. 2.4.

© 2004 by Chapman & Hall/CRC

It is now obvious in view of Fig. 2.4 that 8 kX k < "
" 2 ]0
1]
> 0 > > < kX k  1
" 2 ]1
2]
kX(t
X0 )k < " i > 0 kX0k < "
" 2 ]2
3]
> > : kX0k  3
" 2 ]3
+1]: This means 8 > B"
" 2 ]0
1]
> < B%
" 2 ]1
2]
Ds(") = > 1 B"
" 2 ]2
3]
> > : B%3
" 2 ]3
+1]
which yields Ds =  Ds (") : " 2 ]0
+1] =  (B" : " 2 ]0
1])]  B%1  B" : " 2 ]2
3]]  B%3 = B1  B%1  B3  B%3 = B%3 : Hence, the stability domain Ds of X = 0 of the system equals compact circle B%3 ,

Ds = B% = fX : X 2 <
kX k  3g: Besides, the stability domain Ds equals the strict stability domain Dsc of X = 0, Ds = Dsc . 3

2

2.2.3 De nitions of attraction domains

The notion of the attraction domain has been widely used (cf. 94], 218]). Denition 2.3 (a) The state X = 0 of the system (2.1) has the attraction domain denoted by Da if and only if both (i) and (ii) hold:

(i) for every  2 <+ , there is 2 <+ , = (X0
), such that kX(t X0 )k <  for all t 2 ]
+1 holds provided only that X0 2 Da , and (ii) Da is a neighbourhood of X = 0. (b) The state X = 0 of the system (2.1) has the strict domain of attraction (the strict attraction domain) denoted by Dac if and only if Dac is the largest connected neighbourhood of X = 0, which is subset of Da , Dac Da .

Note 2.3 The condition (i) of the preceding denition can be expressed in an equivalent form (i0 ) , (i0 ) lim kX(t X0 )k : t ! +1] = 0 provided only that X0 2 Da . © 2004 by Chapman & Hall/CRC

x2 α

−α

0

α

x1

−α

Figure 2.5: The state portrait of the system: X_ = (; + jx1 j + jx2j)X (Example 2.4).

Example 2.4 Let the system considered in Example 2.1 be reanalyzed,

dX = (; + jx j + jx j)X
 2 <+ : 1 2 dt Its state portrait is shown in Fig. 2.5. The attraction domain Da of X = 0 of the system is the interior S = X : X 2 <2
jx1j + jx2j < 

of the set

S = X : X 2 <2
jx1j + jx2j  
Da = S

Da = X : X 2 <2
jx1j + jx2j <  : The boundary

@ S = X : X 2 <2
jx1j + jx2j =  is a subset of the set

Se = X : X 2 <2
(X = 0) or (jx1j + jx2j = ) of all equilibrium states. Hence, X(t
 X0)  X0 for every X0 2 @ S . Therefore, @ S \ Da =  and Da is open set. In this example, Da = Dac . The domain Da of attraction of X = 0 of the system is open (bounded connected) set, while its domain Ds = Dsc of stability is closed (also bounded connected) set. © 2004 by Chapman & Hall/CRC

x2 α

−α

Da

α x1

0

−α

Figure 2.6: The attraction domain Da of X = 0 of the system: X_ = (; + jx1 j + jx2 j)X .

They are interrelated so that Da is the interior of Ds and Ds is the closure of Da , hence, Da is subset of Ds , Da = Ds Ds = D%a
Ds  Da :

Example 2.5 Let

8 i kX k  1
> <0 (kX k) = > kX k + 1 : kX k ; 1 i kX k > 1
0

0

0

0

0

and the system motions be dened by X(t X0) = exp (;t) 1 + (X0 )t] X0 : They satisfy the initial condition, X(0 X0)  X0 , and are continuous in t 2 <+ for every X0 2
||X(t ; X0)|| ||X0||+1 ||X0||-1

|| X

||X0|| exp

(t

(-||X 2||+1) 0

; X^

0 ) ||

^ 1< || X0 || 1

t

2 ||X0||+1

1

0

Figure 2.7: The norm of motions dened by X (t X0 ) =1+ (X0)t]X0 exp (;t)
(X0 ) = 0 0k + 1 i kX0 k  1 and (X0 ) = kkX X k ; 1 i kX0 k > 1 (Example 2.5). 0

 k X k + 1 B) kX k > 1 yields kX(t X )k = exp (;t) 1 + kX k ; 1 t kX k. In this case,   X k + 1 kX k exp ; 2 max kX(t X )k : t 2 < ] = kkX k;1 kX k + 1 ! +1 as kX k ! 1 and kX(t X )k ! 0 as t ! +1 for every X 2 1. 

0

0

0

0

0

0

0

+

0

0

0

0

+

0

0

0

Now, we can conclude that 8 fX : X 2 > > < % n Ds (") = > B1  fX : X 2 <
kX k > 1
  > > : kkXX kk +; 11 kX k exp ; kX k2+ 1 < "g \ B"
" 2 ]1
+1]
© 2004 by Chapman & Hall/CRC

8 < fX : X 2
so that

Ds = Ds(") : " 2 < ] =
and

Dsc = Dsc(") : " 2 < ] = ( B" : " 2 ]0
1]])  B% = B% The strict stability domain Dsc of X = 0 is compact hyperball B% . Furthermore, we conclude by examining the motions that the attraction domain Da of X = 0 is the whole state space
1

1

1

Denition 2.4 (a) A set A 
+

0

0

0

and (ii) Da is a neighbourhood of the set A. (b) A set A 
Note 2.4 The condition (i) of Denition 2.4 can be expressed in an equivalent form (i0 ), (i0 ) lim f X(t X0 )
A] : t ! +1g = 0 provided only that X0 2 Da .

Example 2.6 The attraction domain Da(A) of the set A, A = fX : X 2 <
jx j + jx j  1g
2

1

2

of states of the system described by (see Fig. 2.2) dX = (1 ; jx j ; jx j)(4 ; jx j ; 2jx j)X
1 2 1 2 dt is found as (Fig. 2.8) Da (A) = fX : X 2 <2
jx1j + 2jx2j < 4g: © 2004 by Chapman & Hall/CRC

x2

D(A)

2

1 -4

-1

0

4

1

x1

-1

A

-2

Figure 2.8: The attraction domain Da (A) of the set A = fX : X 2 <2
jx1j + jx2 j  1g of states of the system X_ = (1 ; jx1 j ; jx2j)(4 ; jx1 j ; 2jx2 j)X .

The attraction domain Da (A) of the set A is an open subset of its stability domain Ds(A) (Fig. 2.3d). In fact Da (A) = Dac (A) = Ds (A), D% a (A) = Ds(A). Since Da (A) is bounded, then the set A is not globally attractive.

Example 2.7 The set A,

A = X :X 2 <
x +x = 4
2

2 1

2 2

represents a limit cycle of the system 2 ;(1;x2 ;x2)(4;x2 ;x2)(9;x2 ;x2) 3 1 1 2 1 2 1 2 dX = 6 75X dt 4 ;1 ;(1 ; x21 ; x22 )(4 ; x21 ; x22 )(9 ; x21 ; x22) The attraction domain Da (A) of the set A is bounded, open and connected,

Da (A) = X : 1 < x21 + x22 < 9
Da (A) = Dac (A)
which is shown in Fig. 2.9a. Its stability domain Ds (A),

Ds(A) = X : 1  x21 + x22  9 = Da (A)
is shown in Fig. 2.9b. In this example Ds (A)  Da (A).

2.2.4 De nitions of asymptotic stability domains

Asymptotic stability domains were dened in 64], 65], 68]{70], 72] to correspond to the denitions of asymptotic stability of a state and of a set (Section 1.1.3). Denition 2.5 (a) The state X = 0 of the system (2.1) has the asymptotic stability domain denoted by D if and only if both

© 2004 by Chapman & Hall/CRC

x2

3

3

2

2

1

1

0 -3

-2

-1

x2

A

-1

0

x1

1

2

-3

3

a)

-2

-1

-1

x1 1

2

3

-2

-2 -3

A

Da(A)

-3

Ds(A)

b)

Figure 2.9: a) The domain Da (A) of attraction of the set A = fx : x 2 <2
x21 + x22 = 4g b) The domain Ds(A) of stability of the set A (Example 2.7).

(i) it has the stability domain Ds and the attraction domain Da , and (ii) D = Ds \ Da . (b) The state X = 0 of the system (2.1) has the strict domain of asymptotic stability (the strict asymptotic stability domain) denoted by Dc if and only if it has both the strict stability domain Dsc and the strict attraction domain Dac and Dc = Dsc \ Dac .

Comment 2.2 If the state X = 0 of the system (2.1) has both Ds { the stability domain and Da { the attraction domain, then they are neighbourhoods of it. Hence, D { the asymptotic stability domain is also a neighbourhood of X = 0. It can be closed, can be open, which will be illustrated via examples. It need not be connected in general. However, Dc is a connected neighbourhood of X = 0 due to such property of Dsc and Dac , and Dc = Dac \ Dsc . Example 2.8 Let the second order system analyzed in Example 2.1 and Example 2.4 be further considered, dX = (; + jx j + jx j)X
 2 <+ : 1 2 dt It was shown that X = 0 of the system has both the stability domain Ds,

Ds = X : X 2 <2
jx1j + jx2j  
Dsc = Ds
and the attraction domain Da , Da = X : X 2 <2
jx1j + jx2j <  = Ds
Dac = Da : © 2004 by Chapman & Hall/CRC

In view of Denition 2.5, the state X = 0 of the system has also the asymptotic stability domain D,





D = Ds \ Da = X : X 2 <
jx j + jx j <  = Da
D = Dc : In this example D equals Da and they are subsets of Ds . They are exactly the interior Ds of Ds, Da = D = Ds, in this case. 2

1

2

Example 2.9 The zero state of the second order nonlinear system 2 ;(1 ; kX k )(4 ; kX k )(9 ; kX k ) 1 dX 2

6 dt = 4

2

2

3 75X

;1 ;(1 ; kX k )(4 ; kX k )(9 ; kX k ) has the stability domain Ds (Example 2.3),

Ds = B% = X : X 2 <
kX k  3
Dsc = Ds
and the attraction domain Da (Fig. 2.4), Da = B = X : X 2 <
kX k < 1
Dac = Da
Hence, its asymptotic stability domain D, D = Ds \ Da = B = X : X 2 <
kX k < 1
Dc = D
equals Da and they are subsets of Ds , but they are also proper subsets of the interior Ds of Ds. 2

2

2

2

3

2

1

2

1

Example 2.10 In the case of the system dened by its motions, X(t X0 ) = e;t 1 + (X0 )t] X0


8 0
i kX k  1
> <
(X ) = > kX k + 1 : kX k ; 1
i kX k = 1 0

0

0 0

0

the zero state has both the strict stability domain Dsc (Example 2.5), Dsc = B = fX : kX k  1g, Ds =
1

1

1

asymptotically stable, but it is not completely globally asymptotically stable.

© 2004 by Chapman & Hall/CRC

Denition 2.6 (a) A set A 
and (ii) D(A) = Ds (A) \ Da (A). (b) A set A 
Example 2.11 The set A, A = fX : X 2 <
jx j + jx j  1g
2

1

2

of states of the system dX = (1 ; jx j ; jx j)(4 ; jx j ; 2jx j)X
1 2 1 2 dt has both the stability domain Ds (A) (Example 2.2),

Ds (A) = fX : X 2 <
jx j + 2jx j  4g
Dsc (A) = Ds (A)
and the attraction domain Da (A) (Example 2.6), Da (A) = fX : X 2 <
jx j + 2jx j < 4g = Ds (A)
Dac (A) = Da (A): Hence, its asymptotic stability domain D(A) equals Da (A) that is the interior Ds (A) of Ds(A). Notice that X = 0 is unstable, thus it does not have the stability domain 2

2

1

1

2

2

and the asymptotic stability domain.

Comment 2.3 If X = 0 is the unique equilibrium state of the system (2.1) and is stable then as soon as X 62 Ds the motion X(t X ) is either not dened on the whole time axis < or it is not bounded. However, such a statement cannot be applied either to Da and D if X = 0 is also attractive as soon as Da  Ds , or to Dsc if Dsc = 6 Ds. 0

0

+

2.2.5 De nitions of exponential stability domains

In order to explain the relative sense of the denitions of exponential stability domains we shall rst consider the system of Example 2.1 once more.

Example 2.12 The motions of the system dX = (; + jx j + jx j)X
 2 <+ : 1 2 dt © 2004 by Chapman & Hall/CRC

obey the estimate

kX(t X )k   kX k exp (; t)
8t 2 <
0

only for and for

0

+

X0 2 S = fX : X 2 <2
jx1j + jx2j  g


2 ]0

=  ;


with  = 1: We can accept only 2 ]0
. Once has been accepted then is completely determined by =  ; . The bigger , the larger S and the smaller . In other words, the larger set S over which the motions obey the exponential estimate kX(t X0)k  kX0k exp (; t)
8t 2 <+
the smaller rate of the exponential convergence of the motions to the origin. If we wish to nd > 0 in the limiting case = , i.e. S = S = fX : X 2 <2
jx1j + jx2j  g
then we conclude that > 0 does not exist. The same holds for the interior S = fX : X 2 <2
jx1j + jx2j < g of S . The consequence of this is that there does not exist the maximum set S (or S ) for which we can nd obeying the exponential estimate because 2 ]0
 and max ]0
 does not exist, although sup ]0
 =  { exists. Therefore, we cannot speak of the largest (or the maximum) set of system states over which the exponential estimate holds for some  2 1
+1 and 2 ]0
+1. Instead, we can look for the largest set of system states over which the exponential estimate holds with respect to given  2 1
+1 and given 2 ]0
+1. Denition 2.7 (a) The state X = 0 of the system (2.1) has the domain De (
) of exponential stability with respect to (
) if and only if both (i) De (
) is a neighbourhood of X = 0, and (ii) the exponential estimate

kX(t X )k   kX k exp (; t)
8t 2 <
holds provided only that X 2 De (
), where  2 1
+1 and 2 < . 0

0

0

+

+

(b) The state X = 0 of the system (2.1) has the strict domain Dec(
) of exponential stability (the strict exponential stability domain) with respect to (
) if and only if Dec (
) is the largest connected neighbourhood of X = 0 which is subset of De (
), Dec (
) De (
).

© 2004 by Chapman & Hall/CRC

Comment 2.4 The state X = 0 of the system dX = (; + jx j + jx j)X
 2 <+ : 1 2 dt has De (1
) for any 2 ]0
 and it equals S ,

De(1
) = S = fX : X 2 <
jx j + jx j  g
2

=  ;


1

2

Dec(1
) = De (1
):

However, for the set S ,

S = fX : X 2 <
jx j + jx j < g
we cannot nd  2 ]1
+1 and 2 ]0
 for which S satises Denition 2.7, despite S being the asymptotic stability domain of X = 0, S = D. This is clear, 2

because

1

2

S = S : 2 ]0
]


which means that we should nd = min ( ; : 2 ]0
). This minimum does not exist, and inf ( ; : 2 ]0
) = 0. The asymptotic stability domain D = S is not the exponential stability domain with respect to any (
) 2 1
+1]0
. In fact

De (1
)  D
8 2 ]0

and

De(1
) = S = fX : X 2 <
jx j + jx j < g
=  ; : 2

1

2

Denition 2.8 (a) A set A 
X(t X0 )
A]  (X0
A) exp (; t) for all t 2 <+ holds provided only that X0 2 De (A 
), where  2 1
+1 and 2 <+ . (b) A set A 
of exponential stability (the strict exponential stability domain) with respect to (
) if and only if Dec(A 
) is the largest connected neighbourhood of A which is subset of De (A 
), Dec (A 
) De (A 
).

© 2004 by Chapman & Hall/CRC

2.2.6 De nitions of asymptotic stability domains on

Let the system (2.1) be of the Lurie form (Section 1.1.5), dX = AX + Bf(w)
dt w = CX + Df(w):

( )(

N

)

(2.2a) (2.2b)

In order to explain the need for the study of domains of asymptotic stability on Ni ( ) we present the following simple example.

Example 2.13 Let n = 1 and dX = ;sin X: dt This system has innitely many equilibrium points located at X = k, where k is any integer. Hence, X = 0 obviously is not asymptotically stable in the large. However, it is asymptotically stable with the domain of asymptotic stability D = ] ; 
. Over S = D the nonlinearity f, f(X) = ;sin X, belongs to the family N1 (L
M S ) for L = 0
1] and M = ;1
1. If the system is embedded into the class of Lurie systems (2.2), then we can speak only about asymptotic stability of X = 0 for a particular f( ) or for any f 2 N1 (L
M S ), or for any f 2 N0 (L S ). This means that we can look only for the asymptotic stability domain for a particular f e.g. f(X) = ;sin X, or for every f 2 N1 (L
M S ), or for every f 2 N0 (L S ). Let Df denote the asymptotic stability domain of X = 0 of the (Lurie) system (2.2) for a particular nonlinearity f.

Denition 2.9 The state X = 0 of the system (2.2) has the strict] asymptotic stability domain on Ni (L
M S ), which is denoted by Di (L
M S ) Dic (L
M S )], if and only if

a) it has the strict] asymptotic stability domain Df Dcf ] for every f 2

Ni (L
M S )

and b) Di (L
M S ) = \Df : f 2 Ni (L
M S )] is a neighbourhood of X = 0 Dic(L
M S ) = \Dcf : f 2 Ni (L
M S )] is a connected neighbourhood of X = 0], respectively.

This denition was introduced in 68], 69]. It can be extended to sets as follows:

Denition 2.10 A set A 
© 2004 by Chapman & Hall/CRC

and b) Di (L
M A S ) = \Df : f 2 Ni (L
M A S )] is a neighbourhood of the set A  Dic = \Dcf : f 2 Ni (L
M A S )] is a connected neighbourhood of the set A], respectively.

2.3 Domains of practical stability properties 2.3.1 De nitions of domains of practical stability

By following 70] and Section 1.2.2 we accept the following denition for the system (2.3): dX = f(X
i)
f :
Denition 2.11 The system (2.3) has the domain of practical stability with respect to f
XA
Ig, which is denoted by Dps (
XA
I ), if and only if both a) its motions obey

X(t X0 i) 2 XA for every (t
i) 2 <
 I provided only that X0 2 Dps (
XA
I ), and b) the interior Dps (
XA
I ) of Dps (
XA
I ) is non-empty.

When , XA and I are prespecied then we may replace Dps (
XA
I ) by Dps .

Denition 2.12 A set A (of states of the system (2.3)) has the domain of practical stability with respect to f
XA
Ig, which is denoted by Dps (
XA
I  A), if and only if both

a) the system motions obey

X(t X0 i) 2 XA for every (t
i) 2 <
 I provided only that X0 2 Dps (
XA
I  A), and b) Dps (
XA
I  A) is a neighbourhood of the set A.

When , XA and I are known then we may write Dps (A) instead of Dps (
XA
I  A).

Comment 2.5 If we are interested in the practical stability domain of a state X  then we can apply Denition 2.12 by settling A = fX  g.

© 2004 by Chapman & Hall/CRC

2.3.2 De nitions of domains of practical contraction with settling time

As for practical stability domains, we rst introduce the notion of the domain of practical contraction with settling time for the system (2.3) (see Section 1.2.3).

Denition 2.13 The system (2.3) has the domain of practical contraction with the settling time s with respect to f
XF
Ig, which is denoted by Dpc (
s
XF
I ), if and only if both

a) its motions obey

X(t X0 i) 2 XF for every (t
i) 2 T
s  I provided only that X0 2 Dpc (
s
XF
I ), and b) the interior Dpc (
s
XF
I ) of Dpc (
s
XF
I ) is non-empty.

When , s , XF and I are prespecied then we may write Dpc instead of Dpc (
s
XF
I ).

Denition 2.14 A set A (of states of the system (2.3)) has the domain of practical contraction with the settling time s with respect to f
XF
Ig, which is denoted by Dpc (
s
XF
I  A), if and only if both a) the system motions obey

X(t X0 i) 2 XF for every (t
i) 2 T
s  I provided only that X0 2 Dpc (
s
XF
I  A), and b) Dpc (
s
XF
I  A) is a neighbourhood of the set A.

When , s , XF and I are known then we may replace Dpc (
s
XF
I  A) by Dpc (A).

Comment 2.6 If we are interested in the domain of practical contraction of a state X  then we may use Denition 2.14 with A = fX  g.

2.3.3 De nitions of domains of practical stability with settling time

In view of the preceding denition and the notion of practical stability with settling time (Section 1.2.4) we accept the following denition:

© 2004 by Chapman & Hall/CRC

Denition 2.15 The system (2.3) has the domain of practical (contractive) stability with the settling time s with respect to f
XA
XF
Ig, which is denoted by Dp (
s
XA
XF
I ), if and only if a) both 1)

X(t X0 i) 2 XA for every (t
i) 2 <
 I


2)

X(t X0  i) 2 XF for every (t
i) 2 T
s  I

and hold provided only X0 2 Dp (
s
XA
XF
I ), and b) the interior Dp (
s
XA
XF
I ) of Dp (
s
XA
XF
I ) is non-empty.

When , s , XA , XF and I are given then we may write Dp instead of Dp (
s
XA
XF
I ). For the set A we deduce from Denition 2.12 and Denition 2.14 the following 70]:

Denition 2.16 A set A (of states of the system (2.3)) has the domain of practical (contractive) stability with the settling time s with respect to f
XA
XF
Ig, which is denoted by Dp (
s
XA
XF
I  A), if and only if a) both 1)

X(t X0 i) 2 XA for every (t
i) 2 <
 I


2)

X(t X0  i) 2 XF for every (t
i) 2 <
s  I

and hold provided only that X0 2 Dp (
s
XA
XF
I  A), and b) Dp (
s
XA
XF
I  A) is a neighbourhood of the set A.

When , s, XA, XF and I are known and xed then we may write Dp (A) in the sense of Dp (
s
XA
XF
I  A).

© 2004 by Chapman & Hall/CRC

Chapter 3

Qualitative features of stability domains properties 3.1 Introductory comments 3.1.1 De nition of a motion

If a (physical, technical) system is described by a rst order vector dierential equation in (3.1a) dq = r(q
W)
r :
(3.1a)

and by an algebraic vector equation in (3.1b) y = g(q
W)
g :
(3.1b)

then the former describes its internal dynamics that via (3.1b) determines its output behaviour under the inuence of an input (vector function) W. Let yd : < ! <m denote a specic output response of the system (3.1), which is of a particular interest, which is aimed to and, therefore, called a desired output (vector function) of the system.

Denition 3.1 The system (3.1) is in the nominal (desired) regime with respect to yd if and only if y(t)  yd (t). A pair (q
W) is nominal (desired) with respect to yd if and only if the system (3.1) is in the nominal regime with respect to the same yd . Theorem 3.1 In order for a pair (q
W) to be nominal denoted by (qN
WN ) with

respect to a desired output yd of the system (3.1a) it is both necessary and su cient

© 2004 by Chapman & Hall/CRC

that it satises the following identities: dq(t)  rq(t)
W (t)]
dt gq (t)
W(t)]  yd (t):

(3.2a) (3.2b)

rrr

Proof. The statement of Theorem 3.1 follows directly from (3.1) and Denition 3.1. A nominal motion qN with respect to a desired output yd of the system (3.1) is an unperturbed motion in Lyapunov's terminology (Section 1.1.1).

Assumption 3.1 A nominal pair (qN
WN ) with respect to yd is known and it is time invariant.

From now on it is accepted that the Assumption 3.1 holds. This means that (qN
WN ) is elementwise constant solution to (3.2). Let the following change of variables (i.e. translation of coordinate systems) be dened: X = q ; qN
i = W ; WN
f(X
i) = r(qN + X
WN + i) ; r(qN
WN ): (3.3) It is easy to show that after subtracting (3.2a) from (3.1a) and using (3.3) we derive (3.4), dX = f(X
i)
f : X  I !
(i) X is dened in X , X(t X0 i) 2 X , if and only if t 2 I0 for every i 2 I , (ii) X is continuous and dierentiable in t 2 I0 for all i 2 I , (iii) X identically satises the equation (3.4) on I0  I , that is

d dt X(t X0 i)  fX(t X0 i)
i(t)]
and (iv) X fulls the initial condition,

X(0 X0 i)  X0 :

I = 0 = fi : i = 0g for the system (3.7). Hence, we shall use X(t X0 ) in the sense of X(t X0  0) : X(t X0 )  X(t X0 0). In general X0 is a subset of
3.1.2 Existence of motions

There are various theorems on existence of motions of the system (3.7). Their common feature is that they provide su cient conditions (rather than necessary and su cient conditions) for existence of motions. Here will be presented the classical results by referring to 157]. Theorem 3.2 In order for the system (3.7) to have a motion through X0 2 X at t = 0 it is sucient that there exists a compact (closed and bounded) neighbourhood N (X0 ) of X0 , N (X0 )  X , such that the function f is continuous on N (X0 ) : f(X) 2 CN (X0 )]. Then the motion is dened in the time interval ] ; 
, where X0
@ N (X0 )] = max kf(X)k : X 2 N (X0 )]

rrr

Nemytskii and Stepanov 157, Theorem 1.21] established the following result by using the preceding theorem.

© 2004 by Chapman & Hall/CRC

Theorem 3.3 If, as time increases, a given motion X remains in a compact nonempty subset Xc of the interior N (X0 ) of the neighbourhood N (X0 ) on which the conditions of Theorem 3.2 are fullled, then the motion X may be continued for rrr (is dened on) the whole interval <+ = 0
+1]. This theorem is very important for discovering a link between stability of X = 0 and the existence of solutions, which is stated as follows: Theorem 3.4 (a) If X = 0 is stable then X(t X0) exists on <+ for every X0 2 Ds. (b) If X = 0 is attractive and X(t X0) is continuous in t 2 0
(X0
kX0 k)] for every X0 2 Da then X(t X0) exists on <+ for every X0 2 Da . (c) If X = 0 is asymptotically stable then X(t X0 ) exists on <+ for every X0 2 D. (d) If X = 0 is exponentially stable then X(t X0) exists on <+ for every X0 2 De(
). rrr Proof. Theorem 3.3 and (a) the denition of Ds (Denition 2.1, Section 2.2.2) imply directly the statement under (a). (b) the denition of Da (Denition 2.3, Section 2.2.3) yield directly the statement under (b), by noting that  = kX0 k in (X0
) in the same denition. (c) the statement (c) follows from (a) and (b) due to the denition of D (Denition 2.5, Section 2.2.4). (d) The statement under (d) is a direct corollary to Theorem 3.3 in view of the denition of De(
) (Denition 2.7, Section 2.2.5). Several other criteria for the existence of motion follow. The next one is due to Zubov 218, p. 14]. Theorem 3.5 If the function f is dened and continuous on
with bounded norm on
sup kf(X]k : X 2
(3.8)

rrr

In case the function f is continuous on
(3.10)

Since the function 1 +fkf k obeys all the conditions of Theorem 3.5 for f(X) 2 C(
3.1.3 Existence and uniqueness of motions

Denition 3.3 (a) A motion X of a system is backward-time unique through X

0

if and only if it and any other motion X of the same system through X0 obey (3.11) (3.11) X  ( ; t 
X0) = X( ; t 
X0)

for all  in an interval over which both X  ( X0 ) and X( X0 ) are dened and for all t 2 <+ for which both X  ( ; t 
X0) and X( ; t 
X0) are dened. (b) A motion X of a system is forward-time unique through X0 if and only if it and any other motion X  of the same system through X0 obey (3.12) (3.12) X  ( + t 
X0) = X( + t 
X0) for all  in an interval over which both X  ( X0 ) and X( X0 ) are dened and for all t 2 <+ for which both X  ( + t 
X0) and X( + t 
X0) are dened. (c) A motion X of a system is unique through X0 if and only if it is both backwardtime unique through X0 and forward-time unique through X0 .

In order to illustrate the preceding denition several examples follow.

Example 3.1 The rst order system dx = x2=3 dt has solutions x through x0 = 0 at t = 0 determined by

8 > > > < x (t 0) = > > :




1 (t + ) 3
t 2 ] ; 1
;]
8 2 <
+ 3 0
t 2 ;
]
3 1 (t ; )
t 2 
+1
8 2 <
 2 f
g + 3 and the trivial solution x(t 0) = 0
8t 2 <: © 2004 by Chapman & Hall/CRC

x(t;0) xψ(t;0)=[1/3(t-ζ)]3, t∈R+

ζ=0

-2

-3

2

2

3

x(t;0)≡0

-1

t 0

3

1

1

2

3

ξ=0

1

xψ(t;0)=[1/3(t+ξ)]3, t∈R-

Figure 3.1: Solutions of x_ = x2=3 .

The system has innitely many solutions through x0 = 0 at t = 0 in both the forward-time direction and in the backward-time direction. Altogether, it does not have a unique solution through x0 = 0 at t = 0, Fig. 3.1.

Example 3.2 The rst order system 8 > > = ;1
x < 0 dx = sgn x
sgn x < 2 ;1
1]
x = 0
sgn 0 is xed
> dt

> : = 1


x > 0


has solutions determined by

8 > = x + t sgn x
t 2 ] ; jx j
+1
(x 6= 0) 2 <
> < x(t x ) > = 0
8t 2 <
if sgn 0 = 0 ) > : does not exist
if sgn 0 6= 0
x = 0: 0

0

0

0

0

0

Through every (x0 6= 0) 2 < at t = 0 there is the unique solution x(t x0) = x0 + t sgn x0 dened on ] ; jx0j
+1. Through x0 = 0 at t = 0 the system has the unique motion x(t x0) = 0 dened on < if sgn 0 = 0. However, if sgn 0 6= 0 then the system does not have a solution through x0 = 0 at t = 0, Fig. 3.2.

© 2004 by Chapman & Hall/CRC

x(t;x0)

0

x( t;x

)

x0≠0

sgn 0=0 0

t

Figure 3.2: Solutions of x_ = sgn x (Example 3.2). If sgn 0 6= 0 then the system does not have a solution through x0 = 0 at t = 0.

Example 3.3 Let sign x = dene the rst order system

( sgn x
x 6= 0 0


x=0

dx = ;sign x: dt The motions are found in the form ( x0 ; t sign x0
t 2 ] ; 1
jx0j
(x0 6= 0) 2 < x(t x0) = 0
t2<
x0 = 0: Through every x0 2 < at t = 0 the system has the unique motion. The motion x(t x0) is dened on ] ; 1
jx0j if x0 6= 0. The motion x(t 0)  0 is dened on <, Fig. 3.3. It is unique despite f, f(x) = ;sign x, is not Lipschitzian on any neighbourhood of x = 0. Example 3.4 The rst order system dx = ;x1=2 dt © 2004 by Chapman & Hall/CRC

x(t;x0)

x0≠0

x(t;x0) x(t;0)≡0 0

t

Figure 3.3: Motions of the system x_ = ;sign x.

x(t;0)

x0∈R+

x(t;0)

x(t;0)≡0 0

2x01/2

Figure 3.4: Motions of the system x_ = ;x1=2 .

© 2004 by Chapman & Hall/CRC

t

has forward-time unique motions through every x0 2 <+ , which are dened by

8
i i9 p p > > 1 > = jx j ; 2 t
t 2 ;1
2 jx j = > > h >
x 2<
hp < = 0
t 2 2 j x j
+ 1  x(t x ) > > = 0
t2<
x = 0
> > : does not exist
x 2 <; : 2

0

0

0

+

0

0

0 0

Motions are unique through every x0 2 <+ , but not through x0 = 0 because there is not a backward-time unique motion through x0 = 0. They obey xt 
x( 0
0)] = x(t 0
0) = 0 for  2 <+ and t 2 
+1, but not vice versa, Fig. 3.4.

Example 3.5 The rst order system dx = ;x3 dt has unique motions through every x0 2 < at t = 0, which are dened by 8 0
t2<
x0 = 0
> <
x(t x0) = > : (1 + 2txjx0 j2)1=2
t 2 ; 2jx1 j2
+1
(x0 6= 0) 2 <: 0 0





1 They exist on the intervals I = ; 2x
+1 , Fig. 3.5. Notice that I (x ) = 0

0

2 0

0

; 2jx1 j
+1 ! < as jx j ! 0, but for any x 2 (<  <; ) the motion is not 0

2







0

0

+

dened on ;1
; 2jx1 j2 . 0

Denition 3.4 A function f is a Lipschitz function on a compact set S with a nonempty interior S = S if and only if there is L 2 ]0
+1 that satises kf(X ) ; f(X )k  LkX ; X k
1

2

1

2

X 1
X 2 2 S :

(3.13)

The number L is called a Lipschitz constant of the function f on S , the condition (3.13) is the Lipschitz condition and the family of all the functions f obeying the Lipschitz condition on the set S is denoted by Lip (S ), f 2 Lip (S ). If we wish to emphasize the mutual dependence between L and S in Denition 3.4 then we can write S L for S , S L = S . The next criterion for the existence and uniqueness of motions is classical.

Theorem 3.6 In order for the system (3.7) to have a unique motion X through X0 at t = 0 it is sucient that there exists a compact neighbourhood N (X0 ) of X0 rrr on which the function f is Lipschitzian, f 2 Lip N (X0 )]. © 2004 by Chapman & Hall/CRC

t=-

1 2|x0|2

x(t;x0)

x0

x(t;0)≡0

0

t

Figure 3.5: Motions of the system x_ = ;x3.

Comment 3.1 If a function f 2 Lip (S ) then it is dened and continuous on S , but vice versa is not true. For example the function f , f(x) = jxj1=2, is dened and continuous on <. However, there is not N (O) such that f(x) 2 Lip N (O)], i.e. f is not Lipschitzian at x = 0. Comment 3.2 If a function f is dened, continuous and continuously dierentiable in X 2 S , where S = S is nonempty, then f(X) 2 Lip (S ). Theorem 3.6 and comment 3.2 explain the following:

Theorem 3.7 If a function f is continuously dierentiable on a compact set S with the nonempty interior S = S then the system (3.7) has a unique motion X through every X0 2 S at t = 0. rrr Comment 3.3 The function f , f(x) = ;x is continuously dierentiable on <, f(x) 2 C (<), Example 3.5. The system x_ = ;x has a unique motion through every x 2 < at t = 0, Fig.
3.5. However,
for any (x 6= 0) 2 < the motion x is not dened (does not exist) on ;1
; 1 . 2jx j 3

(1)

3

0

0

0

2

Comment 3.4 The function f , f(x) = x = , is dened and continuous on < . It is Lipschitzian on every compact subset S of < with nonempty interior S . Besides, it is continuously dierentiable on < . However, it is neither dierentiable nor 1 2

+

+

© 2004 by Chapman & Hall/CRC

+

f(x)=x1/2

3 2 1

x

0 0

1

2

3

4

5

6

7

8

9

Figure 3.6: The graph of the function f , f (x) = x1=2 .

Lipschitzian at x = 0, Fig. 3.6. Nevertheless, the system has the unique solution x(t 0)  0 through x0 = 0 at t = 0, Fig. 3.4. This illustrates that the Lipschitz condition is sucient but not necessary for the existence and uniqueness of motions.

3.1.4 Continuity of motions in initial conditions

The Lipschitzian property of the function f is important also for continuity of motions X in the initial condition X0 , X0 = X(0). This is shown by what follows 157, Theorem 2.22], the proof of which can be found in 157, p. 14].

Theorem 3.8 Let f(X) 2 Lip N (X )], where N (X ) is a compact neighbourhood of X 2 X , N (X )  X . If a motion X through X at t = 0 is dened and in N (X ) for all t 2 0
T ], then for every " 2 < there is 2 < such that kX  ; X k < implies that a motion through X  at t = 0 is also dened for t 2 0
T ] and kX(t X  ) ; X(t X )k < " for all t 2 0
T ]. rrr 0

0

0

0

+

0

0

+

0

0

Comment 3.5 The conditions of Theorem 3.8 are only sucient for continuity of motions with respect to the initial conditions.

3.1.5 Dierentiability of motions in initial conditions

From Theorem 7.1 by Rouche and Mawhin 178, p. 111], we deduce the following: Theorem 3.9 If X is open and f(X) 2 C (1)(X ) then the solutions of the system (3.4) X(t X0 ) 2 C (21)(<+  X ).

rrr

© 2004 by Chapman & Hall/CRC

3.2 Generalised motions 3.2.1 Motivation

Let a simple rst order unstable linear plant described by y(1) ; y = u be controlled by locally stabilising nonlinear control u, u = k sign e, k 2 <+ , e = yd ; y, where yd is a desired output accepted equal to zero, yd (t)  0. Let x = y, X = (x). Then the overall closed-loop feedback control system is described by dx = x ; k sign x: dt Its motions are found as ( t 2 ] ; 1
0
if jx0j 2 ]0
k
x(t x0)=k sign x0 + (x0 ; k sign x0 ) exp (t)
t 2 <
if x0 = 0 or jx0j 2 k
+1
k sign x0 . If jx j 2 ]0
k then motions do not exist on 
+1 where 0 = ln k sign 0 0 x0 ; x0 because x(t x0) is not dierentiable at t = 0 (Denition 3.2). However, the real system operates well for t 2  0
+1 (see Example 3.11). This motivates us to generalise the notion of motion.

3.2.2 Dini derivatives

Let t 2 <, Tt; be a set of a sequence tk converging to t on the left as k ! +1 if t > ;1, (t = +1 is permitted), Tt; = ftk : tk 2 <
tk < t
k = 0
1
2
: : :
tk ! t as k ! +1g
t 2 ] ; 1
+1]
and, analogously, Tt+ = ftk : tk 2 <
tk > t
k = 0
1
2
: : :
tk ! t as k ! +1g
t 2 ;1
+1: Denition 3.5 (i) A number;  2+< is the partial limit of a function x, x : < ! <, over a sequence set Tt  Tt ], if and only if for every " 2 <+ there is an integer N such that k > N implies jx(tk ) ; j < ", (ii) the symbol  = ;1   = +1] is the partial limit of a function x : < ! < over a sequence set Tt;  Tt+ ], if and only if for every " 2 ]0
+1 there is an integer N such that, respectively, k > N implies x(tk ) < ;";1  x(tk ) > ";1 ], (iii) the greatest the smallest] partial limit of a function x : < ! < over all the sequence sets Tt; is its left upper left lower] limit at t = t , which is 









denoted by







lim sup x(t) : t ! t; ] = lim x(t) : t ! t; ]
 lim inf x(t) : t ! t; ] = lim x(t) : t ! t; ] ]


respectively,

© 2004 by Chapman & Hall/CRC

(iv) the greatest the smallest] partial limit of a function x : < ! < over all the sequence sets Tt+ is its right upper right lower] limit at t = t , which is denoted by lim sup x(t) : t ! t+ ] = lim x(t) : t ! t+ ]


 lim inf x(t) : t ! t+ ] = lim x(t) : t ! t+ ] ]
respectively.

If x(t) = 0 for t = 0 and x(t) = signsin(t;2 )] for t 6= 0 then lim x(t) : t ! 0; ] = lim x(t) : t ! 0+ ] = 1 and lim x(t) : t ! 0;] = lim x(t) : t ! 0+ ] = ;1. Now, we can dene Dini derivatives 143], 217]: Denition 3.6 Let V be a continuous scalar function of X on X , V (X) 2 C(X ) and let the system (3.7) have a solution X through X 2 X at t = 0. Then, (i) the forward-time upper right Dini derivative of V along the solution X at X is  ; V (X) :  ! 0+ 
D+ V (X) = lim sup V X( X)]  (ii) the forward-time lower right Dini derivative of V along the solution X at X is

D V (X) = lim inf +

 V X( X)] ; V (X) 

:!0

+






(iii) the forward-time upper left Dini derivative of V along the solution X at X is

D; V (X) = lim sup

 V X( X)] ; V (X) 



:  ! 0;


(iv) the forward-time lower left Dini derivative of V along the solution X at X is

 ; V (X) :  ! 0; : D; V (X) = lim inf V X( X)] 

(v) The function V has the forward-time left Dini derivative along the motion X at X , which is denoted by Dl V (X), if and only if D; V (X) = D; V (X) and then Dl V (X) = D; V (X) = D; V (X). (vi) The function V has the forward-time right Dini derivative along the motion X at X , which is denoted by Dr V (X), if and only if D+ V (X) = D+ V (X) and then Dr V (X) = D+ V (X) = D+ V (X). (vii) The function V has the forward-time Eulerian derivative along the motion X at X , which is denoted by df V (X)]=dt or by V_ f (X), if and only if it has both Dl V (X) and Dr V (X) and Dl V (X) = Dr V (X), and then df V (X)]=dt = V_ f (X) = Dl V (X) = Dr V (X).

© 2004 by Chapman & Hall/CRC

D V (X) means that both D V (X) and D; V (X) may be used. +

In order to clarify the dierence between the forward-time derivatives and the backward-time derivatives we dene the latter as follows: Denition 3.7 Let V be a continuous scalar function of X on X , V (X) 2 C(X ) and let the system (3.7) have a solution X through X 2 X at t = 0. Then, (i) the backward-time upper right Dini derivative of V along the solution X at X is   V X(; X)] ; V (X) + + :!0
DV (X) = lim sup  (ii) the backward-time lower right Dini derivative of V along the solution X at X is +

 ; V (X) :  ! 0 
DV (X) = lim inf V X(; X)]  +

(iii) the backward-time upper left Dini derivative of V along the solution X at X is   ; DV (X) = lim sup V X(; X)] ; V (X) :  ! 0;




(iv) the backward-time lower left Dini derivative of V along the solution X at X is   V X(; X)] ; V (X) ; : :  ! 0 DV (X) = lim inf ;



(v) The function V has the backward-time left Dini derivative along the motion X at X , which is denoted by l DV (X), if and only if ; DV (X) = ; DV (X) and then l DV (X) = ; DV (X) = ; DV (X) . (vi) The function V has the backward-time right Dini derivative along the motion X at X , which is denoted by r DV (X), if and only if + DV (X) = + DV (X) and then r DV (X) = + DV (X) = + DV (X) . (vii) The function V has the backward-time Eulerian derivative along the motion X at X , which is denoted by dbV (X)]=dt or by V_ b (X), if and only if has both l DV (X) and r DV (X) and l DV (X) =r DV (X), and then df V (X)]=dt = V_ b (X) =l DV (X) = r DV (X) .

Denition 3.6 and Denition 3.7 directly imply the next relationships among backward-time derivatives and forward-time derivatives: ; DV (X) = ;D+ V (X)
; DV (X) = ;D+ V (X)
+ DV (X) = ;D; V (X)
+ DV (X) = ;D; V (X)
l DV (X) = ;Dr V (X)
r DV (X) = ;Dl V (X)
© 2004 by Chapman & Hall/CRC

and

V_ b (X) = ;V_ f (X): These equations and our interest in the forward-time system behaviour justify the usage of the forward-time derivatives only from now on. Hence, we shall use a simplied notation dV (X)=dt, or V_ (X), for the forward-time Eulerian derivative of V along X at X instead of df V (X)]=dt, or V_ f (X), respectively. The preceding clarications were necessary in order to dene generalised motions.

3.2.3 Generalised motions Denition

In the sequel \almost everywhere in t 2 I0" means \for all t 2 I0 except for t belonging to a subset M0 of I0 with zero measure", which is denoted by \8a t 2 I0 ". Besides, \t; " denotes \t ; 0 = lim  :  ! t
 < t]", and \t+ = t + 0 = lim  :  ! t
 > t]".

Denition 3.8 A function X : I0  X0  I ! X is a generalised solution (a generalised motion) of the system (3.4) through X0 at t = 0 if and only if X0 2 X0 and X satisfy (i){(iv): (i) X is dened in X , X(t X0 i) 2 X , if and only if t 2 I0, I0 = (l;
l+ ), for every i 2 I , (ii) for every i 2 I and each X0 2 X0, X is (a) continuous in t 2 I0, (b) dierentiable almost everywhere in t 2 I0. At t 2 M0 , M0 has the zero measure, M0  I0 , at which X is not dierentiable in time t the following holds (c) X has both Dl X(t X0  i) and Dr X(t X0  i) if t is an interior point of I0, t 2 ]l;
l+ , (d) X has Dr X(t X0  i) if l; 2 I0, I0 = l;
l+ ) and t = l; , (e) X has Dl X(t X0  i) if l+ 2 I0, I0 = (l;
l+ ] and t = l+ , (iii) for every i 2 I (a) X identically satises (3.4) almost everywhere on I0,

d X(t X  i) = fX(t X  i)
i(t)]
8 t 2 I
0 0 a 0 dt

© 2004 by Chapman & Hall/CRC

(b) if t 2 M0 is an internal point of I0 then Dl X(t X0 i) = fX(t;  X0  i)
i(t;)]
t = t
Dr X(t X0  i) = fX(t+  X0  i)
i(t+)]
t = t
(c) if l; 2 I0 and t = l; then

Dr X(t X  i) = fX(t  X  i)
i(t )]
t = t = l;
(d) if l 2 I and t = l then Dl X(t X  i) = fX(t;  X  i)
i(t;)]
t = t = l
+

0

+

0

+

+

0

0

+

0

and (iv) X fulls the initial condition,

X(0 X0 i)  X0 : The motivation for introducing the notion of generalised motions will be illustrated by reconsidering the systems x_ = sgn x (Example 3.2) and x_ = ;sgn x (Example 3.3).

Example 3.6 Generalised motions of the rst-order system dx = sgn x dt

are determined by 8= 0
t 2 ] ; 1
;jx0j
) > > x0 2 <
if sgn 0 = 0
> > = x 0 + t sgn x0
t 2 ;jx0j
1
>

> > > does not exist
t 2 <
x = 0
(sgn 0 6= 0) 2 ] ; 1
+1
> > > > does not exist
t 2 ] ; 1
0
) > x = 0
if (sgn 0 6= 0) 2 f;1
+1g
<
t 2 0
1
x(t x ) > = t sgn x > 9 > does not exist
t 2 ] ; 1
;jx j
) > > > > if sgn x = sgn 0
> > > = x + t sgn x
t 2  ;j x j
1 
= (x 6= 0) 2 < > > and > ) > > does not exist
t 2 ] ; 1
;j x j ]
> > sgn 0 6= 0: > > : = x + t sgn x
t 2 ] ;jx j
1
if sgn x 6= sgn 0
>  0

0

0

0

0

0

0

0

0

0

0

0

0

0

They are depicted in Fig. 3.7. Now, we can compare motions with generalised motions as follows:

© 2004 by Chapman & Hall/CRC

0

x(t;x0) x(t; x0*)

sgn 0 ∈]0,1]

x0*>0 -|x0*|

x(t;0)

sgn 0=0

0

x(t;0)

sgn 0=0

x(t;0)≡0

t

^

-|x0|

^x <0 0

sgn 0 ∈[-1,0[

x(t;0) x(t;

x^

0

)

Figure 3.7: Generalised motions of the system x_ = sgn x.

a) (x0 6= 0) 2 <, sgn 0 = 0. In this case the dierence is in the time interval I0 over which motions and generalised motions are dened. I0 = ] ;jx0j + 1 is the interval of denition of motions, and I0 = < is the time interval over which the system has dened generalised motions through every (x0 6= 0) 2 < since sgn 0 = 0. They are unique. b) x0 = 0, sgn 0 = 0. System has unique both motion and generalised motion x(t x0)  0. c) x0 2 <, sgn 0 6= 0. Since x(t _ x0) = sgn x(t x0), then x(t x0) should have the following form: _ x0) = sgn x0 and x(t _ x0 ) = x(t _ x0) = x0 + t sgn x0 . They imply both x(t sgn x(t x0) = sgn(x0 + t sgn x0 ). For them to be equal it is necessary and su cient that: sgn x0 = sgn(x0 + t sgn x0 ) which requires: t 2 0
1 if (sgn 0 6= 0) 2 f;1
+1g and x0 = 0 t 2 ;jx0j
1 if sgn x0 = sgn 0, (x0 6= 0) 2 < and sgn 0 6= 0 t 2 ] ; jx0j
1 if sgn x0 6= sgn 0, (x0 6= 0) 2 < and sgn 0 6= 0.

Example 3.7 The rst order system x_ = ;sign x © 2004 by Chapman & Hall/CRC

x(t;x0)

*

x0∈R + x0

x(t;x0),t∈R

x(t;x*0),t∈]-∞,|x0*|] x(t;x0*)= x(t;0),t∈[|x0*|,+∞[

|x0|

0

t

^

|x0| x(t;x^0),T∈]-∞,|x^0|] x^0∈R -

Figure 3.8: Generalised motions of the system x_ = ;sign x. Through x0 6= 0 there is not a backward-time unique generalised motion.

has generalised motions determined by ( x0 ; t sign x0
t 2 ] ; 1
jx0j] x(t x0) = 0
t 2 jx0j
+1
which are shown in Fig. 3.8. They are forward-time unique and dened on < for every x0 2 <. Besides, they coincide with the motion x(t 0)  0 on jx0j
+1 for every x0 2 <. They are not backward-time unique for x0 6= 0.

Note 3.1 Generalised motions of the system x_ = ;sign x obey xt 
x( 0
0)] = x(t 0
0) for t 2 
+1,  2 <, i.e. in the forward-time direction. They do not obey this equation for t 2 ] ; 1
 i.e. in the backward-time direction. Note 3.2 All the denitions of Chapter 1 and Chapter 2 are valid also in case X therein represents a generalised motion. The same holds for Denition 3.6 and Denition 3.7 (Section 3.2.2). Existence of generalised motions

From the proof by Nemytskii and Stepanov 157, p. 3] of Theorem 3.2 157, Theorem 1.11] and Denition 3.8 it follows:

© 2004 by Chapman & Hall/CRC

Theorem 3.10 In order for the system (3.7) to have a generalised motion through

X0 2 X at t = 0, it is sucient that there exists a compact neighbourhood N (X0 ) of X0 , N (X0 )  X , such that the function f is dened on N (X0 ) and continuous on N (X0 ) except possibly at X0 : f(X) 2 C  N (X0 ) ; fX0 g]. Then, the generalised motion is dened in the interval ] ; 
, where X0
@ N (X0 )] : = maxkf(X)k : X 2 N (X0 )]

rrr

This theorem directly implies the following result.

Theorem 3.11 If, as time increases, a given generalised motion X remains in a compact nonempty subset Xc of the interior N (X0 ) of the neighbourhood N (X0 ) on which the conditions of Theorem 3.8 are fullled, then the generalised motion may be continued for the whole interval <+ .

rrr

Note 3.3 Theorem 3.4 is valid also in the case where X represents a generalised motion of the system (3.7).

Theorem 3.12 If the function f is dened on
0

0

3.2.4 Limit points and limit sets

In the sequel X(t X0 ) will be called a motion or generalised motion when X is a motion or generalised motion, respectively, and as soon as t is arbitrary in I0 rather than xed. By following Nemytskii and Stepanov 157], Zubov 218, p. 21], Bhatia and Szeg(o 10, p. 28], and LaSalle 120, p. 27], and referring to Denition 3.8 we accept the following denition. Notice that the maximal interval on which X(t X0) exists was denoted by I0 = I0(X0  X) = (l;
l+ ). Denition 3.9 (a) A point y is an -limit point of a motion X(t X0) if and only if there is a sequence tn ! l; as n ! +1 such that X(tn  X0) ! y as n ! +1. y is also called a negative limit point of X(t X0). (b) A point z is an !-limit point of a motion X(t X0) if and only if there is a sequence k ! l+ as k ! +1 such that X( k  X0) ! z as k ! +1. z is also called a positive limit point of X(t X0). (c) If X(t X0 ) is a generalised motion then (a) and (b) are still valid.

© 2004 by Chapman & Hall/CRC

Comment 3.7 Let

a) x_ = x2=3, Example 3.1, Fig. 3.1. Every motion x(t 0) has -limit point y = ;1 for  6= 0, and for !-limit point z = +1 for 6= 0. If  = = 0 then y = z = 0. b) x_ = sgn x, Example 3.2, Fig. 3.2, Example 3.6, Fig. 3.7. The -limit point y of x(t x0) is zero, y = 0, for every x0 2 <. The !-limit point z of x(t x0) is z = (+1 sign x0 ) if (x0 6= 0) 2 <, and z = 0 if x0 = 0 and sgn 0 = 0. In all other cases in which the system has generalised motions, the !-limit point z of x(t x0) is z = (+1 sign x0 ). c) x_ = ;sign x, Example 3.3, Fig. 3.3, Example 3.7, Fig. 3.8. If (x0 6= 0) 2 < then both the motion and the generalised motion have the -limit point y = (+1 sgn 0) and the !-limit point z = 0. The motion x(t 0)  0 has y = z = 0. d) x_ = ;x1=2, Example 3.4, Fig. 3.4. If x0 2 <+ then y = +1 and z = 0. If x0 = 0 then y = z = 0. e) x_ = ;x3 , Example 3.5, Fig. 3.5. If (x0 6= 0) 2 <+ then y = (+1 sign x0 ) and z = 0. If x0 = 0 then y = z = 0.

Note 3.4 The preceding analysis illustrates that limit points depend on a (generalised) motion X and an initial X0 , y = y(X0  X)
z = z(X0  X): In the case where a system has a unique motion through X0 2 X0 at t = 0 then the limit points depend on the initial X0 only, y = y(X0 )
z = z(X0 ): Denition 3.10 (a) The set of all negative (;) limit points y of X(t X0) is its negative limit set denoted by L;, L; = L;(X0  X):

(b) The set of all positive (!;) limit points z of X(t X0 ) is its positive limit set denoted by L+ ,

L = L (X  X): +

+

0

Comment 3.8 If X(t X ) is unique through X at t = 0 then L; (X  X) = L; (X )
L (X  X) = L (X ): Comment 3.9 Let a) x_ = x = , Example 3.1, Fig. 3.1.: L;(0 x ) = f;1g and L f0 x g = f+1g, 8 2 < . However, L;(0 x) = L (0 x) = f0g. The origin x = 0 is not either positively or negatively Lagrange stable, or equivalently, the family F (0) is 0

0

0

+

0

0

+

neither positively or negatively precompact.

© 2004 by Chapman & Hall/CRC

0

+

2 3 +

+

0

b) x_ = sgn x, Example 3.2, Fig. 3.2, motions: L; (x0) = f0g, 8x0 2 <, L+ fx0g = f+1g, 8(x0 6= 0) 2 < and L+ (0) = f0g in case sgn 0 = 0 Example 3.6, Fig. 3.7, generalised motions: L;(x0 ) = f0g, 8x0 2 < for which generalised motions exist, L+ fx0g = f+1 sgn x0 g, 8x0 2 < for which generalised motions exist. c) x_ = ;sign x, Example 3.3, Fig. 3.3 and Example 3.7, Fig. 3.8, motions and generalised motions: L; (x0) = f+1 sgn x0 g, 8(x0 6= 0) 2 <, L;(0) = L+ (0) = f0g and L+ (x0 ) = f0g, 8x0 2 <. d) x_ = ;x1=2 , Example 3.4, Fig. 3.4, motions (coincide with generalised motions): L;(x0 ) = f+1g, 8x0 2 <+ , L; (0) = L+ (0) = f0g, L+ (x0) = f0g, 8x0 2 <+ . e) x_ = ;x3 , Example 3.5, Fig. 3.5, motions: L; (x0) = f+1 sign x0 g, 8(x0 6= 0) 2 <, L; (0) = L+ (0) = f0g, L+ (x0) = f0g, 8x0 2 <.

Example 3.8 (Bhatia and Szego 10, p. 30]) The second order system, X = (x1 x2)T
f = (f1 f2 )T
dX = f(X)
dt is determined by 2 ; x2 (1 ; x22)2
f1 (X) = x1(1(1;+xx12))(1 + kX k) 1

8 x  > < 1 ; x + x 1 +1kX k
jx j 6= 1
f (X) = > : sign x
jx j = 1: 1

2 1

2

2

1

1

1

System trajectories are shown in Fig. 3.9. The positive limit set is found as follows for every X0 2 <, X0 6= 0,jx10j < 1:

L (X ) = fX : X 2 <
jx j = 1g: +

0

2

1

This equation shows that the vertical lines through x1 = ;1 and x1 = 1 constitute the positive limit set L+ (X0 ) of every motion X(t X0) for any X0 in the strip between the lines and out of the origin.

© 2004 by Chapman & Hall/CRC

x2

-1

0 1

x1

Figure 3.9: Trajectories of the system of Example 3.8.

3.2.5 Limit sets, Lagrange stability, precompactness and stability domains At rst we present the denition of Lagrange stability 157], 10, p. 46], 152]:

Denition 3.11 (a) A point x 2 X is positively negatively] Lagrange stable relative to X and relative to system (3.7) if and only if the closure 0

0

X(0
l+ (X0  X) X0) of every arc X(0
l+ (X0  X) X0) the closure X(]l; (X0  X)
0] X0) of every arc X(]l; (X0  X)
0] X0)], respectively, is

compact and in X .

(b) A set A0 X0 is positively negatively] Lagrange stable relative to X and relative to the system (3.7) if and only if every point X0 2 A0 is positively negatively] Lagrange stable relative to X and relative to the system (3.7).

© 2004 by Chapman & Hall/CRC

(c) The expression \relative to X" is omitted if and only if

\

\

I (X  X) = 0

X

0

X

]l; (X0  X)
l+ (X0  X) = <:

(d) The expression \and relative to the system (3.7)" is omitted if and only if the system is prespecied.

Note 3.5 Denition 3.11 is general. It holds even if the system (3.7) has multiple (generalised) motions through X0 at t = 0. If the system has a unique (generalised) motion X(t X0) then l (X0  X) = l (X0 ) and there is a unique arc X(0
l+(X0 ) X0 ) X(]l;(X0 )
0] X0)]. In such a case the intersection in (c) taken over all motions X passing through X0 at t = 0 reduces to I0(X0  X), that is, I (X ) = ]l; (X )
l (X ) = <: Besides, Lagrange stability of X 2 X guarantees boundedness of all motions through X 2 X . 0

0

0

0

+

0

0

0

0

In view of Denition 3.11 we slightly generalise the denition of motion precompactness by LaSalle 120, p. 28, Denition 3.1].

Denition 3.12 (a) A family F = fX : X(0 X ) = X g, F = F (X ), of (generalised) motions of the system (3.7) is positively negatively] precompact relative to X if and only if every X 2 F is bounded for all t 2 0
l (X  X)  t 2 ]l;(X  X)
0]] and has no positive negative] limit point on the boundary @ X of X , L (X  X) \ @ X = L;(X  X) \ @ X = ]
8X 2 F
0

0

0

0

0

+

0

0

0

0

+

0

0

0

respectively. (b) A set A0 X0 is positively negatively] precompact relative to X and relative to the system (3.7) if and only if the family F 0 (X0 ) of (generalised) motions of the system is positively negatively] precompact relative to X for every X0 2 A0 respectively. (c) The expression \relative to X" is omitted if and only if X =
© 2004 by Chapman & Hall/CRC

Note 3.6 (LaSalle 120, p. 28]) If X(t X0) is positively precompact relative to X then l+ (X0  X) = +1. Analogously, if X(t X0) is negatively precompact relative to X then l; (X0  X) = ;1. Hence, if X(t X0 ) is precompact relative to X then I0(X0  X) = <. Note 3.7 Denition 3.11 and Denition 3.12 are equivalent as pointed out by La Salle for the case of unique motions 120, p. 28]. Denition 3.11, Denition 3.12 and the denitions of the stability domains (Definition 3.1 { Denition 3.6, Section 2.2) directly imply the following result that concerns the system (3.7) by noting that (generalised) motions are continuous in t 2 I0 (Denition 3.2 and Denition 3.8). Theorem 3.13 (a-1) If X = 0 is stable then its domain Ds of stability is positively (a-2) (b-1) (b-2) (c-1) (c-2)

precompact (positively Lagrange stable). If a set A is stable then its domain Ds (A) of stability is positively precompact (positively Lagrange stable). If X = 0 is attractive then its domain Da of attraction is positively precompact (positively Lagrange stable). If a set A is attractive then its domain Da (A) of attraction is positively precompact (positively Lagrange stable). If X = 0 is asymptotically stable then D = Da  Ds is positively precompact (positively Lagrange stable). If a set A is asymptotically stable then D(A) = Da (A)  Ds (A) is positively precompact (positively Lagrange stable).

Example 3.9 x = 0 of the system x_ = ;x (Example 3.5, Fig. 3.5) is completely

globally asymptotically stable:

3

Ds = Da = D = <:

The whole space < is positively precompact (positively Lagrange stable).

Example 3.10 The set A0 = fX : X 2 <2
jx1j  1g is not positively precompact (positively Lagrange stable) relative to the system of Example 3.8. Example 3.11 The rst order system

dx = x ; k sign x
k 2 <+
dt has generalised motions determined by 8 k sign x + (x ; k signx ) exp (t)
t 2 <
(x = 0 or jx j 2 k
+1)
> 0 0 0 0 0

>

< x(t x0)=> k sign x0 + (x0 ; k signx0 ) exp (t)
t 2 0
t0]
)
jx0j 2 ]0
k
> : 0
t 2 t0
+1
© 2004 by Chapman & Hall/CRC

_x(t;x0)

_x(t;1) ≡ k 0

_x(t;0) ≡ 0

t

_x(t;-1) ≡ −k

Figure 3.10: Generalised motions of the system x_ = x ; k sign x, k 2 <+ , Example 3.11

where t0 = ln k ;kjx j . The generalised motions are shown in Fig. 3.10. In this 0 connection see Section 3.2.1 for system motions. X = 0 is asymptotically stable with stability domains Ds = ;k
k]
Da = ] ; k
k
D = ] ; k
k: They are positively precompact (positively Lagrange stable). They are also negatively precompact (negatively Lagrange stable).

Example 3.12 The rst order system

dx = x ; x2sign x = x(1 ; jxj) dt has generalised motions determined by x(t x0) = jx j + (1 ; xjx0 j) exp (;t)
8x0 2 <: 0 0 They are shown in Fig. 3.11. Let A0 = fx : x 2 <
jxj  1g: The set A0 is completely asymptotically stable in the large. Its stability domains are Ds (A) = Da (A) = D(A) = <: They are positively Lagrange stable (positively precompact). However, they are not negatively Lagrange stable (negatively precompact) because jx(t x0)j ! +1 as t ! t0 = ln jxj0xj ;j 1
jx0j 2 ]1
+1
0 © 2004 by Chapman & Hall/CRC

_x(t;x0) x(t;x^ 0 ),t∈]t 0 ,+∞[

x^0

_x(t;1)≡0

1

t0

_x(t;0)≡0

0

t _x(t;-1)≡-1

-1

_x(t;x^ 0 ),t∈]-∞,t 0 [

Figure 3.11: Generalised motions and motions of the system x_ = x ; x2 sign x.

so that

L;(x ) = f+1 sign x g
jx j 2 ]1
+1: 0

0

0

Notice that

L (x ) = fsign x g
jx j 2 <
L; (0) = L (0) = f0g
+

0

0

+

0

+

and

L; fx g = L fx g = fsign x g
jx j = 1: 0

+

0

0

0

3.2.6 Invariance properties of sets

By following LaSalle 120, p. 28, Denition 4.1], we dene invariance properties of sets at follows:

Denition 3.13 (a) A set A, A 
0

0

+

+

0

0

0

0

0

for all system generalised motions X(t X0 ) through X0 at t = 0.

© 2004 by Chapman & Hall/CRC

0

(b) A set A, A 
This denition incorporates the classical denition of invariance properties of sets 157, p. 330], 10, p. 20], 120, p. 28].

Denition 3.14 (a) A set A, A 
0

+

both positively and negatively invariant relative to the system. (c) The expression \relative to the system (3.7)" is omitted if and only if the system is prespecied.

Example 3.13 Let

a) x_ = x2=3
Example 3.1, Fig. 3.1. The set O = f0g is neither positively nor negatively invariant. b) x_ = ;x1=2
Example 3.4, Fig. 3.4. The set O = f0g is invariant. c) x_ = sgn x
Example 3.6, Fig. 3.7. The set O = f0g is invariant i sgn 0 = 0. d) x_ = x ; k sgn x
k 2 <+ Example 3.11, Fig. 3.10. The sets O = f0g, A1 = fx : x 2 ] ; k
kg, A2 = fx : x = kg, A3 = fx : x = ;kg, A4 = fx : jxj  kg, A5 = fx : x 2 k
+1g, A6 = fx : x 2 ]k
+1g, A7 = fx : x 2 ;k
;1g, A8 = fx : x 2 ] ; k
;1g, and < are invariant. Notice that A1 = A4
A4 = A1
A2  A3 = @ A4 :

e) x_ = x(1 ; jxj)
Example 3.12, Fig. 3.11. The invariant sets are O = f0g, A1 = fx : ;1 < x < 1g, A2 = f1g, A3 = f;1g, A4 = fx : ;1  x  1g, A5 = fx : x 2 1
+1g, A6 = fx : x 2 ]1
+1g, A7 = fx : x 2 ;1
;1g, A8 = fx : x 2 ] ; 1
;1g and <. Note 3.8 (LaSalle 120, p. 28]) If a set A is precompact relative to X and weakly invariant then it is invariant. Denition 3.15 (a) Generalised motions X(t X0) of the system (3.7) are continuous in X0 2 X for any t 2 I0 = I0(X0  X), which is denoted by X(t X0 ) 2 C0 (I0  X0), if and only if for every ("
t) 2 <+  I0 there is 2 <+ , = (" t X0  X), such that kX  ; X0 k implies kX(t X  ) ; X(t X0 )k < " for all motions X(t X0 ) or equivalently, for any sequence Xn converging to X0 , Xn ! X0 as n ! +1, and every ("
t) 2 <+  I0 there is a natural number m such that n > m implies kX(t Xn ) ; X(t X0)k < " for all motions X(t X0 ), which is denoted by X(t Xn ) ! X(t X0) as Xn ! X0 . © 2004 by Chapman & Hall/CRC

(b) Generalised motions X(t X0 ) of the system (3.7) are continuous in t 2 I0(X0  X) = I0 for every X0 2 X , which is denoted by X(t X0 ) 2 Ct (I0 X0 ), if and only if for every motion X(t X0 ) and every ("
t
X0) 2 <+  I  X0 there is 2 <+ , = (" t X0 X), such that jt ; tj < implies kX(t  X0 ) ; X(t X0 )k < " or equivalently, for every motion X(t X0 ) and every sequence tn converging to t, tn ! t as n ! +1, there is a natural number m such that n > m implies kX(tn  X0) ; X(t X0)k < ", which is denoted by X(tn  X0 ) ! X(t X0 ) as tn ! t.

Note 3.9 C(I  X )  C (I  X ) \ Ct(I  X )]. Theorem 3.14 If the system (3.7) has generalised motions X(t X ) continuous in X 2 X for every t 2 I , X(t X ) 2 C (I  X ), then the system has a unique rrr generalised motion X(t X ) through every X 2 X at t = 0. Proof. Let X(t X ) 2 C (I  X ) and (X
"
t) 2 X  <  I be arbitrarily chosen. Let X and X be two generalised motions of the system through X at t = 0. Let X  = X . Hence, kX  ; X k = 0 < , = min f (" t X  X i ) : i = 1
2g (Denition 3.15a). Since X(t X ) 2 C (I  X ) then (Denition 3.15a) kX i (t X  ) ; X k (t X )k = kX i (t X ) ; X k (t X )k < ", i
k 2 f1
2g. Letting " ! 0 it follows that kX i(t X ) ; X k (t X )k = 0 for every (t X ) 2 I  X and i
k 2 f1
2g, which proves the uniqueness of X(t X ) through X at t = 0. Lemma 3.1 If the system (3.7) has generalised motions X(t X ) continuous in X 2 X for every t 2 I (X  X), X(t X ) 2 C (I  X ), then the motions obey 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

+

0

0

0

2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

the group property (3.14),

0

0

0

8t 2 I (X  X)
8 2 I (X  X)
( + t) 2 I (X  X): (3.14) rrr Proof. Let X(t X ) 2 C (I  X ). Hence X(t X ) is unique for every X 2 X (Theorem 3.14) so that I (X  X) = I (X ) (= I , when X is xed). Let X 2 X , t 2 I and  2 < be arbitrarily chosen provided ( + t) 2 I . The theorem can be Xt X( X0 )] = X(t +  X0)


0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

proved in two ways.

(a) Let at rst the equation (3.7) be integrated from 0 to , X( X0 ) = X0 + and now from 0 to  + t, X( + t X0) = X0 +

Z

fX(  X0 )]d = X( 0
X0)


0

Z 0

t

+

fX(  X0 )]d = X( + t 0
X0)


(3.15) (3.16)

We can present (3.16) as follows due to additivity of the integral and (3.15), X( + t 0
X0) = X( 0
X0) +

© 2004 by Chapman & Hall/CRC

Z

t

+

fX(  0
X0)]d


(3.17)

Since X(t 0
X0) = Xt 
X( X0 )] then

Z

t

+

fX(  0
X0)]d =

Z

In this equation we replace by u + ,

Z

t

+

t

+

f fX  
X( X0 )]gd =

Zt 0

f fX  
X( X0 )]gd :

(3.18)

f fXu +  
X( X0 )]gdu:

(3.19)

The system (3.7) is time invariant so that Xu +  
X( X0 )] = Xu 0 X( 0
X0)]
which with (3.17) through (3.19) yields (3.20), X( + t 0
X0) = Xt 0 X( 0
X0)]


(3.20)

and proves the theorem. (b) The statement of the theorem results from uniqueness of generalised motions. The following results slightly generalise those by Nemytskii and Stepanov 157, p. 331], Bhatia and Szeg(o 10], and LaSalle 120, p. 29].

Theorem 3.15 If the system (3.7) has generalised motions X(t X0) continuous in X0 2 X0 for every t 2 I0(X0  X), X(t X0) 2 C0(I0  X0 ), then (a) The closure A of a positively negatively] weakly invariant set A is positively negatively] weakly invariant, respectively. (b) The closure A of a weakly invariant set A is weakly invariant. (c) If a set A is invariant and precompact relative to X then its closure A is invariant. (d) If a set A is invariant and A  X then its closure A is invariant.

rrr

Proof. Let the system have generalised motions X(t X ) 2 C (I  X ). (a) Let A be positively negatively] weakly invariant. If X 2 A then X(t X ) 2 A for all t 2 (< \ I )  t 2 (<; \ I )], due to positive negative] invariance of A and A A. Let X 2 @ A and Xn 2 A, Xn ! X as n ! +1. Then, X(t X ) 2 C (I  X ) implies X(t Xn ) ! X(t X ) for every t 2 I . This and X(t Xn ) 2 A for every t 2 (< \ I )  t 2 (<; \ I )], due to positive negative] invariance, imply X(t X ) 2 A for every t 2 (< \ I )  t 2 (<; \ I )], respectively. 0

0

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+

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(b) The statement under (b) results from (a) in view of Denition 3.14a.

© 2004 by Chapman & Hall/CRC

+

0

(c) Let the set A be invariant and precompact relative to X . Then for any Xn 2 A both I0(Xn ) = < and L;(Xn ) L+ (Xn )] \ @ X = . Let X0 2 @ A and Xn be dened as in (a) so that for every t 2 <, X(t Xn ) ! X(t X0) as n ! +1, which implies L (Xn ) ! L (X0 ). Therefore, L; (X0 )  L+ (X0 )] \ X = . This shows that X(t X0 ) is precompact relative to X , which together with (b) proves that X(t X0) 2 A for all t 2 <. (d) Let the set A be invariant and A  X . Since X is open, then A is compact, which with invariance of A and A A prove precompactness of X(t X0) for every X0 2 A. This and (c) imply invariance of A. What follows broadens slightly a theorem by Bhatia and Szeg(o 10, p. 22, Theorem 1.2.35].

Theorem 3.16 Let Ai  X , i = 1
2
: : :
m, and let the system (3.7) have generalised motion X(t X ) 2 C (I  X ). Then, 0

0

0

0

(a) if Ai is positively negatively] weakly invariant, i = 1
2
: : :
m, then Ai : i = 1
2
: : :
m] is positively negatively] weakly invariant, respectively, (b) if \ Ai : i = 1
2
: : :
m] 6=  and Ai is positively negatively] weakly invariant, then \ Ai : i = 1
2
: : :
m] is positively negatively] weakly invariant, respectively, (c) if Ai  Aj , Aj ; Ai 6= , and Ai and Aj are positively negatively] weakly invariant, then (Aj ; Ai) is also positively negatively] weakly invariant, respectively. (d) All the statements (a){(c) hold also if \weakly" is omitted in each case.

rrr

Proof. The proof repeats that by Bhatia and Szeg(o 10, Theorem 1.2.35]. Notice that the generalised system motions are unique (Theorem 3.14).





(a) The statement (a) is obvious due to Ai   Ai : i = 1
2
: : :
m . (b) Let P = \ Ai : i = 1
2
: : :
m] 6=  and Ai be positively negatively] weakly invariant so that X0 2 P implies X(t X0 ) 2 Ai , t 2 (<+ \I0)  t 2 (<; \I0)], because P Ai , i = 1
2
: : :
m. Hence, X(t X0 ) 2 P , t 2 (<+ \ I0)  t 2 (<; \ I0)], respectively. (c) Let Ai  Aj , Aj ;Ai 6=  and both be positively negatively] weakly invariant. If the statement were wrong then there would exist X0 2 (Aj ; Ai) implying X(t X0 ) 62 (Aj ; Ai ) at some  2 (<+ \ I0)   2 (<; \ I0)]. Since X0 2 Aj yields X(t X0 ) 2 Aj , for all t 2 (<+ \ I0)  t 2 (<; \ I0 )], then, altogether, X(t X0 ) 2 Ai , for all t 2 (
+1\I0)  t 2 (] ;1
] \I0)], which contradicts the invariance property of Ai due to X0 62 Ai . Hence, such X0 2 (Aj ; Ai ) does not exist. The statement is proved.

© 2004 by Chapman & Hall/CRC

(d) Since all Ai  X then, under the conditions of (a) or (b) or (c), A is positively negatively] precompact, hence, positively negatively] invariant, which holds also for Ai : i = 1
2
: : :
m], for P and for Aj ; Ai . Therefore, the proofs of (a){(c) are true for all t 2 <+  t 2 <; ], respectively, which implies the statement (d). This theorem is the key-stone for the next one.

Theorem 3.17 Let A be an open set, A  X and generalised motions of the system (3.7) X(t X0) 2 C0(I0  X ). (a) If A is positively negatively] weakly invariant then its boundary @ A is positively negatively] weakly invariant. (b) If A is positively negatively] invariant then its boundary @ A is also positively negatively] invariant. (c) If A is invariant then its boundary @ A is also invariant. rrr

Proof. Let A be open, A  X and X(t X ) 2 C(I  X ). (a) Let A be positively negatively] weakly invariant. Now, (a) of Theorem 3.15 and (c) of Theorem 3.16 prove positive negative] weak invariance of @ A because @ A = A ; A. 0

0

0

(b) The statement results from (d) of Theorem 3.15 and from (c) and (d) of Theorem 3.16. (c) The statement (b) and Denition 3.14b imply the statement under (c).

3.2.7 Invariance properties of limit sets

At rst we broaden Theorem 5.1 by LaSalle 120] to the system (3.7) with generalised motions. Theorem 3.18 Let generalised motions X(t X0) of the system (3.7) be continuous in X0 2 X0 for every t 2 I0 (X0  X), X(t X0 ) 2 C0(I0  X0). For every X0 2 X rrr the positive limit set L+ (X0  X) is closed and weakly invariant. Proof. Let X(t X0) 2 C0(I0 X0 ). Then, X(t X0 ) is unique, 8X0 2 X0, (Theorem 3.14), so that L+ (X0  X) = L+ (X0 ). Let us rst prove closeness, for which the proof of (i) of 1.3.13 Theorem by Bhatia and Szeg(o in 10] is repeated except one detail which will be explained in the sequel. Let y 2 L+ (X0 ) for arbitrarily chosen X0 , let yk 2 L+ (X0 ), yk ! y as k ! +1. It must be shown that y 2 L+ (X0 ). For each k there exists a sequence tkn ! l+ (X0 ) as n ! +1, and X(tkn X0 ) ! yk . We may assume without loss of generality that X(tkn X0 )
yk ] < 1=k holds for each k. In (the proof of i) of) Theorem 1.3.13 in 10], Bhatia and Szeg(o request, in addition, tkk > k, which is not

© 2004 by Chapman & Hall/CRC

possible in this framework as soon as l+ (X0 ) < +1. Then considering the sequence tn , tn = tnn, we see that tn ! l+ (X0 ) and X(tn  X0) ! y as n ! +1, since X(tn  X0)
y]  X(tn  X0)
yn ] + (yn
y) < n1 + (yn
y) ! 0: From (1=n) + (yn
y)] ! 0 as n ! +1 it now follows X(tn  X0)
y] ! 0 as n ! +1. Hence y 2 L+ (X0 ) that proves closeness of L+ (X0 ). In order to prove weak invariance we repeat the proof of Theorem 5.1 of Chapter 2 by LaSalle in 120]. Let y 2 L+ (X0 ) \ X , and let t 2 I0 (y). Since L+ (X0 ) \ X is nonempty then l+ (X0 ) = +1, and there is a sequence tn such that tn ! +1 and X(tn X0 ) ! y as n ! +1. From X(t X0) 2 C0(I  X0) it follows for t 2 I0X(tn  X0)] that (Lemma 3.1) Xt X(tn X0 )] = X(t + tn  X0) ! X(t y) as n ! +1. Therefore, X(t y) 2 L+ (X0 ) and L+ (X0 ) is weakly invariant.

Theorem 3.19 Let generalised motions X(t X0) of the system (3.7) be continuous in X0 2 X0 for every t 2 I0(X0  X), X(t X0 ) 2 C0(I0  X0 ). In order for L+ (X0 ), X0 2 X0 is arbitrary, to be nonempty, in X and compact it is both necessary and rrr sucient that X(t X0 ) is positively precompact relative to X . Proof. Let X(t X0) 2 C0(I0  X0). Then X(t X0) is unique for any X0 2 X0 (Theorem 3.14). Necessity. Let L+ (X0 ) be nonempty, in X and compact, X0 2 X0 is arbitrary.

Since X(t X0 ) ! L+ (X0 ) as t ! +1 then X(t X0 ) is bounded because X(t X0 ) 2 C0(I0  X0 ) \ Ct(I0  X0)] and L+ (X0 ) is bounded, and X(t X0 ) does not have a positive limit point on the boundary of X because L+ (X0 ) is in X . Hence, X(t X0 ) is precompact relative to X (Denition 3.12).

Suciency. Let X(t X0) be positively precompact relative to X . Then, it is bound-

ed and does not have a positive limit point on the boundary of X (Denition 3.12). Hence, L+ (X0 ) is in X , nonempty and bounded. It is also closed (Theorem 3.18), thus it is compact.

Note 3.10 Theorem 3.18, Theorem 3.19 and the next theorem hold for L;(X  X) 0

provided \positive" and \positively" are replaced by \negative" and \negatively", respectively.

Theorem 3.20 Let generalised motions X(t X ) of the system (3.7) be continuous in X 2 X for every t 2 I (X  X), X(t X ) 2 C (I  X ). If X is positively precompact relative to X then L (X ) is connected for every X 2 X . rrr 0

0

0

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0 +

0

0

0

0

0

0

0

0

Proof. Let X(t X ) 2 C (I  X ). Hence, X(t X ) is unique 8X 2 X . L (X ) is nonempty, in X and compact (Theorem 3.19). If L (X ) is assumed non-connected, then there are two disjoint subsets Li (X )  L (X ), i = 1
2, +

0

0

0

© 2004 by Chapman & Hall/CRC

0

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0

+

0

+

0

+

0

0

0

L (X )  L (X ) = L (X ). The subsets are also nonempty and compact. There are 
 2 < such that N 
L (X )] \ N 
L (X )] = . In view of Denition 3.10 there are two sequences tn and n , y 2 L (X ) and y 2 L (X ) such that X(tn  X ) ! y and X( n  X ) ! y as n ! +1. From this and X(t X ) 2 Ct(I X ) it follows that there is a sequence n, tn < n < n such that X(n X ) 2 f(N 
L (X )]  N 
L (X )]) ; (L (X )  L (X ))g. There exist sequences n and n such that X(m + n  X ) ! y and X(m + n  X ) ! y as n ! +1, for arbitrary natural number m. In view of Lemma 3.1, X(m + n  X ) = X n X(m  X )] ! y as n ! +1 and X(m + n  X ) = Xn X(m  X )] ! y as n ! +1. This and X(t X ) 2 Ct(I  X ) imply existence of a sequence n, tn < n < n, and w 2 f(N 
L (X )]  N 
L (X )]) ; (L (X )  L (X ))g such that X(m + n  X ) = Xn X(m  X )] ! w. Hence, w 2 L (X ) which contradicts w 2 f(N 
L (X )]  N 
L (X )]) ; (L (X )  L (X ))g because L (X )  L (X ) = L (X ). Hence, L (X ) and L (X ) are not disjoint, i.e. L (X ) is connected. + 1

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3.3 System regimes 3.3.1 Forced regimes and the free regime

Denition 3.16 (a) The system (3.4) is in a forced regime if and only if there is t 2 < such that i(t) 6= 0. +

(b) The system (3.4) is in a free regime if and only if i(t) = 0 for all t 2 <+ .

The system (3.4) can be either in a forced or in a free regime. However, the system (3.7) is in a free regime.

3.3.2 Periodic regimes

Denition 3.17 The system (3.4) is in a periodic regime if and only if there is

 2 <+ such that its generalised motions obey

X(t +  X0  i)  X(t X0  i): The minimal  is called period. It is denoted by T . The generalised motion X(t X0 i) is called periodic (with period T ).

In literature it can be found that  = 0, i.e. T = 0, is permitted, which means that time-invariant (elementwise constant) X(t X0  i)  const is considered as a periodic motion.

© 2004 by Chapman & Hall/CRC

3.3.3 Stationary regimes and stationary points

Denition 3.18 The system (3.4) is in a stationary regime if and only if its generalised motions through X0 obey X(t X0  i) = X0
8t 2 <+ : The point X0 is called a stationary point (stationary state, steady state), and it is denoted by Xs , X0 = Xs .

Theorem 3.21 In order for X  2 X0 to be a stationary point of the system (3.4) it is necessary and sucient that both (a) the system has forward-time unique generalised motion X(t X   i) through X  , and (b) fX 
i(t)] = 0
8t 2 <+ .

rrr

Proof. Necessity. Let X  = Xs be a stationary point of the system (3.4). Then, in

view of Denition 3.18, any two generalised motions X 1 (t X   i) and X 2(t X   i) obey X k (t X   i) = X 
8t 2 <+
k = 1
2: Hence, the system (3.4) has the forward-time unique generalised motion X(t X   i) = X  through X  , which obeys d X(t X   i) = 0
8t 2 < : + dt This and (3.4) imply (b). Suciency. Let (a) and (b) hold and (t) = X  for all t 2 <+ . Hence,

(0) = X 
and d dt (t) = 0 = f(t)
i(t)]
8t 2 <+ : The last result shows that ( ) is a generalised motion of the system (3.4), (t) = X(t 0 X  i) = X  , for all t 2 <+ . Since X(t 0 X  i)  X  is unique then X  is a stationary point.

© 2004 by Chapman & Hall/CRC

3.3.4 Equilibrium regimes and equilibrium points

Denition 3.19 The system (3.4) is in an equilibrium regime if and only if its generalised motions in the free regime obey X(t X0  0) = X0
8t 2 <+ : The point X0 is called an equilibrium point (an equilibrium state), and it is denoted by Xe , X0 = Xe . Theorem 3.22 In order for X  2 X0 to be an equilibrium point (an equilibrium state) of the system (3.4) it is necessary and sucient that both (a) the system has forward-time unique generalised motion X(t X   i) through X  in the free regime, and (b) f(X 
0) = 0:

rrr

Proof. Theorem 3.22 results directly from Theorem 3.21 due to i(t)  0 (Denition 3.19). A direct consequence of Theorem 3.22 is the following result due to (3.5)

Corollary 3.1 In order for X  = 0 to be an equilibrium point of the system (3.4) it is necessary and sucient that the system has forward-time unique generalised motion X(t 0 0)  0 through X  = 0 in the free regime. Note 3.11 Theorem 3.21 and Theorem 3.22 directly apply to the system (3.7). Note 3.12 When we study equilibrium states and their properties it is the same to study them for the system (3.4) or for the system (3.7) due to (3.6) and Denition 3.19. The next two results generalise Theorem 2.10 and Corollary 2.11 by Nemytskii and Stepanov 157, p. 332].

Theorem 3.23 If, for any " 2 < , there exists a point y 2 N ("
X  ), such that the right half-trajectory X(<  y 0)  N ("
X  ), then X  is an equilibrium point of the system (3.4) as soon as system generalised motions X(t X  0) 2 C (<  X ). rrr +

+

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0

Proof. Let generalised motions of the system (3.4) X(t X  0) 2 C (< X ). Let for any " 2 < there exist a point y 2 N ("
X  ) such that the right half-trajectory X(<  y 0)  N ("
X  ). Let it be assumed that X  is not an equilibrium state of 6 X  . Let X 
X( X   0)] the system. Hence, for some  2 < , X( X   0) = 0

+

+

© 2004 by Chapman & Hall/CRC

+

0

+

0


d = d > 0. From X(t X  0) 2 C (<  X ) it follows that there is " 2 0
2 such 0

0

+

0

that t 2 0
] and (z
X  ) < " < d2 imply

X(t z 0)
X(t X   0)] < 2d :

By the condition of the theorem there is y 2 N ("
X  ) such that X(<+  y 0)  N ("
X  ): The triangle property of a distance function yields X 
X( X   0)] < X 
X( y 0)] + X( y 0)
X( X   0)]
which with the above results (in which we set z = y) leads to X 
X( y 0)] + X( y 0)
X( X   0)] < " + 2d < d
so that X
X( X   0)] < d: This contradicts existence of  2 <+ as assumed and proves the theorem. The requirement for X(t X0  0) 2 C0(<+  X0) can be omitted in case a point X  is stable. This is explained by the following statement. Theorem 3.24 If X  2 X0 is stable point of the system (3.4) then it is an equilib-

rium point of the system rrr Proof. Let X  2 X0 be stable point of the system (3.4). Then (Denition 1.1 of the Section 1.1.1 and Note 1.2), for any " 2 <+ there is = (") 2 ]0
"] such that (X0
X  ) < implies X(t X0  0)
X ] < " for all t 2 <+

and for all the generalised motions of the system. Let "n 2 <+ be a sequence converging to zero as n ! +1. Then, ("n ) ! 0 as n ! +1. Since (X 
X  ) < ("n ) for every n 2 f1
2
: : :g then for any generalised motion X(t X   0) holds X(t X   0)
X  ] < "n for all t 2 <+ : As n ! +1, "n ! 0 so that X(t X   0)
X ] = 0 for all t 2 <+ and for every generalised motion X(t X   0). Equivalently, every generalised motion X(t X   0) obeys X(t X   0) = X  for all t 2 <+ which proves that X  is an equilibrium state of the system (Denition 3.19).

© 2004 by Chapman & Hall/CRC

Theorem 3.25 Let generalised motions of the system (3.4) X(t X  0) 2 C (<  X ). If for any " 2 < , there exists a point y 2 N ("
X  ) such that lim X(t y 0) : t ! +1] = X  2
+

0

0

+

Proof. Let the conditions of the theorem hold. From the denition of a limit, for any " 2 <+ , there is  2 <+ such that X(t y 0)
X  ] < " for all t 2 
+1: Let z = X( y 0), z 2 N ("
X  ), so that X(<+  z 0)  N ("
X  )


(3.21)

in view of Xt X( y 0) 0]  X(t +  y 0) (Lemma 3.1). The conditions of Theorem 3.23 are satised due to (3.21), which proves that X is an equilibrium state of the system. Theorem 3.25 is general. It can be linked with attraction of X  2 X0 as follows.

Theorem 3.26 Let generalised motions of the system (3.4) X(t X  0) 2 C (<  X ). If a point X  2 X is attractive then it is an equilibrium point of the system. rrr 0

0

0

+

0

Proof. Let X(t X  0) 2 C (<  X ) and X  2 X be attractive. Then, there is 0

0

+

0

$ > 0 such that (y
X  ) < $ implies

0

lim X(t y 0) : t ! +1] = X  :

(3.22)

Let " 2 <+ be arbitrarily chosen and (y
X  ) < min($
"). Then, (3.22) holds. Hence, all the conditions of Theorem 3.25 are satised, which completes the proof.

Note 3.13 Theorems 3.23{3.26 hold obviously for the system (3.7).

3.4 Invariance properties of sets of equilibrium states The set of all the equilibrium states denoted by Se has interesting invariance properties.

Theorem 3.27 Let S be any nonempty subset of Se { the set of the equilibrium states of the system (3.4), S Se. Then the set S is positively invariant. rrr © 2004 by Chapman & Hall/CRC

Proof. Let (S 6= ) Se . Let X 2 S be arbitrary. Then X = Xe, i.e. X(t X ) = X
8t 2 <
0

0

0

0

+

due to Denition 3.19. Hence, X(t X0) 2 S
8t 2 <+ :

Note 3.14 Theorem 3.27 is valid also for the system (3.7) due to (3.6) and Denition 3.19.

Theorem 3.28 If the system (3.7) has generalised motions X(t X0) continuous in X0 2 X0 for every t 2 <, X(t X0) 2 C0(<X0 ), with I0(X0 ) = < for all X0 2 X0, then every nonempty subset S of the set Se of its equilibrium points is invariant.

rrr

Proof. Let I (X ) = <, 8X 2 X and X(t X ) 2 C (<  X ). Then, Lemma 3.1

yields

0

0

0

0

0

0

0

Xt X( X0 )] = X(t +  X0 )
8t 2 <
8 2 <
8X0 2 X0 :

(3.23)

Let (S 6= ) Se be arbitrary. Let X0 2 Se be also arbitrary. Then X0 = Xe so that (Denition 3.19) X(t X0 ) = X0
8t 2 <+ :

(3.24)

Let us assume that there is  2 <; such that X( X0 ) 6= X0 :

(3.25)

Since (3.23) is true for any t 2 < then we may take 2 <+ such that +  2 <+ , which with (3.24) implies X( +  X0) = X0 :

(3.26)

(3.23) and (3.26) show that X( +  X0 ) = X  X( X0 )] = X0


2 <
( + ) 2 < : +

+

(3.27)

Since 2 <+ and the system has unique generalised motions (Theorem 3.14) then Denition 3.19, (3.24) and (3.27) imply X( X0 ) = X0
which disproves (3.25) and shows that X( X0 ) 2 S
8 2 <; : This, (3.24) and X0 2 S complete the proof. This theorem directly implies the next result.

© 2004 by Chapman & Hall/CRC

Corollary 3.2 If the system (3.7) has generalised motions X(t X0) continuous in X0 2 X0 for every t 2 <, X(t X0 ) 2 C0(<  X0 ), with I0(X0 ) = < for all X0 2 X0 then the set Se of the equilibrium states cannot be reached in nite time by any generalised motion X(t X0 ) as soon as X0 2 (X0 ; Se ).

Note 3.15 If we reconsider the system x_ = ;sign x (Example 3.7, Fig. 3.8), then we conclude that Se = f0g is the set of its equilibrium states and it is singleton (i.e. xe = 0 is its unique equilibrium state). For any x 2 < the generalised motion x(t x ) reaches Se = f0g in nite time that equals jx j. This does not contradict Corollary 3.2 because the system generalised motions are not continuous in x 2 ] ;
 for any 2 < and for every t 2 <. The same is valid for the system 0

0

0

of Example 3.1.

0

+

3.5 Dynamical and generalised dynamical systems We shall present at rst the classical denition of dynamical systems 10], 120], 157], 218]. Then, it will be discussed from the point of view of the preceding results.

3.5.1 De nition and properties of dynamical systems Denition of dynamical systems

In the next denition \dynamical system" means \time-invariant continuous-time dynamical system". Denition 3.20 A system with motions (or generalised motions) X is a dynamical system if and only if its (generalised) motions have the properties P1 through P3: P1 Existence: Every system (generalised) motion X is dened in
X(0 X0) = X0
8X0 2
P2 Continuous Dependence: Every (generalised) motion X is continuous in (t
X0) 2 < 
Properties of dynamical systems Comment 3.11 What follows is due to Nemytskii and Stepanov 157, p. 327],

Zubov 218, p. 20], and La Salle 120, p. 28]. Property P2 can be interpreted in either of the following two equivalent ways: a) If (tn
Xn ) 2 < 
statement on motions, which is broadened to generalised motions. Theorem 3.29 For any (X

") 2
dynamical system. rrr Comment 3.12 Property P2 concerns two dierent (generalised) motions through two dierent initial points X and y at two dierent instants s and t, X(s y) and X(t X). However, Theorem 3.29 concerns such two (generalised) motions at the same time t 2 <. It in fact proves Note 3.9, that is that X(t X0) 2 C(I0  X0) implies both X(t X0) 2 Ct (I0  X0) and X(t X0 ) 2 C0(I0  X0).

Note 3.16 In view of Theorem 3.29 and Comment 3.12, we conclude that Theorems 3.15{3.20, 3.23, 3.25 and 3.28 and Corollary 3.2 hold for dynamical systems. Theorems 3.21, 3.22, 3.24, 3.26 and 3.27 are valid for dynamical systems because those theorems are general. From Property P2 we deduce uniqueness of motions.

Theorem 3.30 A dynamical system has a unique (generalised) motion X through every X0 2
Proof. The theorem can be proved in two ways. a) Let a system be dynamical. Definition 3.9 is valid. In view of Property P , Theorem 3.13 holds. Let it be assumed that the statement of Theorem 3.30 is false. Then, there exists (
X ) 2 <  0. Let (
") 2 ]
+1]0
 and let = (" 
X ) 2

0

0

1

0

2

1

0

2

0

0

1

0

2

0

0

obey Theorem 3.13. Since > 0 and (X0
X0) = 0 then (X0
X0 ) < so that X 1( X0 )
X 2( X0 )] < ", due to Theorem 3.13. The last inequality contradicts X 1( X0 )
X 2( X0 )] =  > ", which is the consequence of the assumption that  > 0. Hence,  = 0, or X 1 ( X0 ) = X 2 ( X0 ). This further contradicts the existence of  2 < at which the (generalised) motions X 1 and X 2 dier. Therefore, X 1(t X0 ) = X 2 (t X0) = X(t X0 ) for all t 2 < and every X0 2
© 2004 by Chapman & Hall/CRC

r+s+t 0

t

r+t

s

s+t

t

Figure 3.12: A graphical explanation of Group Property P3: x = x(s x0 ), x(t x ) = xt x(s x0 )] = x(s + t x0 ). The same holds for any r
s
t 2 < and any x0 2 <.

Comment 3.13 Bhatia and Szego 10, p. 11] proved that every motion X of a dynamical system is a commutative group characterised by the following axioms, the rst three of which are axioms of a group:

(i) (ii) (iii) (iv)

Xt X(0 X0)]  X(t + 0 X0)  X(t X0 ): Xt X(;t X0 )]  X(t ; t X0 )  X(0 X0): X(t Xs X(r X0 )])  X(t Xs + r X0])  X(t + s + r X0): Xt X(s X0 )]  X(t + s X0 )  X(s + t X0)  Xs X(t X0 )]:

Comment 3.14 Group Property P3 means that the (generalised) motion at time (t + s), which passes through X0 at t = 0, equals the motion at time t, which passes through X  at t = 0, provided for X  is taken X(s X0 ) achieved at time s from X0 at t = 0, Fig. 3.12. Comment 3.15 Let us consider the systems of the preceding examples. 2=3 a) dX dt = X was analysed in Example 3.1, Fig. 3.1. It is not a dynamical system because it does not have a unique motion (or generalised motion) through X0 = 0 at t = 0, i.e. not through every X0 2 < at t = 0 (Theorem 3.14).

b) dX = sgn X was considered in Example 3.2, Fig. 3.2, and Example 3.6, dt Fig. 3.7. Its motions through any (X0 6= 0) 2 < at t = 0 are not dened on <, and its generalised motions do not have Group Property P3 which will be shown as follows: let t = 3jX0j 2 <+ , s = ;2jX0 j 2 <; and sgn 0 2 f;1
0
+1g. Then, for X0 2 <+ , the generalised motion is determined by

© 2004 by Chapman & Hall/CRC

X(s X0 ) = 0 and X(t X0 ) = X0 + t sgn X0 = X0 + 3jX0j sgn X0 = 4X0. Further, X(s+t X0 ) = X(;2jX0 j+3jX0j X0) = X0 +jX0 j sgn X0 = 2X0 , and Xt X(s X0 )] = X(t 0) = X(3jX0j 0) = X0 + 3jX0j sgn X0 = 4X0 Hence, Xt X(s X0 )] 6= X(s + t X0 ). The system is not a dynamical system.

c) dX = ;sgn X , Example 3.3, Fig. 3.3, and Example 3.7. Fig. 3.8. Let dt X0 2 <+ , s 2 ] ; 1
;2X0, t = ; 2s . Then, X(t X0 ) = 0, Xs
X(t X0 )] =     X(s 0) = 0, X(t + s X0 ) = X ; 2s + s X0 = X 2s  X0 = X0 ; 2s sign X0 = X0 ; 2s > 2X0. Hence, Xs X(t X0 )] 6= X(t + s X0 ). The Property P3 is not realised. The system is not a dynamical system.

dX

d) dt = ;X 1=2 of Example 3.4, Fig. 3.4, does not obey Property P1. It is not a dynamical system.

(e) dX = X 3 is considered in Example 3.5, Fig. 3.5. It does not satisfy Properdt ty P
1 because motions
(which coincide with generalised motions) do not exist 1 on ;1
; 2X 2 for every (X0 6= 0) 2 <. The system is not a 0 dynamical system.

Example 3.14 The rst order system, n = 1

8 dX = ; sat X
sat X = < X
jX j  1
: sign X
jX j 2 <
jX j > 1 dt +

has motions dened by 8 (1 + ln jX j ; t) sign X
t 2 ] ; 1
ln jX j] 9 = > 0 0 0 >

> > 
jX j 2 ]0
1
exp ( ; t) X
t 2  ln j X j
+ 1  > > < X(t X ) = > 0
t2<
X = 0
> 9 > (jX j ; t) sign X
t 2 ] ; 1
jX j ; 1] = > > > : exp (;t+jX j;1) sign X
t 2 jX j ; 1
+1 
jX j 2 1
+1: 0

0

0

0

0

0

0

0

0

0

0

0

The motions are dened and continuous in (t
X0 ) 2 <  <, and dierentiable in t 2 I0 for all X0 2 <. The system has a unique motion through every X0 2 < at t = 0, Fig. 3.13. Hence, the system motions possess Property P1 and Property P2. They do not possess Property P3, which can be veried as follows. Let for example X0 2 ]0
1, s 2  ln jX0j
+1 and t 2 (ln jX0 j ; s)
+1. Then, X(s X0 ) = exp (;s) X0 ) jX(s X0 )j  1
t + s 2  ln jX0j
+1
© 2004 by Chapman & Hall/CRC

_x(t;x0)

1 _x(t;0) ≡ 0

t

-1

Figure 3.13: Motions of the system x_ = ;sat x (Example 3.8).

X(t + s X0 ) = exp (;s ; t) X0
s 2  ln jX0j
+1) t 2 <
Xt X(s X0 )] = exp (;t) X(s X0 ) = exp (;t) exp (;s) X0 = exp (;t ; s) X0
for t 2  ln jX0j
+1
but Xt X(s X0 )]=(1 + ln jX(s X0 )j ; t)sign X(s X0 ) = (1 + ln jX0j ; t ; s)sign X0
for t 2 ;1
ln jX0j: Hence, Xt
X(s X0 )] 6= X(t + s X0 )
t 2 ;1
ln jX0j: This illustrates that the motions do not possess Property 3. Hence, the system is not a dynamical system (Denition 3.9).

Comment 3.16 The function f of the system X_ = ;sat X , f(X) = ;sat X , is dierentiable in X 2 (< ; f;1
1g). However, the motions are dierentiable in time t for all t 2 < and every X 2 <. This means they are dierentiable also at time t when jX(t  X )j = 1, which further means sat X(t  X ) = sign X , (X = 6 0) 2 <. However, at that moment f is not dierentiable with respect to X = X(t  X ). This illustrates that dierentiability of f on
0

0

0

0

0

© 2004 by Chapman & Hall/CRC

0

3.5.2 De nition and properties of generalised dynamical systems Denition of generalised dynamical systems

Theorems 3.14{3.28 suggest generalisation of Denition 3.20.

Denition 3.21 A system is a generalised dynamical system if and only if its gen-

eralised motions X have the properties P1 (Denition 3.20), P20 and P3 (Denition 3.20), where P20 Continuous Dependence: Every (generalised) motion X is continuous in t 2 < for every X0 2
X(t X0 ) 2 Ct(< 
Properties of generalised dynamical systems Note 3.17 Theorems 3.14{3.28 hold for generalised dynamical systems provided I  X = < 
0

0

0

0

3.6 Stability properties and invariance properties of sets At rst we generalise Theorem 1.5.24 by Bhatia and Szeg(o 10, p. 54] to systems that need not be dynamical systems (in the sense of the classical mathematical Denition 3.20). In what follows we consider the system (3.7) that has generalised motions X(t X0).

Theorem 3.31 If a closed set A, A 
0

0

system (3.7) obeys (3.29), X(t X0) 2 N ("
A) © 2004 by Chapman & Hall/CRC

for all t 2 < and every " 2 <+


(3.29)

because (X0
A) = 0 < M (") for every " 2 <+ . From (3.28) and (3.29) it follows that every generalised motion through X0 2 A satises (3.30), X(t X0 ) 2 \N ("
A) : " 2 <+ ] = A for all t 2 <+


(3.30)

which proves the theorem. (b) The theorem can be proved also in the following way. Let a closed set A, A 
0

of the system (3.7) fulls (3.32),

X(t X0 )
A] < " for all t 2 <+ and every " 2 <+


(3.32)

because (X0
A) = 0 < M ("). Let " ! 0+ which with (3.32) and (3.31) yield X(t X0 ) 2 A for all t 2 <+ : Theorem 2.6.8 by Bhatia and Szeg(o 10, p. 137] is broadened to hold also for nondynamical systems.

Theorem 3.32 Let A, A 
and

N ("
A) = N ("
Ai) : i 2 I ]
8" 2 < : +

(3.33)

Necessity. Let A be stable. Let j 2 I be arbitrarily chosen. Let " 2 <+ be such

that

N ("
Aj ) \ N ("
Ak ) = 
8(k 6= j) 2 I : (3.34) Such " exists because all Ai are compact and I is nite. Let " 2 ]0
" ] be arbitrarily chosen. Then, X 2 N  M (" A)
A] implies (Denition 1.3, Section 1.1.1.): (3.35) X(t X ) 2 N ("
A)
8t 2 < : Let X  2 N  M (" A)
Aj ] be arbitrarily chosen. Since M (" A)  "  " then 0

0

+

(3.33){(3.35) imply

X(t X  ) 2 N ("
Aj )
8t 2 <+ : © 2004 by Chapman & Hall/CRC

For " 2 ]"
+1 it su ces to accept (" Aj ) = M ("  A). Hence, for every " 2 <+ there is (" Aj ) 2 <+ obeying (X0
Aj ) < (" Aj ) ) X(t X0 )
Aj ] < "
8t 2 <+ : This means stability of Aj because X(t X0) is also arbitrary generalised motion of the system (3.7). Suciency. Let Ai be stable for any i 2 I . Let " 2 <+ be arbitrarily chosen and

(" A) = min  M (" Ai ) : i 2 I ]: Since M (" Ai ) 2 <+ for every i 2 I , and I is nite, then (" A) 2 <+ . Let X0 2 N  (" A)
A]. Since Ai -sets are pairwise disjoint and A = Ai : i 2 I ], then there is j 2 I such that X0 2 N  (" A)
Aj ]. The denition of (" A) yields N  (" A)
Aj ] N  M (" Aj )
Aj ]. Hence, X0 2 N  M (" Aj )
Aj ] which guarantees X(t X0) 2 N ("
Aj ), 8t 2 <+ , due to stability of Aj . This result and (3.33) imply further X(t X0) 2 N ("
A), 8t 2 <+ , which means stability of A because X(t X0) is arbitrary generalised motion of the system (3.7).

3.7 Invariance features of stability domains properties Invariance properties of stability domains of sets hold also for stability domains of equilibrium states, which is obvious if we dene a set A = fXe g. With this in mind it su ces to consider only invariance properties of stability domains of sets.

Theorem 3.33 Let the system (3.7) be a generalised dynamical system. Let a set A, A 
Then there is " 2 <+ such that X(t X0 ) 2 N ("  A) for all t 2 <+ . Hence, X0 2 Dsc ("  A) yields X(t X0 ) 2 Dsc ("  A) Dsc(A) for all t 2 <+ and proves positive invariance of Dsc (A). The next results broaden the validity of the classical theorems on properties of the domain of attraction from the framework of dynamical systems to generalised dynamical systems.

Theorem 3.34 Let the system (3.7) be a generalised dynamical system. Let a set A, A 
0

0

+

+

0

1

+

2

+

0

0

0

1

0

+

0

+

3

+

0

+

+

0

inition 2.4 (Section 2.2.3) and the properties P1 and P20 of generalised dynamical systems (Denition 3.21). The system (3.7) has unique generalised motions (Denition 3.21 and Property 20  and Theorem 3.14), which possess Property 3, X(t +  X0 ) = Xt X( X0 )], 8(t
) 2 <  <, 8X0 2
Comment 3.17. Since @Da (A) of Da (A) is invariant when the system (3.7) is a generalised dynamical system then obviously

lim fX(t X0 )
A] : t ! +1g > 0 for X0 2 @ Da : © 2004 by Chapman & Hall/CRC

There is an interesting relationship among domains of stability properties 81]:

Theorem 3.35 Let a set A, A 
+

0

0

+

0

0

and

D(A) = Dc (A) = Da (A) = Dac (A): rrr Proof. Let a set A, A 
X it follows that for any X 2 Da (A) and any t 2 < the point y = X(t X ) is connected with X . This and existence of 2 < such that X(  X ) 2 Dac (A) due to lim fX(t X )
A] : t ! +1g = 0 imply that X is connected with X(  X ). Hence, X 2 Dac (A) and Da (A) Dac (A). By denition, Dac (A) Da (A). Therefore Da (A) = Dac (A). Let X 2 Dac (A) = Da (A) and " = 2 2 < be dened as above so that for all t 2 < , X(t X ) 2 N (" A), which together with lim fX(t X )
A] : t ! +1g = 0 and X(t X ) 2 Ct < Da (A)] imply existence of 2 < such that X(  X ) 2 Dsc(" A), connectedness of X(t X ) at any t 2 < with X(  X ) and X(t X ) 2 Dsc(" A) Dsc (A) for all t 2 < . Hence, X 2 Da (A) = Dac (A) implies X 2 Dsc (A) and Da (A) = Dac (A) Dsc. By the denitions Dsc (A) Ds(A), Dc (A) = Dsc(A) \Dac (A) and D(A) = Ds(A) \Da (A). Therefore, D(A) = Da (A) = Dac(A) = Dc (A) Dsc (A) Ds (A). Comment 3.18. This theorem explains why many authors use the term \region of 0

+

0

0

0

0

+

0

0

0

0

0

+

0 +

0

0

0

+

+

+

0

0

0

0

0

+

0

0

0

+

0

0

+

0

0

0

0

+

0

0

+

0

0

0

+

+

0

0

+

0

0

+

0

attraction" for the \domain of asymptotic stability". It also justies the use of \the asymptotic stability domain" for \the strict asymptotic stability domain".

Comment 3.19. Various other qualitative features of domains of stability properties were discovered and described by Bhatia and Szego 10], Genesio et al. 53] and Chiang and Thorp 30]. We shall particularly consider stability of equilibrium states on the boundaries of domains of stability properties.

3.8 Features of equilibrium states on boundaries of domains of stability properties At rst we establish a theorem discovering that the boundary of attraction domain cannot contain either a stable or an attractive equilibrium state.

© 2004 by Chapman & Hall/CRC

Theorem 3.36 If a set A  0 for all t 2 <+ that contradicts X0 2 Da (A). Therefore, Xe cannot be stable. Xe cannot be attractive. If Xe were attractive then there would exist $ > 0 such that (X0
Xe ) < $ implies lim fX(t X0)
Xe ] : t ! +1g = 0. Let X0 2 N ($
Xe) \ Da (A)]. Then lim fX(t X0 )
A] : t ! +1g > 0 that contradicts X0 2 Da (A). Hence Xe 2 @ Da (A) cannot be attractive. This theorem yields the following:

Theorem 3.37 If a set A 
© 2004 by Chapman & Hall/CRC

Chapter 4

Foundations of the Lyapunov method 4.1 Introductory comment Denitions of stability properties and of their domains are stated via (generalised) motions of systems. A test of a stability feature via its denition demands knowledge of system (generalised) motions (for innitely many dierent initial points X0 from a neighbourhood of an equilibrium state or a set). Determination of the (generalised) motions further requires solving the system's mathematical model, which is rarely possible analytically in the closed form for nonlinear systems described by nonlinear rst-order vector dierential equation (4.1): dX = f(X): (4.1) dt Papers by Poincar&e 165], 166] inspired Lyapunov 127] to pose the problem of testing stability properties by using the equation (4.1) directly rather than by solving it. If f(X) = AX the system is linear and stability properties are tested via real parts of the eigenvalues of the matrix A, and eventually also via their multiplicities. However, when f is nonlinear, then it was necessary to discover another, essentially dierent, method. It was discovered and established in 1892 by Lyapunov 127] and has been well-known as the direct method of Lyapunov, or the second method of Lyapunov, or simply as the Lyapunov method. It is based on the concept of sign denite functions v :
_ at The behaviour of v along X(t X0) is now studied via the sign of v(X) every X 2 B . This immediately poses questions of the sense, origin and construction of a function v for a given system (4.1). These questions will be mathematically claried in the next chapter. Their excellent physical clarication in the framework of conservative systems can be found in the book by Rouche and Mawhin 178, pp. 3{4]. In what follows the classic Lyapunov method and the related results will be briey synthesised. For extended studies see 1]{10], 14]{17], 22]{33], 36]{38], 40]{42], 45]{47], 49], 51], 53]{56], 59], 64], 66]{70], 72], 73], 80], 87]{90], 94]{98], 100]{102], 104], 106]{108], 112]{124], 126], 128]{132], 134]{140], 147]{149], 151]{155], 158]{160], 167], 170], 171], 174]{193], 196]{204], 210]{218].

4.2 Sign de nite functions 4.2.1 Sign semi-de nite functions Denition 4.1 A function v :
a) positive semi-denite with respect to a set A, A 
v(X) 2 CN (A)]
2) the function v is non-positive on the interior A of the set A:

v(X)  0
8X 2 A
3) the function v vanishes on the boundary @ A of the set A:

v(X) = 0
8X 2 @ A
4) the function v is non-negative on N (A),

v(X) 0
8X 2 N (A) ; A]
and 5) there exists y 2 N (A) at which the function v has a positive real value

9y 2 N (A)
v(y) 2 < : +

b) positive semi-denite with respect to a set A, A 
© 2004 by Chapman & Hall/CRC

c) positive semi-denite with respect to a set A, A 
Example 4.1 Let a function v be dened by 1) v(X) = aT X. It is continuous on
X  n  4) v(X) =  i xi is positive semi-denite in the whole. i =1

5) v(x) = sin jxj is positive semi-denite on ;
].

( 0


jxj  1
is positive semi-denite with respect to A = jxj ; 1
jxj 1
fx : x 2 <
jxj  1g in the whole.

6) v(x) =

(

0
jxj  1
, is positive semi-denite with respect (jxj ; 1)(4 ; jxj)
jxj 1
to A = B1 on the set S = fx : x 2 <
jxj  4g, but not in the large.

7) v(x) =

8) v(x)  0 is not positive semi-denite because there are not a neighbourhood N of x = 0 and y 2 N such that v(y) > 0.

Denition 4.2 A function v :
(b) A matrix H 2
© 2004 by Chapman & Hall/CRC

The next theorem is a slight modication 88, p. 22], of the classical criteria for semi-denite properties 50, Vol. 1, p. 307). Let H = (hij ) 2
  hi1j1  :::jr =  hi2j1 Hij11ij22:::i r  |  h ir j 1

hi1 j2 hi2 j2 | hir j2

::: ::: ::: :::



hi1 jr   hi2 jr  
|   hir jr 

8r = 1
2
: : :
n
8ik 2 f1
2
: : :
ng
8jk 2 f1
2
: : :
ng
8k = 1
2
: : :
r:

Theorem 4.1 In order for a matrix H = H T 2
:::jr are non-negative, 1) all its principal minors Hij11ij22:::i r

8r = 1
2
: : :
n
8ik 2 f1
2
: : :
ng
8jk 2 f1
2
: : :
ng
8k = 1
2
: : :
r:

:::jr 0
Hij11ij22:::i r

and 2) at least one of its principal minors is positive,

9r = f1
2
: : :
n g  9ik 2 f1
2
: : :
ng
9jk 2 f1
2
: : :
ng
9k = 1
2
: : :
r :::jr > 0: Hij11ij22:::i r

rrr

Note 4.1 The necessary and su cient conditions for negative semi-denitness of a matrix H 2
4.2.2 Sign de nite functions Denitions of sign denite functions Denition 4.4 A function v :
v(X) 2 CN (A)]


© 2004 by Chapman & Hall/CRC

2) the function v is non-positive on the interior A of the set A:

v(X)  0
8X 2 A


3) the function v vanishes on the boundary @ A of the set A,

v(X) = 0
8X 2 @ A
4) the function v has positive values on N (A) out of the closure A of the set A,

v(X) > 0
8X 2 N (A) ; A]
b) positive denite with respect to a set A, A 
Note 4.2 A necessary condition for (global) positive deniteness of a function v with respect to a set A (on a set S ) is its (global) positive semi-deniteness with respect to the set A (on the set S ), respectively. Example 4.2 Let a function v be dened by

X  n  1) v(X) =  ixi . It is not positive denite because it vanishes on the i =1

hyperplane 1x1 + 2x2 + : : : + nxn = 0, which means that there is not a neighbourhood N of X = 0 such that v(X) > 0 for all (X 6= 0) 2 N . 2) v(x) = sin jxj is positive denite on ];
. It is not positive denite on ;
] because v(x) = 0 for jxj = . ( 0
jxj  1 , is globally positive denite with respect to the 3) v(x) = jxj ; 1
jxj 1 set A = fx : x 2 <
jxj  1g. However, it is not positive denite because v(x) = 0 on A that means there is not a neighbourhood N of x = 0 on which v(x) > 0 out of the origin. ( 0
jxj  1
4) v(x) = is positive denite with respect (jxj ; 1)(4 ; jxj)
jxj 1
to the set A = fx : x 2 <1
jxj  1g = ;1
1] on the set S = fx : x 2 <1
jxj < 4g = ] ; 4
4. It is not positive denite on S = ;4
4] because v(x) = 0 for jxj = 4. © 2004 by Chapman & Hall/CRC

k X

5) v(X) = i jxi j is positive denite i k = n and i 2 <+ for every i=1 i = 1
2
: : :
k = n. Under these conditions it is globally positive denite, too. Denition 4.5 A function v :
The criterion for positive deniteness of a square symmetric matrix has the following simple form 50, Vol. 1, p. 306]: Theorem 4.2 (Positive deniteness criterion) In order for a matrix H = (hij ) = H T 2 0
8k 2 1
2
: : :
n: H11 22:::k  | | : : : |   h h : : : h  k1 k2 kk

rrr

Note 4.3 The necessary and su cient conditions for negative deniteness of a matrix H = H T reduce to the necessary and su cient conditions for positive deniteness of the matrix (;H) (Denition 4.6). More precisely: Theorem 4.3 In order for a matrix H = (hij ) = H T 2
  h h h h (;1)k   | |  hk hk 11

12

21

22

1

2

::: ::: ::: :::

    > 0
8k 2 1
2
: : :
n: |   hkk 

h1k h2k

rrr

Note 4.4 If a matrix H 2
(negative) denite i its symmetric part Hs = 1=2(H + H ) is positive (negative) denite, respectively. This follows from the fact that   X T HX = X T H + 12 H T ; 21 H T X = X T HsX + X T Has X = X T Hs X © 2004 by Chapman & Hall/CRC

because

;  X T Has X = 12 X T H ; H T X = 21 X T HX ; (X T H T X)T  0:

Properties of sign denite functions Denition 4.7 (a) A set V (A) is the largest connected neighbourhood of a set A 
Lemma 4.1 If v(X) 2 C(S ) and S is an open connected set, then X 2 S determines uniquely the value v(X) of v. rrr Proof. Let S be an open and connected set and v(X) 2 C(S ). Let it be assumed that some y 2 S does not determine uniquely v(y), i.e. that there are and

2 <, = 6 , such that v(y) = and v(y) = . Let " = = j ; j. From the conditions of the lemma follows the existence of = (" y) 2]0
"] ensuring that kX ; yk < implies jv(X) ; v(y)j < ", i.e. both jv(X) ; j < " and jv(X) ; j < ". Since X = y obeys kX ; yk < then jv(X) ; j = j ; j < " = = j ; j, which is a contradiction because = 6 was assumed. Hence, = , or in other words there is not y 2 S as supposed. 2

1

2

1

1

2

1

4

1

1

1

1

1

2

2

1

2

1

2

4

1

2

2

A direct consequence of this lemma is the following Lemma 4.2 If v(X) 2 C(S ), 1 and 2 2 < are such that V i (A)  S , and S is open connected set, S
rrr Proof. Let v(X) 2 C(S ), and 2 <, 6= , V i (A)  S , S = S be connected. Let it be assumed that @ V1 (A) \ @ V2 (A) 6= . Then there is y 2 @ V1 (A) \ @ V2 (A)], i.e. y 2 S , such that v(y) = and v(y) = , which contradicts Lemma 4.1. Hence, @ V1 (A) \ @ V1 (A) = : 1

2

1

2

1

2

Rouche and Mawhin established 178, p. 9] the following: Lemma 4.3 Let v : B ! < be a positive denite function, and 0 = inf v(X) : kX k = ]. For every 2 ]0
0, there exists an open connected neighbourhood N of the origin such that v(X) < on N and v(X) = on @ N . As ! 0+ , the sets N enclose each other and their diameters tend to 0. rrr

Note 4.5 In Lemma 4.3 the statement \the sets N enclose each other" means N1  N2 and @ N1 \ @ N2 =  for 0 < < < . Note 4.6 From Denition 4.7 and Lemma 4.3 it follows that N = V in Lemma 4.3 for 2 ]0
. 1

0

© 2004 by Chapman & Hall/CRC

2

0

4.2.3 Comparison functions

What follows on comparison functions is due to Hahn 94, p. 7 and p. 95].

Denitions of comparison functions Denition 4.8 (a) A function ' : 0
! < is a comparison function of the class K  if and only if 1) ' is continuous on 0
: '( ) 2 C(0
), +

0



2) ' vanishes at the origin: '(0) = 0, 3) ' is strictly monotonously increasing on 0
: 0  1 < 2 <  imply 0  '( 1 ) < '( 2 ). (b) K 0 = K if and only if  = +1.

Example 4.3 Let a function  be dened by a)

8 0
=0 > < ( ) = >   1  : j j 1 + cos 
2 < : +

It is not strictly monotonously increasing on 0
 for any  > 0. Notice that ( ) 2 C(<+ ) and (0) = 0. b) ( ) = sin i
 i 2 <+. It is a comparison function of the class K 0 for any  2 0
2 . c) ( ) = k
2 <+
k 2 <+ . It is a comparison function of the class K.

Denition 4.9 A function ' : < ! < is a comparison function of the class K< if and only if ' 2 K and lim '( ) : ! +1] = +1. +

+

Properties of comparison functions 'I denotes the inverse function to ',

'I '( )] 
''I ()]  : If '( ) =  then = 'I (). '2  '1 denotes the function composition, that is '2  '1 ( ) = '2 '1 ( )]:

© 2004 by Chapman & Hall/CRC

Lemma 4.4 (a) If 'i 2 K, i = 1
2, then '  ' 2 K. (b) If ' 2 K  and '() =  then the inverse function 'I is dened at least on 0
] and 'I 2 K  . If, in addition, ' is dened for all 0 and if lim '( ) : ! +1] =  then 'I is not dened on ]
+1]. (c) If 'i 2 K  , i = 1
2, ' ( ) > ' ( ) on 0
] and ' () =  then 'I () < 2

0

1



0

0

]



1

2

1

1

'I2 () on 0
]. (d) If ' 2 K<, hence lim '( ) : ! +1] = +1, then 'I 2 K<.

rrr

4.2.4 Positive de nite functions and comparison functions

Denitions 4.4 and 3.6 Criterion by Rouche and Mawhin 10, p. 10] directly yield the following result. Lemma 4.5 In order for a function v : S ! <, S 
1) there is a neighbourhood N (A) S , (N (A)= S ), such that v(X) 2 CN (A)], 2) v(X)  0 for all x 2 A, and v(X) = 0 for every X 2 @ A, 3) there are  2 <+ obeying N (
A) S (N (
A)  S ) and ' 2 K 0

such that v(X) '(X
A)] for all X 2 N (
A) (' 2 K 0 ] such that v(X) '(X
A)] for all X 2 S ), respectively rrr

Example 4.4 Let




1) v(x) = sin jxj and S = ] ; 
,  2 ]0
. Then, '( ) = sin  2 K is a comparison function satisfying v(x) '(x
O)] = '(jxj) = sin  jxj because sin jxj sin  jxj for all x 2 ] ; 
. Notice that there is not a comparison function ' 2 K 0] for sin jxj because sin jxj ! 0 as jxj !  that requires '() = 0, which is impossible for ' 2 K 0] . However, v is positive denite on ] ; 
. 2) v(X) = (kX k ; 1) 1 ; sign (1 ; kX k)]. It is globally positive denite with respect to the set A = B1 . The function '( ) = 2 K< obeys v(X) '(X
A)] for all X 2
No ' 2 K can satisfy it. However, if we take  2 <+ arbitrary large then there is ' 2 K 0 ] obeying v(X) = (kX k ; 1) 1 ; sign (1 ; kX k)] exp (1 ; kX k) ' (X
A)]. Such a comparison function ' 2 K is dened by ' (X
A)] = (X
A) exp (;) where  2 ]0
1] can be arbitrarily accepted and (X
A) = 1=2(kX k ; 1) 1 ; sign (1 ; kX k)]. This explains why the closure S of the set S is used in Lemma 4.5.

4.2.5 Radially unbounded and radially increasing positive de nite functions

Denition 4.10 A positive denite function v :
Note 4.7 A comparison function ' 2 K< is radially unbounded because '( ) ! +1 as ! +1 (Denition 4.10). Lemma 4.6 In order for a globally positive denite function v :
;  ; because v(X) = kX ;k ; 1 1 ;sign 1 ;;kX k 2(X
A) for all X 2
1

2

1

2

1

By following 73], 88] we introduce:

Denition 4.11 A function v :
© 2004 by Chapman & Hall/CRC

1

2

1

2

b) radially increasing on the boundary @ S of a set S 
Example 4.6 Let ;  ;  ; 1) v(X) = kX k ; 1 1 ; sign 1 ; kX k 1 ; exp 1 ; kX k . It is radially increasing with respect to the set A = B . However, it is not radially increasing on any neighbourhood N of X = 0. It is radially unbounded globally positive denite with respect to B .  ;  ; 2) v(X) = 1 ; sign 1 ; kX k 1 ; exp 1 ; kX k . It is radially increasing globally positive denite with respect to A = B . But it is neither radially unbounded nor radially increasing on any neighbourhood N of X = 0. 1

1

1

3)

8 0
X=0 > <  
v(X) = > : kX k " + sin kX1 k
" 2 <
(X 6= 0) 2
2

+

It is not radially increasing on any neighbourhood N of X = 0, although it is radially unbounded globally positive denite. By following the proof of Theorem 24.1 by Hahn 94, p. 99], we generalise it to sets as follows.

Lemma 4.7 Let v : S ! < be positive denite with respect to a compact set A 
Example 4.7 Let, by following Hahn 94, p. 99], v(X) =  (X
(X
A)A+) 1 , X 2 < , and A be a compact set in < . Then v(X) = > 1 is not closed. In fact such a v does not exist because v(X) = yields  (X
A) = ; ; 1 < 0, which is not 2

2

2

2

2

possible.

Lemma 4.8 Let v : S ! < be positive denite on a set S 
+

0

1

2

2

1

+

1

2

bitrary and t2 > t1. Let vZ(ti )] = i , i = 1
2. Hence, 2 < 1 . Now, Lemma 4.1, Lemma 4.2 and Lemma 4.7 imply that the hypersurfaces are closed and do not intersect, which with the monotonous decreasing of vZ(t)] prove that v(X) = 1 encloses v(X) = 2 . If t2 = +1 then Z(t2 ) 2 @ A so that vZ(t2 )] = 2 = 0 due to positive deniteness of v on S . Thus, the statement of the lemma is true for 0  2 < 1 < .

Lemma 4.9 Let v : S ! < be radially increasing positive denite function on a set S 
1

2

2

1

Proof. Let all the conditions of the lemma hold. Then v(X) = is closed for

2 ]0
 due to Lemma 4.7. Radial increasing of v on V (A) ; A] and Lemma 4.1 prove now that v(X) = 1 encloses v(X) = 2 for 0 < 2 < 1 <  because if it were otherwise then there would exist X 2 @ V2 and 2 ]0
1 such that  (X ;y)] 2 @ V1 , that would mean v (X ; y)] < v(X ; y) and would contradict radial increasing of v (Denition 4.11d). © 2004 by Chapman & Hall/CRC

4.3 Uniquely bounded sets 4.3.1 De nition of uniquely bounded sets

Uniquely bounded sets and O-uniquely bounded sets were discovered in 67] and further studied in 66], 80], 84].

Denition 4.12 (a) A set S 
(X U  Z) (X ; Z)] 2 @ U , (b) A set S 
(X Z) if and only if U and  are xed. Note 4.9 From (X U  Z)(X ; Z)] 2 @ U (Denition 4.12-3) it follows:

(X U  Z)  (X ; Z) U ]
+

+

0

0

+

and

(X Z)  (X ; Z)] in case U is xed. If  = 1 is also xed then it is omitted. Denition 4.12 will be illustrated by several examples taken from the reference 66].

Example 4.8 Starlike sets with respect to 0 dened by Spivak 189] seem at rst glance equivalent to O-uniquely bounded sets. However, they are dierent. The set S dened by S = fX : X 2 <
(kX k < 4
x < 0) _ (kX k < 4
x > 0
x 2 ] ; 2
1) _ _ (x = 0
x 2 ] ; 2
1)g is starlike with respect to 0 but it is not O-uniquely bounded because the ray through (0
0) and (0
2) has innitely many common points with the boundary of the set S . 2

1

2

2

© 2004 by Chapman & Hall/CRC

1

2

1

2

Example 4.9 At rst glance the family of uniquely bounded sets seems to be the

family of starlike sets dened by Hocking and Young 103]. To show that the former is a proper subfamily of the latter let the following starlike set be considered, S = X : X 2 <2
(x1 ; 2)2 + x22  4 _ (x1 + 2)2 + x22  4 : It is not uniquely bounded. There is not a point Z 2 S that satises Denition 4.12.





Example 4.10 Let S = X : X 2
Example 4.11 Let S = X : X 2
2

quely bounded.

2

2

4.3.2 Properties of uniquely bounded sets

Property UB. Let U 
u(X ; Z) = 0 for (X ; Z) 2 N i X = Z , if N =
5) there is  2 <+ such that both 1) and 2) hold: 1) u(X ; Z)   for (X ; Z) 2 N i (X ; Z) 2 U
2) u(X ; Z) =  for (X ; Z) 2 N i (X ; Z) 2 @ U :

Lemma 4.10 (a) For a bounded subset U of an open set N , U N , to be uniquely bounded it is both necessary and sucient that it possesses the Property UB.

(b) For a bounded subset U of an open set N , U N , to be O-uniquely bounded it is both necessary and sucient that it possesses the Property UB with Z = 0.

rrr Proof. Necessity. (a) Let a bounded subset U of an open set N , U N , be uniquely bounded. Denition 4.12 may be used. Let u(X ; Z) =  (X U  Z)]; =  (X ; Z U )]; . 1

1

© 2004 by Chapman & Hall/CRC

1) Denition 4.12 yields (Z U  Z) = +1. Hence, u(Z ; Z) = 0. 2) (X U  Z) ! 0 as kX k ! +1 in case N = 0, 8(X ; Z) 2 N , X 6= Z. Since u(X ;Z) ! 0 as X ! Z and u(Z ;Z) = 0 then, altogether, u(X ;Z) 2 C(N ). 4) Let (X ; Z) 2 N be arbitrary, X 6= Z. Let i 2 <+ , i = 1
2, 1 < 2, be such that i(X ; Z) 2 N . Hence, (1X U  Z) > (2X U  Z), where (iX U  Z) i(X ; Z)] 2 @ U , i = 1
2. The last inequality implies u1(X ; Z)] < u2(X ; Z)]. This and u(Z ; Z) = 0 prove radial increasing of u on N with respect to Z. 5) Since (X U  Z) = 1 i (X ; Z) 2 @ U , it now immediately results (due to 1) and 4)) u(X ; Z)  1 for (X ; Z) 2 N i (X ; Z) 2 U and u(X ; Z) = 1 for (X ; Z) 2 N i (X ; Z) 2 @ U , which completes the proof if we set  = 1. (b) If U is O-uniquely bounded then Z = 0 (Denition 4.12). Suciency. (a) Let all the conditions of the lemma hold. Let (X U  Z) = u(X ; Z)];1 = (X ; Z U ). Radial increasing of u on N from u = 0 with respect to Z implies radial decreasing of on N from (Z U  Z) = +1, that is (1X U  Z) >

(2 X U  Z), i 2 <+ , 1 < 2, i (X ; Z) 2 N , i = 1
2. It now follows that

(y U  Z) = 1 for (y ; Z) 2 N i (y ; Z) 2 @ U due to 5.2) and i(X ; Z)] 2 @ U , i = 1
2, i 1 = 2 = (X U  Z), 8(X ; Z) 2 N , X 6= Z. Hence, U is uniquely bounded (Denition 4.12).

(b) The conditions under (a) hold for Z = 0. Therefore, U is O-uniquely bounded (Denition 4.12). This lemma justies the following denition 67], 66], 80]. Denition 4.13 (a) A function u :
© 2004 by Chapman & Hall/CRC

4.3.3

-uniquely bounded sets and positive de nite functions

O

Lemma 4.11 Let u :
1 < 2 ,

+

4) the hypersurface u(X) = is closed for every 2 <+ for which U S ,

5) if S =
Proof. Let u :
to Denition 4.13b.

1) Positive deniteness of u on S results from 1), 3), and 4) of Property UB and S = N (Denition 4.13b). 2) If U S then it is bounded since 2 <+ and due to 1), 3){5) of Property UB, Lemma 4.10b and Denition 4.10b. Connectedness follows from Denition 4.12b because for any X and y 2 U , (X) 2 U and (y) 2 U , as soon as  2 0
1] and  2 0
1]. Hence, X and y are linked via semi-lines (x) and (y) for  and  2 <+ . 3) The statement under 3) results from 3){5) of Property UB, Lemma 4.10b and Denition 4.13b. 4) The hypersurface u(X) = is closed for 2 <+ for which U S because of 3) and 4) of Property UB, Lemma 4.10b and Denition 4.13b. 5) S = N =
4.3.4 De nition of uniquely bounded neighbourhoods of sets

Denition 4.14 (a) A set S 
1) S = Nu(A) is a bounded neighbourhood of the set A, 2) the point w is in the interior A of the set A: w 2 A, 3) for every (X ; w) 2 (
Example 4.12 Let A = fX : X 2 <
(x ; 2) + x  4] _ (x + 2) + x  4]g and S = fX : X 2 <
x + (x ; 6)  100g. The set S is uniquely bounded with any Z 2 S, hence O-uniquely bounded, too. However, it is not a uniquely bounded neighbourhood of A, although it is a neighbourhood of A, because the set A does 2

2

2 1

2

2

2

1

2 2

2

1

2 2

not obey the conditions 2) and 3) of Denition 4.14.

Example 4.13 The set S = fX : X 2 <
x + (x ; 6)  100g is a uniquely bounded neighbourhood of the set A = fX : X 2 <
(x + 2) + x  4g 2

(Denition 4.14).

2 1

2

2

2

2

1

2 2

4.3.5 Properties of uniquely bounded neighbourhoods of sets

Lemma 4.12 (a) In order for a set S 
(X A w)  (X A
S  w) (Denition 4.12). 2) (X ; w) 2 A. Since S = Nu (A) and A A  Nu (A) then for any V 2
(X A Z) = V (X
w) A
S  w]l(X
V
w) and by Z = w. +

+

+

© 2004 by Chapman & Hall/CRC

Altogether, it now follows that for every (X ; Z) 2 S , X 6= Z, there is exactly one

= (X A Z) 2 <+ such that (X ; Z) 2 @ A with Z = w. (b) The statement under (b) results directly from (a) and Denition 4.14b. Lemma 4.12 is illustrated by Examples 4.12 and 4.13.

Lemma 4.13 Let a set A 
8 0
X = w
> < v(X) = > (X ; w
A) ; (X w) n : 1 ; 1 (X w)X w] (X w)(X ; w)
A]
(X 6= w) 2 <

9 > = = > 

= v(X ; w) a) is globally positive denite with respect to the set A, b) is radially increasing with respect to the set A, and c) v(X) = 1 for all (X ; w) 2 @ Nu (A).

rrr Proof. a) From the proof of a-3 of Lemma 4.10 it follows that (X w) 2 
(X w)X(X w)w] 2 ]0
1 for all X 2  0 for all (X ; w) 2 (
Now, 2) through 4) prove global positive deniteness of v with respect to A. b) Notice that (X ;w w)  (X w), (X ;w w)  (X w), and (X w)X w]  (X ; w)(X ; w)] (Denition 4.12 and Note 4.9). Hence, v(X) = v(X ; w). Since ( X w) > ( X w), since unique boundedness of A guarantees  (X ; w)
A] <  (X ; w)
A], ( X w) X w]  ( X w) X w] and ( X w) (X ; w)
A]  ( X w) (X ; w)
A] because ( X w)  (X ; w)]  ( X w) (X ; w)] 2 @ Nu (A) for every i 2 < , i = 1
2,  <  , and for all (X = 6 w) 2
2

1

2

1

1

1

1

2

1

2

2

+

2

2

2

1

1

2

c) Let (X ; w) 2 @ Nu (A). Then (X w) = 1 and sign (X
A) = 1. Hence,

; (X w) 1 = 1
v(X) = 11 ;

(X w) 1 because (X w) 2 ]0
1. In view of the proof of su ciency of the conditions of Lemma 4.10, we get the following corollary to Lemma 4.13. In Corollary 4.1 the notation is in the sense of Denition 4.13(d).

Corollary 4.1 Let a set A 
1

1

2

8 > > > > < v(X) = > > > > :

2

+

2

0


X = w


u1 (X) ; 1 u12 u;2 1(X)X] u1(X) u1 2u;2 1(X)X] ; 1 (X ; w
A)
(X 6= w) 2
a) is globally positive denite with respect to the set A, b) is radially increasing with respect to the set A, c) v(X) = 1 for all (X ; w) 2 @ Nu (A).

Example 4.14 LetpA = fX : X 2
1

© 2004 by Chapman & Hall/CRC

2

1

2

S = Nu (A) with  = 4pnp= 4; . The function p ;v dened in Lemma 4.13 has the 1

2

following form due to = nu1 1 and  = 4 nu2 1:

8 0
X = 1
> > > pn > > > 1 ; < (X kXp; 1k
v(X) = > pn ; 1
A) n 4 !!  1T jX ; 1j (X ; 1)
A 1 ; !! 4pn > > ! !! > (X ; 1 ) > : ! 1T jX ; 1j


(X 6= 1) 2
or,

8 0
X = 1
> > > < 4(kX ; 1k ; pn) (X v(X) = > (4kX
pn ; 1
A) T jX ; 1j) ; 1 k ; 1 4 >  1T jX ; 1j (X ; 1)
A > :
(X 6= 1) 2 0 for all (X ; 1) 2 (
p

4 n 1T jX ; 1j (X ; 1) 2 @ S





 pn (X ; 1) = 4pn =  , so that due to 1T  1T j4X ; 2  1j p

!! > 0 for (X ; 1) 2 (
p For (X ; 1) 2 ( n. Hence, 1 ; kX ;n 1k > 0. Besides, p
(X ; 1
A) > 0 and  1T j4X n; 1j (X ; 1)
A > 0 for all (X ; 1) 2 (
8 0
X = w
> > < p 4(kX ; 1k ; n)(Xp; 1)
A] =>

X 6= w: > : (4kX ; 1k ; 1T jX ; 1j) 1T4jXn; 1j (X ; 1)
A

Unique boundedness of A ensures that  is increasing in  for X 6= 1. This and v(X) =

8 0
X = 1
> > >  > pn  < 4 kX ; 1k ;  (X ; 1)
A] => > pn

(X 6= 1) 2 4 > : (4kX ; 1k ; 1T jX ; 1j) 1T jX ; 1j (X ; 1)
A

imply v(1 X) < v(2 X) for all i 2 <+ , 1 < 2, and every (X ; 1) 2 ( 0 so that pn) 4( k X ; 1 k ; pn ; 1
A)  = 1 for all (X ; 1) 2 @ S : v(X) = (4kX ; 1k ; 4pn)  4(X  4pn (X ; 1)
A

This veries the last property of the function v as specied in Lemma 4.13-c.

4.4 Dini derivatives and the Lyapunov method 4.4.1 Fundamental lemmae on Dini derivatives

Dini derivatives are dened in Section 3.2.2. For the following result see McShane 143, p. 200] and LaSalle 120, p. 30]. Lemma 4.14 Let  : a
b] ! <. If (i)  is left lower semicontinuous on a
b (i.e. lim inf (t) : t ! ; ] ( ) for each 2 a
b) and (ii) D+ (t)  0 for all t 2 a
b except at the most countable number of points, then  is nonincreasing on a
b, hence dierentiable almost everywhere, and

Zt a

D ()d (t) ; (a)
8t 2 a
b: +

© 2004 by Chapman & Hall/CRC

rrr

This lemma leads to the following result.

Lemma 4.15 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X , X
0

0

0

0

0

0

0

+

or (ii), (ii) D+ v(X)  0 for all X 2 S . Then, (a)  is nonincreasing on 0
b0 where b0 is the rst time when X(t X0 ) is not in S : X(t X0) 2 S for all t 2 0
b0 and X(b0  X0) 62 S , (b)  is dierentiable almost everywhere on 0
b0, and (c) either

(i) or

(ii)

Zt

D vX( X )] vX(t X )] ; v(X )
8t 2 0
b 
+

0

Zt 0

0

0

0

0

D vX( X )] vX(t X )] ; v(X )
8t 2 0
b 
+

0

0

0

0

rrr Proof. Let the system (4.1) have generalised motions X(t X ) for each X 2 X . Hence, X(t X ) 2 Ct(I  X ) (Denition 3.8, Section 3.2.3). Let v,  and S obey respectively.

0

0

0

0

0

0

the conditions of the lemma. Altogether,  is continuous on 0
b0]. Lemma 4.14 and (i) or (ii) guarantee (a) and (b). The condition (i) implies (c-i) due to Lemma 4.15. The condition (ii) implies (c-ii) due to the same lemma because D+ v(X) D+ v(X) by denition and D+ v(X)  0 on S in view of (ii).

Note 4.10 It is important to note that Lemma 4.15 1) does not guarantee positive invariance of the set S , and 2) does not guarantee validity of the result (c) for t b0 . Therefore, it is important to nd conditions for positive invariance of the set S . The following result by Yoshizawa 217, p. 3], simplies determination of Dini derivatives.

© 2004 by Chapman & Hall/CRC

Lemma 4.16 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X
0

0

0

0

+

and 2.

+



D v(X) = lim inf +

 vX + f(X)] ; v(X) 

:!0

+



: rrr

Applications of Lemma 4.16 will be illustrated by an example in which we shall make use of a function ! : <  < ! < dened by

8 ;1
(x 2 <; and x_ 2 <) or (x = 0 and x_ 2 <;)
> > < !(x
x) _ = > 0
(x = 0 and x_ = 0)
> : +1
(x 2 <+ and x_ 2 <) or (x = 0 and x_ 2 <+ ):

With the function ! we associate  :
Example 4.15 Let v : < ! < , v(x) = jxj, and n = 1 so that the system (4.1) takes the scalar form

1

1

dx = f(x): dt Let the system have generalised motion x(t x0) 2 C0(I0  X0), X0 X <1 . The function v is globally Lipschitzian. We may apply Lemma 4.16,  jx + f(x)j ; jxj  + + D v(X) = lim sup :!0 = 

8  ;x + f(x)] ; (;x)  > lim sup :  ! 0
> >  > > (x 2 <; ) or (x = 0
f(x) 2 <; )
> >   > < lim sup x + f(x) ; x :  ! 0
 = > > (x = 0
f(x) = 0)
> >   > x + f(x) ; x > lim sup :!0
>  > : (x 2 < ) or (x = 0
f(x) 2 < ): +

+

+

+

© 2004 by Chapman & Hall/CRC

+

Hence, after cancelling x and (;x), and using x_ = f(x), we get

8 ;x_
(x 2 <; ) or (x = 0
x_ 2 <; )
> > < D jxj = > 0
(x = 0
x_ = 0)
> : x_
(x 2 < ) or (x = 0
x_ 2 < ): +

+

+

Now we can make use of the function ! so that

D jxj = !(x
x)_ x_ = !(x
x_ )f(x) = !x
f(x)]f(x): Obviously, D jxj = D jxj. Example 4.16 Let v :
+

+

1

v(X) = bT jX j = b1 jx1j + b2jx2j + : : : + bn jxnj
with b = (b1 b2 : : :bn)T > 0 elementwise (i.e. bi > 0, 8i = 1
2
: : :
n). Now, by referring to Example 4.15 we derive

D v(X) = bT D jX j = +

+

n X i=1

bi D+ jxij =

n X i=1

_ X: _ bi !(xi
x_ i)x_ i = bT (X
X)

Altogether, along generalised motions of the system (4.1), _ X_ = bT (X
X)f(X) _ D (bT jX j) = bT (X
X) = bT X
f(X)]f(X)
+

and D+ (bT jX j) = D+ (bT jX j). Let V ;1 ( ) = fX : X 2
4.4.2 LaSalle principle

LaSalle established the invariance principle 120, p. 30], that will be called the LaSalle principle.

Theorem 4.4 (LaSalle principle) Let 1) v(X) 2 C(X ), X
+

and 3) the system (4.1) have generalised motions X(t X0 ) 2 C0(I0  X0), X0 X .

© 2004 by Chapman & Hall/CRC

Then, for some , L+ (X0 ) \ X  M \ V ;1 ( ), where M is the largest weakly invariant set in E0 = fX : X 2 X
D+ v(X) = 0g or E0 = fX : X 2 X
D+ v(X) = 0g], respectively. If, in addition to 1){3) 4) X(t X0 ) is precompact relative to X , then X(t X0) ! M \ V ;1 ( ) as t ! +1, where M is the largest invariant set in E0. rrr

Proof 120]. Let at rst the conditions 1) through 3) hold. Let it be assumed that y 2 L (X ) \ X . Then l (X ) = +1, and there is a sequence tn such that X(tn  X ) ! y and tn ! +1 as n ! +1. Now, vX(tn  X )] ! v(y) as n ! +1 (due to 1)) since vX(t X )] = (t) is nonincreasing with respect to t due to 2) (Lemma 4.15) then vX(t X )] v(y) for all t 2 < and there is such that vX(t X )] ! as t ! +1. Hence, for such , v(y) = for each y 2 L (X ). Now, L (X ) is weakly invariant, and therefore L (X )  E , and hence in M . This proves L (X ) \ X  M \ V ; ( ). If, in addition to 1){3), the condition 4) also holds, then L (X ) is invariant and L (X )  M \ V ; ( ). Since X(t X ) ! L (X ) as t ! +1, then X(t X ) ! M \ V ; ( ) as t ! +1. +

+

0

0

0

0

0

0

+

+

0

0

+

+

0

0

+

+

0

1

+

+

0

0

0

0

0

0

1

1

LaSalle deduced the following results from the invariance principle (Theorem 4.4) 116, pp. 31, 33]:

Corollary 4.2 120] Let 1) v(X) 2 C(X ), X
+

3) the system (4.1) have generalised motions X(t X0 ) 2 C0(I0 X0 ), X0 X


Note 4.11 Since the system (4.1) has generalised motion X(t X0) 2 C0(I0  X0) then it is unique (Theorem 4.14, Section 3.2.6). Lemma 4.17 120]. Let 1) S be a positively invariant open set in X with the property that each solution starting in S is bounded and has no positive limit points on the boundary of S , 2) v(X) 2 C(S ), © 2004 by Chapman & Hall/CRC

3) 4) 5) and 6)

D v(X)  0 or, D v(X)  0] for all X 2 S , +

+

the system (4.1) have generalised motions X(t X0 ) 2 C0(I0  S ), M0 = M \ S  S ,

M is precompact. Then, M is attractive set and S Da (M ). 0

0

0

If, in addition to 1){5), 6) v is constant on the boundary of M0 , then M0 is asymptotically stable set.

rrr

Comment 4.1 Eective applications of the preceding results by LaSalle need 1) adequate choice, construction of the function v, 2) eective test of precompactness, and 3) eective test of positive invariance.

4.4.3 Dini derivatives, positive de niteness, positive invariance and precompactness

Lemma 4.18 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X
N (A). Hence, the hypersurface v(X) = is closed, which holds also for v(X) =  that bounds V (A) as soon as  2 ]0
] due to Lemma 4.2. Let it be assumed that V (A) is not positively invariant. Then, there are X 2 V (A) and 2 I for which X(  X ) 2 @ V (A). From X(t X ) 2 Ct(I  X ) \ C (I  X )] (Denition 3.8, Section 3.2.3), X 2 V (A), V (A) =  V (A) and X(  X ) 2 @ V (A) 0

0

0

0

0

0

+

+

+

0

0

0

0

0

0

0

0

0

0

+

0

0

0

0

© 2004 by Chapman & Hall/CRC

0

0

0

follows the existence of a sequence tn , tn 2 0
, tn ! as n ! +1, such that vX(tn;1 X0)] < vX(tn  X0)]. This inequality contradicts the conditions 1) and 2) due to Lemma 4.15 and X(tk  X0 ) 2 V (A), k = n ; 1
n. The contradiction is a consequence of the assumption that V (A) is not positively invariant, which proves its positive invariance. This further implies vX(t X0 )]  v(X0 ) < for all t 2 <+ (Lemma 4.15). Hence, L+ (X0 )  V (A), i.e. L+ (X0 ) \ @ V (A) =  that now proves positive precompactness of V (A) relative to itself.

Denition 4.15 A function v obeying Lemma 4.18 will be called Lyapunov function.

Lemma 4.18 can be linked with Lemmae 4.5 and 4.7 to yield the following result.

Lemma 4.19 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X
0

0

0

0

0

1) v be positive denite function with respect to A on a neighbourhood N (A) of A, 2)  2 <+ be the greatest number for which -neighbourhood N (
A) is in N (A): N (
A) N (A), 3) ' 2 K 0 ] be such that v(X) '(X
A)] for all X 2 N (
A), 4)  = '() 2 <+ , and 5) D+ v(X)  0 or, D+ v(X)  0] for all X 2 N (A). Then the set V (A) is positively invariant and positively precompact relative to itself for each 2 ]0
. rrr

Proof. Let all the conditions of Lemma 4.19 hold, where the condition 3) and 4) result from the condition 2) due to Lemma 4.5. The set V (A) is bounded for every 2 ]0
 in view of Lemma 4.7. Hence, all the conditions of Lemma 4.18 are fullled, which proves Lemma 4.19.

4.5 Stability theorems Following Lyapunov 127], Halanay 95], Yoshizawa 217], Bhatia and Szeg(o 10] and Hahn 94] we shall outline synthesised results on stability conditions.

© 2004 by Chapman & Hall/CRC

4.5.1 Stability of a set

Theorem 4.5 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X
0

0

0

0

0

+

0

0

0

0

0

0

+

+

0

0

+

0

0

]

+

+

0

+

0

+

0

+

From the denition of v it follows (iii) v(X) '(X
A)] for all X 2 N (A). The results (i){(iii) prove positive deniteness of v with respect to A on N (A). The condition 1) is necessary. Let  2 <+ be arbitrarily small and X  = X(t+ t
X) = X( 0
X) = X( X). Now, v(X  ) = sup ('fX(t +  +  t + 
X  )
A]g :  2 <+ ): From X(t X0) 2 Ct<+  N (A)] \ C0 <+  N (A)], Theorem 3.14 (Section 3.2.6), time invariance of the system (4.1) and the denition of X  follows X(t +  +  t + 
X  ) = Xt +  +  t + 
X( X)]  X(t +  +  t
X)  X( +  0
X) = X( +  X), so that for 1 =  + , 1 2 
+1 due to  2 <+ , v(X  ) = vX( X)] = sup ('fX(1  X)
A]g : 1 2 
+1) 

 sup ('fX(  X)
A]g :  2 < )=v(X)
8X 2N (A) 1

© 2004 by Chapman & Hall/CRC

1

+

where the inequality holds because  2 <+ . Hence,  ; v(X) :  ! 0+   0
8X 2 N (A): D+ v(X) = lim sup vX( X)]  The condition 2) is also necessary. Suciency. Let " 2 <+ be arbitrary,  2 <+ be such that N (
A) N (A) and

' 2 K 0 ] obey v(X) '(X
A)], 8X 2 N (
A) (Lemma 4.7). Let  = '(), = (") 2 <+ and 2 ]0
 obey N (
A) V (A) N ("
A). Now Lemma 4.19, all the conditions of which are satised, guarantees X(t X0 ) 2 V (A) ) X(t X0) 2 N ("
A)
8t 2 <+
8X0 2 N (
A): This proves the stability of A (Denition 1.3, Section 1.1.1).

4.5.2 Stability of X = 0

Let A = O = f0g in Theorem 4.5 that takes the following form.

Theorem 4.6 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), O  X X X
0

0

0

0

0

and 2) D+ v(X)  0 for all X 2 N .

rrr

4.5.3 Comment

Theorem 4.5 and Theorem 4.6 present necessary and su cient stability conditions in terms of the existence of a function v with specied properties. However, they do not provide an algorithm to construct the function v for a given system (4.1).

4.6 Asymptotic stability theorems We shall synthesise results by Lyapunov 127], Malkin 130], Halanay 95], Yoshizawa 217], Bhatia and Szeg(o 10] and Hahn 94] on asymptotic stability via the LaSalle principle.

© 2004 by Chapman & Hall/CRC

4.6.1 Asymptotic stability of a set

Theorem 4.7 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X
0

0

0

0

0

+

and 3) D+ v(X) = 0 for all X 2 A.

rrr Proof. Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X
0

0

0

0

0

Necessity. Let A be asymptotically stable, hence stable and attractive (Deni-

tion 1.6, Section 1.1.3). Let " 2 <+ be arbitrary and = (" A) 2 <+ obey Denition 1.3, Section 1.1.1. Let $ > 0 and (X0
 A) = 2 <+ obey Denition 1.5, Section 1.1.2. Now let  = min (
$) 2 <+ ,  2 ]1
+1 be arbitrary and xed, X 2 N (
A) ; A] be arbitrary, = 'I f;2'(X
A)]g 2 <+ for ' 2 K 0"]. v(X) = sup ('fX(t +  t
X)
A]g 11++  :  2 <+ ): The right hand side of this equation is independent of t because X(t +  t
X)  X( 0
X)  X( X) due to time-invariance of the system (4.1). This claries that v does not depend on time t. Notice that () = 11++  2 C(<+ )
sup () :  2 <+ ] = : These facts and X(t X0 ) 2 Ct<+ N (
A)] \ C0 <+ N (
A)] due to N (
A)

N (
A) and Denition 1.3, Section 1.1.1, prove (i) v(X) 2 CN (
A)]. Stability of A implies X(t X0 )
A] = 0 for all t 2 <+ and each X0 2 A (Theorem 3.31, Section 3.6). Hence, (ii) v(X) = 0 for all X 2 A. The denition of v implies directly (iii) v(X) '(X
A)] for all X 2 N (
A). Let N (A) = N (
A). Therefore, N (A) is a connected neighbourhood of A. Altogether, (i){(iii) show that v is positively denite with respect to A on N (A), © 2004 by Chapman & Hall/CRC

which proves necessity of 1). From the denitions of = (X
 A) (Denition 1.5, Section 1.1.2) it follows X(t X)
A] < for all t 2 ]
+1
which together with the denition of yield 'fX(t X)
A]g < '( ) = '('I f;2'(X
A)]g) = ;2 '(X
A)] for all t 2 ]
+1. Hence, ;2 'fX( X)
A]g 11++  < f '(X
A)]g < '(X
A)] for all t 2 ]
+1. This result shows that there is ^ 2 0
] such that  v(X) = 'fX(t + ^  t
X)
A]g 11++^ ^ : Let  2 <+ be arbitrarily small and X  = X(t +  t
X)  X( 0
X)  X( X). Then, by using Theorem 3.14 (Section 3.2.6), we derive v(X  ) = sup ('fX(t +  +  t + 
X  )
A]g 11++  :  2 <+ ) = = sup ('fX(t +  +  t
X)
A]g 11++  :  2 <+ ) = = sup ('fX( +  X)
A]g 11++  :  2 <+ ) = : = 'fX( + ^  X)
A]g 11++^ ^ Notice that (see Halanay 95]) for  =  + ^ 2 
+ ]  <+   1 + ^ = 1 +  1 ; ( ; 1) 1 + ^ 1+ (1 + ^ )(1 + ) > 0: Therefore, for  =  + ^ , v(X  ) = vX( X)] =





( ; 1) = 'fX(t +  t
X)
A]g 11++ 1 ;  (1 + ^ )(1 + )   sup('fX(t +  t
X)
A]g 11++  :  2 <+ )  ; 1)  = 1 ; (1 +(^ )(1 + )

 ; 1)  : = v(X) 1 ; (1 +(^ )(1 + ) © 2004 by Chapman & Hall/CRC

Now, we set v(X) on the left hand side of the preceding inequality, divide the new one by  2 <+ and take lim sup ( ) of both sides so as to get the following, by using ^ =  ;  and  2 
+ ] ! 0+
] as  ! 0+ ,  ; v(X) :  ! 0+   D+ v(X) = lim sup vX( X)]    ;1 +  lim sup ; (1 +  ; )(1 v(X) :  ! 0  + ) 1)v(X)
8X 2 N (A):  ; (1(+ ;)(1 + )

Positive deniteness of v with respect to A on N (A),  2 ]1
+1 and 2 <+ linked with the preceding result prove D+ v(X) < 0 for all X 2 N (A) ; A]. Hence, 2) is necessary. Let X 2 A. Then X(t X) 2 A for all t 2 <+ that yields vX(t X)] = 0 for all t 2 <+ and

 ; v(X) :  ! 0  =lim sup  0 ; 0 :  ! 0  =0 D v(X) =lim sup vX( X)]   +

+

+

for all X 2 A, which veries necessity of 3). Suciency. Let the conditions of the theorem hold. Then all the conditions of

Theorem 4.5 are satised implying stability of the set A. Let  2 <+ and  2 <+ be dened as in Lemma 4.19, which is possible due to positive deniteness of v with respect to A on N (A) (Lemma 4.5). Now all the conditions of Lemma 4.19 are satised so that V (A) is positively invariant and positively precompact relative to itself for any 2 ]0
. Let $ 2 <+ be such that N ($
A) V (A) for arbitrary

2]0
. Then, X0 2 N ($
A) implies X(t X0 ) 2 V (A) for all t 2 <+ . From the conditions 2) and 3) follows E0 = fX : X 2 V (A)
D+ v(X) = 0g = A so that M A and M0 = M \ V (A) A. With these facts and S = V (A) we conclude that the conditions 1){6) of Lemma 4.17 are satised (M0 is compact because it is closed by the denition and subset of compact set A). Hence, M0 is attractive and V (A) Da (M0 ), which proves attraction of A due to M0 A and V (A) Da (A). A is stable and attractive. Therefore, it is asymptotically stable. Notice that M0 = M0 A = A and v(X) = 0 on A guarantee v(X) = const: on @ M0 so that all the conditions 1){6) of Lemma 4.17 are fullled, which proves immediately (in another way) asymptotic stability of A.

Note 4.12 The classical form of Theorem 4.7, when it is not proved via the LaSalle

principle diers from the form given here for the condition 2). The classical form of condition 2) reads as follows: 2 0 ) there is a positive denite function  with respect to A on N (A) such that D+ v(X)  ;(X) for all X 2 N (A):

© 2004 by Chapman & Hall/CRC

Necessity of this condition is proved essentially in the proof of necessity of the conditions of Theorem 4.7 because  dened by 1)v(X) (X) = (1(+ ;)(1 ; ) is positive denite on N (A) and obeys the above condition 20 ). For su ciency we have just to conclude that the above condition 20) guarantees validity of the condition 2) of Theorem 4.7. Altogether, Theorem 4.7 is valid nevertheless we use the condition 2) or 20 ).

4.6.2 Complete global asymptotic stability of sets

Barbashin and Krasovskii 6] extended the asymptotic stability conditions to those for global asymptotic stability by requiring radial unboundedness of the function v. This will be rst stated in the framework of sets in an extended form related to complete global asymptotic stability. Theorem 4.8 Let the system (4.1) have generalised motions X(t X0 ) 2 C0(I0 
v is globally positive denite with respect to A, v is radially unbounded with respect to A, D+ v(X) < 0 for all X 2 
D v(X) = 0 for all X 2 A. rrr Proof. Let the system (4.1) have generalised motions X(t X ) 2 C (I 
0

0

0

of the necessity part of Theorem 4.7, which is then to be repeated so that necessity of the conditions 1), 3) and 4) immediately results. From (iii) of the proof of the necessity part of Theorem 4.7 follows now also necessity of the condition 2) due to ' 2 K<.

Suciency. Let all the conditions of Theorem 4.8 hold. Hence, all the conditions of

Theorem 4.7 are valid for N (A) =
8X 2
vf (X) '(kX k) for all X 2
a) D+ vf (X) < 0 for all (X 6= 0) 2
Let

H = H T 2 <(n+m)
(n+m)
- = diag f1 2 : : : m g 2 <m
m E (L) = (I ; DL);1 C
E : L ! <m
n
F (L) = A + BLE (L)
F : L !
rrr

  R (L) = I E T (L) L H I E T (L) L T + E T (L) L-E (L)
R : L !
© 2004 by Chapman & Hall/CRC

Q (L
M) = (I ; DM);1 CF (L)
Q : L  M ! <m
n


  T (L
M) = F T (L) QT (L
M)M H I E T (L) L T + E T (L) L-Q (L
M) +   + I E T (L) L H F T (L) QT (L
M) M T + QT (L
M) -LE (L) :

Theorem 4.13 Let Assumption 4.1 hold. For the functional family F , (4.7) F = 1



vf

; : vf (X) = X T f T

1

 Zw ; T T T T H X f + 2 f (w) -dw
f 2 N (L
M)
1

0

(4.7)

to be a Lyapunov functional family for the system (4.3) on N1 (L
M) it is both necessary and sucient that 1) the matrix R(L) is positive denite for every L 2 L, and 2) the matrix T(L
M) is negative denite for every (L
M) 2 L  M.

rrr

Proof. Let f 2 N1 (L
M) be arbitrary. Notice that f(w) = N(w)w, w = E(N)X for N = N(w), f (1) (w) = fw (w)w(1), w(1) = CX (1) +Df (1) = CF(N)X +Dfw w(1) so that w(1) = Q(N
fw )X. Furthermore, for every X 2
;

;

T (L) = F T (L) H + C T L-C + H + C T L-C F(L)
H 2
Theorem 4.14 For the functional family F (4.9), F = 0



vf

: vf (X) = X T HX + 2

Zw 0

0

f T (w)-dw




f 2 N0 (L)


(4.8) (4.9)

to be a Lyapunov functional family on N0 (L) for the system (4.3) in case D = 0 it is both necessary and sucient that 1) the matrix (H + C T L-C) is positive denite for every L 2 L, and 2) the matrix T(L) (4.8) is negative denite for every L 2 L.

rrr

Note 4.14 Theorems 4.13 and 4.14 present linear systems stability like conditions

for absolute stability of Lurie nonlinear systems. Various other purely algebraic criteria for absolute stability of Lurie systems were established in 68], 69]. One of them will be presented. It does not require matrix deniteness tests for all L 2 L. However, the whole set of its conditions is only su cient.

© 2004 by Chapman & Hall/CRC

Theorem 4.15 In order for the functional family F0 (4.9) to be a Lyapunov functional family on N0 (0
K]) for the system (4.3) in the case D = 0, it is necessary that the conditions 1), 2a) and 2b) hold and is sucient that all the conditions 1) through 2d) are valid: 1) Real parts Re i (A) of all eigenvalues i (A) of the matrix A are negative,

Re i (A) < 0 for every i = 1
2
: : :
n
2) The matrices H and - are such that a) b) c) d)

H is positive denite, ;H + C T K-C is positive denite, ;2K ;1 ; -CB ; BT C T - is positive denite, h T ; ; ; A H +HA+ AT C T -+C T +HB 2K ;1; -CB ; B T C T - ;1 AT C T i + C T + HB T is negative denite.

rrr

Proof. Let D = 0 in (4.3) and L = 0
K]. Necessity. f(w)  0 is in N0 (L), which reduces (4.3a) to X_ = AX implying the

necessity of 1), and reduces vf to vf (X) = X T HX that yields necessity of 2a) due to 2) of Denition 4.16 by accepting F0 (4.9) to be Lyapunov functional family on N0 (L) for the system (4.3). For the same reasons f(w) = Kw 2 N0 (L) implies the condition 2b) by noting that then vf (X) = X T (H + C T K-C)X. Suciency. Notice that vf obeys

min X T HX
X T (H + C T K-C)X]  vf (X) 

 max X T HX
X T (H + C T K-C)X] for every X 2
(4.10a)

P = AT C T - + C T
P 2
(4.10b)

Y T Y = ;
Y 2 <m
m
XY = HB + P
X 2
(4.10c) (4.10d) (4.10e)

The condition 2c) implies det ; 6= 0 so that det Y 6= 0 that enables solving the equation (4.10d) for X, X = (HB+P)Y ;1 . The last equation and (4.10e) yield N = AT H+HA+(HB+P )(Y T Y );1 (HB+P)T = AT H+HA+(HB+P );;1 (HB+P)T due to (4.10c). This result, (4.10a) and (4.10b) imply

; ; N = AT H + HA + AT C T - + C T + HB 2K ;1 ; -CB ; B T C T - ;1

;AT C T - + C T + HB T :

The condition 2d) shows that N is negative denite. Now, we determine v_ f (X) = X_ T HX + X T H X_ + f T -w_ + w_ T -f along motions of the system (4.3) with D = 0, and add on and subtract from the right hand side the term  = 2f T (w ; K ;1 f) by observing that  0 for all w 2 <m due to f 2 N0 (L) and L = 0
K]. After carrying out all calculations and using the equations (4.10) we get v_ f (x) = X T NX ; kX T X ; Y f(w)k2 ;   X T NX
8X 2
Matrix Popov criterion

The method by Popov 168] is based on the use of the frequency matrix function F of the linear part of the Lurie system that will be considered for the case D = 0, that is, dX = Ax + Bf(w)
(4.11a) dt w = CX: (4.11b) The matrix transfer function G of the linear part of the system with input f and output (;w), G(s) = C(A ; sI);1 B
yields the matrix frequency p function F of the linear part of the system, F (j!) = G(s)js=j! , ! 2 <, j = ;1, that is F (j!) = C(A ; j!I);1 B: Its hermitian part He F(j!) is now determined by  He F (j!) = 21 F (j!) + F T (;j!) : The following lemma is known as the Yakubovich{Kalman lemma 168]:

© 2004 by Chapman & Hall/CRC

Lemma 4.20 Let 1) the matrix A have all eigenvalues with negative real parts, that is Re i (A) < 0 for every i = 1
2
: : :
n
2) the pair (A
B) be controllable, that is

;

rank B AB A2 B : : : An;1B = n


3) the pair (A
C) be observable, that is

;

rank C T AT C T (AT )2 C T : : : (AT )n;1C T = n


4) the matrices ; and P be dened by (4.10a) and (4.10b), respectively,
m be such that ; is positive denite, 5) - 2 <m + and 6) N be negative denite. For the existence of a unique positive denite matrix H = H T and matrices X and Y satisfying the Lurie matrix equations (4.10c){(4.10e) it is necessary and sucient that the matrix K ;1 + He (I + j2!-)F (j!)] is positive denite for every ! 2 <+ and ! = +1.

rrr

The following theorem represents the matrix Popov criterion.

Theorem 4.16 Let the Lurie system (4.11) have the following properties: 1) the real parts of all eigenvalues of A are negative, that is Re i (A) < 0 for all i = 1
2
: : :
n
2) the pair (A
B) is controllable, that is

;

rank B AB A2 B : : : An;1B = n


3) the pair (A
C) is observable, that is

; rank C T AT C T (AT )2 C T : : : (AT )n;1C T = n


For the existence of a Lyapunov functional family on N0 (0
K]) for the
m yielding positive system (4.11) in the form of the family F0, (4.9), with - 2 <m + denite ;, (4.10a), it is necessary and sucient that K ;1 + He (I + j2!-)F(j!)] is positive denite for every ! 2 <+ and for ! = +1.

© 2004 by Chapman & Hall/CRC

rrr

Proof. Let the system (4.11) have the properties 1){3). Let ; and P be dened by (4.10a) and (4.10b), respectively, - 2 <m
m be such that ;, (4.10a), is positive denite.

+

Necessity. Let there exist Lyapunov functional family on N0 (0
K]) in the form F0, (4.9), for the system (4.11). Let f(w) = 0, f 2 N0(0
K]). Hence, H is positive denite. The equations (4.10) and

 = 2f T (w) w ; K ;1 f(w)] 0 for all w 2
4.7 Exponential stability of X = 0 4.7.1 Krasovskii criterion

Krasovskii 116, pp. 70-74], established the criterion for exponential stability of X = 0 of the system (4.1) satisfying Assumption 4.2. The Krasovskii criterion will be presented as Theorem 4.17 without proof.

Assumption 4.2 There are an open hyperball B


1) f is continuously dierentiable on B : f(X) 2 C (1)(B ), © 2004 by Chapman & Hall/CRC

+

 @fi 

2) the Jacobian matrix function fX = @x :
kf(X)k < for all X 2 B : 

Theorem 4.17 (a) Let Assumption 4.2 hold. For X = 0 of the system (4.1) to be exponentially stable it is necessary and sucient that there exist an open hyperball B
satisfy 1){4): 1) v is continuously dierentiable on B : v(X) 2 C (1) (B ), 2) 1kX k2  v(X)  2 kX k2 for every X 2 B , 3) v_ (X)  ;3 kX k2 for every X 2 B , 4) kgrad v(X)k  4kX k for every X 2 B . (b) Let Assumption 4.2 hold for $ = +1, that is B =
rrr

4.7.2 Yoshizawa criterion

Yoshizawa 217, Theorem 11.6, p. 47 Theorem 19.2, p. 93] generalised the Krasovskii criterion to the systems (4.1) satisfying Assumption 4.3 The function f, (4.1), has the following properties: 1) f(0) = 0, 2) there is an open hyperball B
© 2004 by Chapman & Hall/CRC

4.8 Stability domain estimates Through this section we consider the system (4.1). For various approaches to and results on stability domain estimates see 25], 30], 49], 51], 53], 54], 68], 70], 72], 89], 90], 98], 106], 107], 113], 115], 119], 120], 124], 128], 151], 158], 159], 180], 181], 184]{187], 197], 204], 210]{216].

4.8.1 De nitions of stability domain estimates

If the necessary and su cient conditions for a set to be the stability domain of another set, or of X = 0, cannot be eectively applied, then we are looking for as large as possible estimates of stability domains in the sense of the following denition:

Denition 4.17 (a) A set S , S
1) S is a connected neighbourhood of the set A, 2) S is a subset of Dsc(A): S Dsc (A). (b) Esc(A) will be denoted by Esc: Esc (A) = Esc if and only if A = O = f0g.

Notice that we allow any motion X starting from Esc (A) at t = 0 to leave Esc(A), X(t X0) 2 Esc (A) at some t 2 <+ although X0 2 Esc(A), rather than to demand positive invariance of Esc (A) by denition because it is not crucial. The crucial property is that Esc (A) Dsc (A), i.e. that for every X0 2 Esc (A) there is " 2 <+ such that X(t X0 )
Esc(A)] < " for all t 2 <+ .

4.8.2 Estimates of the stability domain of a set

Theorem 4.19 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X 0,  2 < and a function v : N (
A) ! < satisfying 1){5), 1) S is an open connected neighbourhood of A and subset of N (
A): A  S

N (
A), 2) v is positive denite with respect to A on N (
A), 3) V (A) = S , 4) V (A) is bounded for every 2 ]0
], 0

0

0

0

+

and

© 2004 by Chapman & Hall/CRC

+

0

0

5) D+ v(X)  0 for all X 2 N (
A).

rrr

Proof. Let all the conditions of the theorem be valid. Then the set A is stable (Theorem 4.5). Let " 2 < be arbitrary,  2 < obey V (A) N ("
A) \ S ]. The conditions 2){5) of the theorem guarantee positive invariance of V (A) (Lemma 4.18) so that for any X 2 V (A), X(t X ) 2 V (A) for all t 2 < , hence X(t X ) 2 N ("
A) for all t 2 < . This implies V (A) Dsc (" A) and V (A)  Dsc (A). Hence, S \ Dsc (A) =6 . Let now X 2 S be arbitrary. Let 2 v(X )
]. The denition of V (A) and the conditions 2){4) guarantee V1 (A)  V2 (A) and @ V1 (A) \ @ V2 (A) = , 0 < <   so that X 2 V (A). Hence, V (A) is also positively invariant. Let " 2 < be such that V (A) N ("
A). Such " exists because V (A) is bounded (condition 4). Altogether, X 2 V (A) implies X(t X ) 2 V (A) for all t 2 < , hence X(t X ) 2 S \ N ("
A)] for all t 2 < . This proves the positive invariance of S and V (A) Dsc ("
A) Dsc (A), 8 2 ]0
], hence S Dsc(A). +

+

0

0

0

+

+

0

1

2

0

0

+

0

0

+

0

+

If we cannot verify the existence of generalised motions of the system (4.1) then we can apply what follows.

Theorem 4.20 Let A 
+

and 4) D+ v(X)  0 for all X 2 S .

rrr

Proof. Let all the conditions of the theorem hold and " 2 < be arbitrary. Let  2 < obey V (A) N ("
A) \ S ]. The condition 2) guarantees boundedness of V (A) for every 2 ]0
,  = sup v(X) : X 2
+

1

2

0

0

+

0

0

0

+

0

+

1

1

© 2004 by Chapman & Hall/CRC

1

neighbourhood with common centre w of unique boundedness, w 2 A, and with a generating function u2 satisfying u2 (X) = 2 on @ Nu (A), i 2 <+ , i = 1
2. Let v :
8 0
X=w > > > < u (X) ;  (X ; w
A) u  u; (X)X] v(X) = > ; ; u (X) > u  u (X)X] ;   u (X)(X ; w)
A] > :
(X 6= w) 2
1

2 2

1

1

1

1

2 2

1

1

2

2

1

In order for the set Nu(A) to be an estimate of the strict domain Dsc(A) of stability of A: Nu (A) = Esc (A) it is sucient that D+ v(X)  0 for all X 2 Nu (A).

rrr

Proof. Under the conditions of the theorem all the conditions of Lemma 4.13 are fullled. The conditions of Lemma 4.13, u (X) =  on @ Nu (A), Nu (A) = V 2 (A) and D v(X)  0 on Nu (A) show that all the conditions of Theorem 4.19 are met for  =  and S = Nu (A). 2

2

+

2

In contrast to Theorems 4.19 and 4.20, Theorem 4.21 denes the function v. Notice that every estimate Esc(A) of Dsc (A) is also an estimate of Ds (A).

4.8.3 Estimates of the stability domain of X = 0

Theorem 4.22 Let the system (4.1) have generalised motions X(t X ) 2 C(I  X ), (X = 0) 2 X X 0,  2 < and a function v : N (
0) ! < satisfying 1){5): 1) S is an open connected neighbourhood of X = 0 and subset of N (
0): O  S N (
0), 2) v is positive denite on N (
0), 3) V = S , 4) V is bounded for every 2 ]0
], 0

0

0

+

and 5) D+ v(X)  0 for all X 2 N (
A).

0

0

+

rrr

Proof. Since O = f0g is connected we may set A = O in Theorem 4.19 that then reduces to Theorem 4.22.

In case we wish to avoid testing the existence of the system's generalised motions, then we can use the following criterion.

© 2004 by Chapman & Hall/CRC

Theorem 4.23 For a set S
+

2) v is globally positive denite and radially increasing, 3) V = S , and 4) D+ v(X)  0 for all X 2 S .

rrr

Proof. Let A = O. Now, Theorem 4.20 becomes Theorem 4.23. Theorems 4.22 and 4.23 leave open the problem of nding functions v obeying their conditions. This can be overcome under the following conditions. Theorem 4.24 Let N0 be O-uniquely bounded neighbourhood of X = 0 with a generating function u satisfying u(X) =  on @ N0 ,  2 <+ . Let v :
rrr

Proof. Lemmae 4.11 and 4.22 imply Theorem 4.24 for A = O.

4.9 Asymptotic stability domain and estimates In what follows the system (4.1) is assumed.

4.9.1 Classical approach

The classical approach to determination of the asymptotic stability domain is based on the Lyapunov method used via the existence of a v function with appropriate properties.

Zubov theorem

The Zubov theorem 218, Theorem 19, pp. 52-53], was most probably the rst result of an exact determination of the asymptotic stability domain Ds of X = 0. It has had a great theoretical signicance and inuence on the study of the Lyapunov function construction problem 3], 38], 46], 101], 102], 131], 196]. For other approaches to this problem see 4], 5], 7], 26], 28], 31], 32], 36], 46], 47], 59], 97], 100]{102], 104], 112], 117], 123], 126], 131], 132], 153]{155], 160], 167], 170], 171], 174]{176], 182], 183], 190], 199]{203]. The Zubov theorem takes the following form in this framework:

© 2004 by Chapman & Hall/CRC

Theorem 4.25 (Zubov theorem) Let the function f of the system (4.1) be continuously dierentiable on a closed hyperball B for some  2 < . In order for an open invariant set S , S B , containing a certain neighbourhood of a closed invariant set M, M  B , to be the domain of asymptotic stability of the asymptotically stable set M, it is necessary and sucient that there exist two functions v +

and  having the following properties:

1) The function v is dened and continuous on S , and the function  is dened and continuous on 0 for X 2 0 it is possible to nd quantities 1 and 1 such that

v(X) < ; 1 for (X
M) 2
and

(X) > 2 for (X
M) 2 : 4) The functions v and  tend to 0 as (X
M) ! 0. 5) If there exists a point y 62 M, y 2 @ S then lim v(X) : (X
y) ! 0] = ;1. 6) dv(X) = (X) 1 + v(X)].

rrr dt The Zubov theorem is valid for smooth systems described by continuously dierentiable function f in (4.1). The theorem assumes that the set M is asymptotically stable. Furthermore, the theorem is expressed in terms of the existence of the functions v and . It does not determine any guideline for a choice of either. The Zubov theorem is directly applicable to the state X = 0. It su ces to set M = f0g. For the proof of the Zubov theorem see 218, p. 53].

Knobloch{Kappel theorem

Knobloch and Kappel derived several interesting results in 114] for a special class of systems described by (4.1). They are those which obey the following assumption:

Assumption 4.4 The function f, (4.1), has the following properties: 1) f(0) = 0, 2) f is dened with continuous partial derivatives on
© 2004 by Chapman & Hall/CRC

Theorem 4.26 (Knobloch{Kappel theorem) Let Assumption 4.4 hold. Let

: 0 for all X = 6 0, 3) the function has a positive lower limit on every subset of the set fX : kX k "g, " > 0. Then, there is a unique function v that is continuously dierentiable on D: v(X) 2 C (D) and obeys grad v(X)]T f(X) = ; (X) for all X 2 D and v(0) = 0: (2)

(1)

Besides, v satises the following conditions: (a) v(X) > 0 for X 6= 0, and (b) lim v(X) : X ! @ D
X 2 D] = 1.

rrr

The proof is given in 114, the proof of Theorem 10.3, p. 171]. Notice that the conditions of Assumption 4.4 assume knowledge that X = 0 is asymptotically stable. Notice also that the choice of a function obeying 1) through 3) of Theorem 4.26 is arbitrary, that it is not stated in terms of the existence of v and assumes knowledge of D.

Vanelli{Vidyasagar results

Vanelli and Vidyasagar established a series of very interesting results on generation of a system Lyapunov function 198]. The main results will be presented without proofs that are given in 198]. The Vanelli{Vidyasagar approach and all their results in 198] are based on the following assumption.

Assumption 4.5 (a) The system (4.1) has a unique solution X(t X ) through each initial state X 2
0

(b) The solution X(t X0) depends continuously on X0 . (c) The state X = 0 is asymptotically stable equilibrium state of the system. The condition (c) of Assumption 4.5 shows that X = 0 has the domain D of asymptotic stability and the domain Da of attraction so that (Theorem 3.35, Section 3.7) Da = D = Dac = Dc . © 2004 by Chapman & Hall/CRC

Theorem 4.27 Let Assumption 4.5 hold. If a set S
are such that: (i) v(0) = 0, v(X) > 0 for all X 2 (S ; O): v is positive denite on S ,

 v(X(t X)) ; v(X)



: t ! 0 is well dened at all X 2 S , (ii) Dr v(X) = lim t and satises Dr v(X) = ;(X) for all X 2 S , and (iii) v(X) ! +1 as X ! @ S and/or as kX k ! +1, then S = D. rrr +

This theorem provides su cient conditions for a set S to be the asymptotic stability domain D. Under an additional requirement on the system, the conditions are also necessary: Theorem 4.28 Let Assumption 4.5 hold and f be Lipschitz continuous on D.

Then, in order for an open set S containing O to be the domain of asymptotic stability of X = 0, it is necessary and sucient that there exist a continuous function v : S ! <+ and a positive denite function  such that (i){(iii) of Theorem 4.27 are satised. rrr

Notice that this theorem requires knowledge of the Lipschitzian property of the function f on D, which essentially requests knowledge of asymptotic stability domain itself. Both Theorem 4.27 and Theorem 4.28 state the conditions in the form of the existence of a function v with the described properties. The same conclusion is valid for the next theorem that is based on the following important lemma: Lemma 4.21 Let Assumption 4.5 hold. Suppose v is a continuous function on some hyperball B such that v(0) = 0, and Dr v is negative denite. Then v is

rrr

positive denite.

Theorem 4.29 Let Assumption 4.5 hold. Suppose f is Lipschitz continuous on D, and suppose v is a continuous function on some closed hyperball B such that v(0) = 0 and Dr v is negative denite on B  . Then there exists a (maximal Lyapunov) function vm that agrees with v on B  and obeys (i){(v): (i) vm : 0 for all X 2 (D ; O), (iii) vm (X) < +1 i X 2 D, (iv) vm (X) ! +1 as X ! @ D and/or kX k ! +1, (v) Dr v is well dened and negative denite over D. rrr +

Vanelli and Vidyasagar used 198] the preceding results by themselves to develop a numerical computer aided procedure to generate a system Lyapunov function. It is explained in Section 4.9.4.

© 2004 by Chapman & Hall/CRC

4.9.2 De nition of asymptotic stability domain estimate

The classical results do not provide a complete solution to the problems of an exact generation of a system Lyapunov function v and of an exact determination of the asymptotic stability domain. Therefore we are often satised with the best possible estimate.

Denition 4.18 (a) A set S , S
1) S is a connected neighbourhood of the set A, 2) S is a subset of Dac (A), S Dac (A). (b) Eac (A) will be denoted by Eac , Eac(A) = Eac if and only if A = O.

Denition 4.19 (a) A set S , S
1) S is a connected neighbourhood of the set A, 2) S is a subset of Dc (A), S Dc (A). (b) Ec (A) will be denoted by Ec , Ec(A) = Ec if and only if A = O.

Note 4.15 Ec (A) = Esc(A) \ Eac (A) (Denitions 4.17{4.19).

4.9.3 Estimates of the asymptotic stability domain of a set

Theorem 4.30 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X 0,  2 < and a function v : N (
A) ! < satisfying 1){6), 1) S is an open connected neighbourhood of A and subset of N (
A): A  S = S N (
A), 2) v is positive denite with respect to A on N (
A), 3) V (A) = S , 4) V (A) is bounded for every 2 ]0
], 5) D v(X)  0 for all X 2 N (
A), 0

0

0

0

0

0

+

+

+

and 6) A is the largest invariant set in E0 = fX : X 2 N (
A)
D+ v(X) = 0g.

rrr

© 2004 by Chapman & Hall/CRC

Proof. Let all the conditions 1){6) hold. Then all the conditions of Theorem 4.18 are satised, which proves that S is an estimate of Dsc (A). From the proof of Theorem 4.19 it follows that S is positively invariant. The same was proved to hold for V (A) for any 2 ]0
] (Lemma 4.18). Hence, S is positively precompact relative to itself. This, the conditions 1), 2), 5), and 6), together with the existence of generalised motions prove X(t X ) ! A as t ! +1 for every X 2 S . Since S is positively invariant we may now conclude that S Da (A). Altogether, S Dac (A) \ Dsc(A) = Dc (A) because Da (A) = Dac (A) (Theorem 4.35, 0

0

Section 3.7).

Verication of the condition 6) can be di cult. To avoid it we state the following result. Theorem 4.31 Let the system (4.1) have generalised motions X(t X0 ) 2 C0(I0  X0), X0 X 0,  2 <+ and a function v : N (
A) ! <+ satisfying 1){6), 1) S is an open connected neighbourhood of A and subset of N (
A): A  S = S N (
A), 2) v is positive denite with respect to A on N (
A), 3) V (A) = S , 4) V (A) is bounded for every 2 ]0
], 5) D+ v(X) = 0 for all X 2 A, and 6) D+ v(X) < 0 for all (X 62 A) 2 N (
A).

rrr

Proof. The conditions 5) and 6) show that E = fX : X 2 N (
A)
D v(X) = 0g = A. This shows that A can be the largest invariant set in E , which together with conditions 1){5) imply S Dc(A) (Theorem 4.30). +

0

0

If we prefer to avoid verication of the existence of system generalised motions then we can use any of the following results.

Theorem 4.32 Let A 
© 2004 by Chapman & Hall/CRC

+

3) 4) and 5)

V (A) = S , D v(X)  0 for all X 2 S , +

A is the largest invariant set in E = fX : X 2 S ,D v(X) = 0g. +

0

rrr

Proof. Let all the conditions be valid. Then S is an estimate of the strict stability domain of A, S = Esc(A), due to the conditions 1){4) (Theorem 4.20). Besides, S is positively invariant and positively precompact relative to itself (the proof of Theorem 4.20), which together with the conditions 1), 2), 4), and 5) proves that X(t X ) ! A as t ! +1 for every X 2 S (Lemma 4.18). Hence, S is an estimate of the strict attraction domain of A, S = Eac (A). Altogether, S = Esc (A)\ Eac (A) = Ec (A). 0

0

The test of invariance of A can be omitted by sharpening the requirement on D+ v.

Theorem 4.33 Let A 
+

+

and 5) D+ v(X) < 0 for every (X 62 A) 2 S .

rrr

Proof. Let all the conditions hold. Hence, S = Esc(A) is an estimate of the strict stability domain of A (Theorem 4.20). It is also positively invariant and

precompact relative to itself (the proof of Theorem 4.20), which together with the conditions 1), 2), 4), and 5) prove (Lemma 4.18) that S = Eac (A). Finally, Ec = Esc(A) \ Eac (A) = Ec(A). The preceding results leave open the problem of nding an appropriate function v.

Theorem 4.34 Let a set A 
1

2

+

© 2004 by Chapman & Hall/CRC

1

2

2

+

8 0
X=w > > > < (X ; w
A) u (X) ;  u  u; (X)X] v(X) = > ; ; (X)(X ; w)
A] u (X) u  u (X)X] ;   u > > :
(X 6= w) 2
1

2 2

1

1

1

1

2 2

1

1

2

2

1

In order for the neighbourhood Nu (A) to be an estimate of the strict domain Dc (A) of asymptotic stability of A, Nu (A) = Ec (A), it is sucient that 1) D+ v(X) < 0 for all X 2 Nu(A) ; A], and 2) D+ v(X) = 0 for all X 2 A.

rrr

Proof. Let all the conditions hold. Lemma 4.13 is therefore valid, which shows that all the conditions of Theorem 4.34 are satised for S = Nu (A). Note 4.16 When the system (4.1) has generalised motions X(t X ) 2 C (I  X ) and Dc (A) then D(A) = Dc (A). Evidently Ec (A) is an estimate of both Dc (A) and D(A). 0

0

0

0

4.9.4 Estimates of the asymptotic stability domain of X = 0 General results Theorem 4.35 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X 0 and  2 < and a function v : N (
O) ! < satisfying 1){6), 1) S is an open connected neighbourhood of O and subset of N (
O), O  S = S N (
O), 2) v is positive denite on N (
O), 3) V = S , 4) V is bounded for every 2 ]0
], 5) D v(X)  0 for all X 2 N (
O), 0

0

0

0

0

0

+

+

+

and 6) O is the largest invariant set in E0 = fX : X 2 N (
O)
D+ v(X) = 0g.

rrr

© 2004 by Chapman & Hall/CRC

Proof. Let A = O. Theorem 4.30 becomes Theorem 4.35. If we wish to avoid verication of the condition 6) then we can use the following result.

Theorem 4.36 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X 0,  2 < and a function v : N (
O) ! < satisfying 1){6), 1) S is an open connected neighbourhood of O and subset of N (
O), O  S = S N (
O), 2) v is positive denite on N (
O), 3) V = S , 4) V is bounded for every 2 ]0
], 5) D v(X) = 0 for X = 0, 0

0

0

0

0

0

+

+

+

and 6) D+ v(X) < 0 for all (X 6= 0) 2 N (
O).

rrr

Proof. For A = O Theorem 4.31 takes the form of Theorem 4.36. The condition for the existence and continuity in X0 2 X0 of system generalised motions can be omitted if v is also radially increasing.

Theorem 4.37 For a set S
+

2) v is globally positive denite and radially increasing, 3) V = S , 4) D+ v(X)  0 for all X 2 S , and 5) O is the largest invariant set in E0 = fX : X 2
rrr

Proof. Theorem 4.37 follows from Theorem 4.32 for A = O. The test of condition 5) can be avoided as explained by the following result.

© 2004 by Chapman & Hall/CRC

Theorem 4.38 For a set S
+

2) v is globally positive denite and radially increasing, 3) V = S , 4) D+ v(X) = 0 for X = 0, and 5) D+ v(X) < 0 for every (X 6= 0) 2 S .

rrr

Proof. Theorem 4.33 reduces to Theorem 4.38 in the case A = O. The next Theorem relies on properties of O-uniquely bounded sets, which determine the form of v.

Theorem 4.39 Let N0 be an O-uniquely bounded neighbourhood of X = 0 with a generating function u satisfying u(X) =  on @ N0 ,  2 <+ . Let v :
rrr

Proof. Lemma 4.11 and Theorem 4.38 imply directly Theorem 4.39. Constructive computer Lyapunov function generation

In a series of papers 148], 149], 151] Michel et al. developed an algorithmfor a computer construction of a system Lyapunov function and estimation of the asymptotic stability domain D for systems of the form (4.1) with f(X) 2 C (1)(
and Tong 22], 23] is used by accepting an initial polyhedral neighbourhood W0 of the origin of
h

i

Wk = H  (Mkj; Wk ) : j = 0
1
: : :
1
+1

1

where HS ] is the convex hull of the set S . A possible choice of W0 is such that the set E (W0) of all extreme points of the set W0 obeys

E (W ) = fwi : wi = (xi xi : : : xin)T 2
1

1

2

1

0

such that 23]:

h

i

h

i

H  (Mkj; Wk ) : j = 0
1
: : :
1 = H  (Mkj; Wk ) : j = 0
1
: : :
r : 1

1

The number r is the smallest natural number  obeying

h

i

h

i

Mk;1H  (Mkj;1Wk ) : j = 0
1
: : :
 H  (Mkj;1Wk ) : j = 0
1
: : :
 : The set W  determines k k as kX k = inf f : 0
X 2 W  g, which is accepted for a system Lyapunov function, v(X) = kX k : Hence, @ W  = fX : X 2 1 then X is out of W  . Let 2 <+ . Then, for arbitrarily large ,

D v(X) < 0 for all (X 6= 0) 2 W  +

along motions of the linearised system X_ = AX. We now determine the largest value denoted by M such that

D v(X) < 0 for all (X 6= 0) 2 M W  +

© 2004 by Chapman & Hall/CRC

holds along motions of the system (4.1). Finally we conclude that the function v, v(X) = kX k , is positive denite on M W  and its Dini derivative D+ v(X) taken along motions of the system (4.1) obeys D+ v(X) < 0 for all (X 6= 0) 2 M W  . We may conclude that the set M W  is an estimate of the asymptotic stability domain of the system (4.1) with f(X) 2 C (1)(
Vanelli{Vidyasagar approach

What follows further exposes the approach by Vanelli and Vidyasagar, which is based on their results presented in Section 4.9.1 198]. It assumes validity of Assumption 4.5 as well as the following:

Assumption 4.6 The function f(4:1) is analytic and can be expressed in the form f(X) =

1 X i=1

fi (X)


where fi is a homogenous function of degree i.

Example 4.17 (a) f (X) = AX represents a homogenous function of degree i = 1. (b) f (X) = XX T 1 represents a homogenous function of the second degree, i = 2. 1

2

(c) f3 (X) = X T XX represents a homogenous function of degree i = 3.

A tentative Lyapunov function v for the system (4.1) is accepted in the form of a real rational function with n(X) and d(X) denoting its numerator and denominator polynomial, respectively, v(X) = n(X) (4.13) d(X)
where n and d are assumed in the next form that enables us to use a recursive procedure, n(X) =

1 X i=2

d(X) = 1 +

ri (X)
so that n_ (X) = 1 X i=1

"X 1 i=2

_ = qi(X)
so that d(X)

#T

_ grad ri(X) X


"X 1 i=2

#T

(4.14a)

_ (4.14b) grad qi(X) X:

The functions ri and qi are homogenous of degree i. From (4.1), (4.13) and (4.14) we get

8" 1 # T "X #"X #T 9 1 1 1 < X #"X = ri (X) grad qi(X)  f(X): grad ri(X) ; v_ (X)= : 1 + qi(X) i i i i =1

=2

© 2004 by Chapman & Hall/CRC

=2

=1

(4.15) Let G = GT 2
kX ;2 X k;1 

i=2

i=1 j =2

(grad ri )T fk+1;i +

= ;X T GX



qi(grad rj )T ; rj (grad qi)T fk+1;i;j =

"

2qk;2 +

kX ;3 i=1

!

qi qk;2;i
k 3:

(4.17b)

Theorem 4.40 (Vanelli{Vidyasagar theorem) Let Assumptions 4.5 and 4.6 hold, and X = 0 of the linear system (4.18),

X_ = f1 (X) = AX (4.18) be asymptotically stable. Let G = GT 2
"

z(X) = 1 +

mX ;2 i=1

#

2

q1(X) = d(X)]2:

Now, (4.19) and (4.20) yield along solutions of (4.1),

2

0

(4.20)

1

3

" mX m m mX ;2 ; 2 !X X 1 (grad rj )T ; @ rj A (grad qi)T 5 f: v_ m (X)= z 4 1 + qi i=1 j =2 j =2 i=1

1 The condition \ 1+ ( )+ 1 in 198]. q

X

:::

© 2004 by Chapman & Hall/CRC

+ m ( )] = 0 on some neighbourhood of q

X

6

X

(4.21)

= 0" is not stated

Expanding the right-hand side of (4.21) and using Assumption 4.6, we have

8

< v_m (X) = 1z :(grad r2)T + : : : + (grad rm )T + +

mX ;2 X m  i=1 j =2

91 =X qi(grad rj )T ; rj (grad qi)T  fk = k =1

8 1 1< X = z :(grad r )T + : : : + (grad rm )T + k 9 mX ; X m  = + qi(grad rj )T ; rj (grad qi)T  fk = i j 8 2k " m < X X = 1 :(grad r )T f + (grad r )T fk; + 4 (grad rj )T + 2

=1

2

=1

=2

z

+

2

1

2

k=3

1

!

j; X 2

j =3

i=1

(qi(grad rj ;1)T ; rj ;1(grad qi)T ) fk;j +1

8

2 "

39 = 5 +

1 m <X X fk;1(grad r2 )T + 4 (grad rj )T + + z1 : j =3 k=m+1

+

9 !3 = (qi(grad rj ; )T ; rj ; (grad qi)T ) 5 fk;j  :

j; X 2

i=1

1

1

(4.22)

+1

Substituting the right hand sides of equations (4.17) into (4.22) we obtain " "X !# m m kX ;3 X 1 T T v_ m (X) = z ;X GX ; X GX 2qk;2 + qi qk;2;i + k=3 k=3 i=1 + 1z (terms of degree n + 1): (4.23) The numerator of the rst term is  ;X T GX 1 + 2(q1 + q2 + : : : + qm;2 ) + q12 + 2q1q2 + q1qm;3 +



+q2 qm;4 + : : : + qm;3 q1 : (4.24) The terms in the parentheses are the terms and cross terms of z of degree up to and including m ; 2. We add to and subtract from it the term z: 1+2

m X

k=3

qk;2 +

m kX ; X

3

k=3 i=1

"

qiqk;2;i = z ; z ; 1 ; 2

© 2004 by Chapman & Hall/CRC

m X

k=3

qk;2 +

m kX ; X

3

k=3 i=1

!

qiqk;2;i :

(4.25) The terms in the brackets on the right hand side of (4.25) are of a degree not less than m ; 1. We use (4.25) to rewrite (4.23) in view of (4.24) as follows: v_ m (X) = ;X T GX + z1 (terms of degree m ; 1): (4.26) From (4.26), we see that v_m is negative denite over a small neighbourhood of the origin. Lemma 4.21 now implies positive deniteness of vm due to the condition that 1 + q1(X) + : : : + qm (X)] 6= 0 on some neighbourhood of X = 0.

Vanelli{Vidyasagar recursive algorithm for generating estimates of Dc 198] Step 1. Solve the Lyapunov matrix equation (4.27),

AT H + HA = ;G
(4.27) for arbitrarily chosen positive denite G = GT . Then, v2 (X)  r2(X) = X T GX
(m = 2): Step 2. Find the linear system representation form for m 3 of (4.17b) by taking unknown coe cients as elements of an unknown vector y, that is, set (4.17b) in the equivalent form (4.28), Am y = bm
m 3: (4.28) Step 3. Let em (y) = kyk2 and solve the problem of minimizing em (y) under the constraint (4.28). Let y be the solution. Step 4. Use y to nd coe cients of rm and qm;2 . If em (y ) is su ciently small,

go to the next step otherwise, take m + 1 as the new value for m and repeat Steps 2{4.

Step 5. If em (y )  ", go to Step 6 otherwise, solve the constrained minimization problem: min em (y) : y] under (4.28). Let  be the optimal value of vm . Then, S1 = fX : vm (X) <  g

is an approximation of D, S1 D. Step 6. The set

(

S = X: 2

mX ;2 i=1

)

qi (X) > ;1

is an approximation of Dc , which may not be necessarily contained in Dc (that ^ = 0). happens if there is X^ 2 S2 such that vm (X) © 2004 by Chapman & Hall/CRC

Example 4.18 198] The following system is known as the state{space representation of the Van der Pol equation,

x_ 1 = x2
x_ 2 = x1 ; x2 + x21 x2: Then, for m = 4, e4 = 0:12295
+ r3 (X) + r4 (X) v4 (X) = r2 (X) 1 + q1(X) + q2(X)
r2 (X) = 32 x21 ; x1x2 + x22
r3 = 0
r4 = ;0:3186x41 + 0:7124x31x2 ; 0:1459x21x22 + 0:1409x1x32 ; 0:03769x42
q1 = 0
q2 = ;0:2362x21 + 0:31747x1x2 ; 0:1091x22


 = 5:4413
S1 = fX : v4 (X) <  g  Dc : The strict asymptotic stability domain Dc and its approximation S1 are shown in Fig. 4.1.

Estimate of the domain of asymptotic stability with nite attraction time Denition 4.20 (a) The state X = 0 of the system (4.1) has the domain of asymptotic stability with nite attraction time a = a (X ), which is denoted by D
, if and only if

0

1) it has the asymptotic stability domain D, 2) there is nite time a = a (X0 ) such that X(t X0 ) = 0 for all t 2  a
1 provided only that X0 2 D
, 3) D
is a neighbourhood of X = 0. (b) A set S , S
Note 4.17 From Denitions 2.1, 2.3 and 2.5 of Section 2.2 and Denition 4.20 follows D
D. © 2004 by Chapman & Hall/CRC

x2

0

x1

Figure 4.1: The strict asymptotic stability domain Dc (|) and its approximation S1 (- - -) for the Van der Pol system (Example 4.18) (reprinted from 198] with kind permission from Elsevier Science Ltd, UK).

Theorem 4.41 For a set S
+

2) v is positive denite and radially increasing, 3) V = S , 4) D+ v(X)  ;v(X)] for all X 2 S .

rrr

Proof. Let all the conditions hold. Hence, Theorem 4.38 is satised, which shows S = E { an estimate of D. Integrating D vX(t X )]  ;fvX(t X )]g for any X 2S we get the maximal generalised solution vX(tX )] as vX(tX )] = v(X )] ; ; (1 ; )t for t 2 0
a(X )], and vX(t X )] = 0 for t 2  a (X )
+1, where +

0

0

0

0

0

0

0

0

1


v(X )

0

0 1 ;  2 0
+1 for every X0 2 S . The function v is globally positive denite (hence, positive denite on S ) because it is both positive denite and radially increasing. Positive deniteness of v on S implies now X(t X0) = 0 for all t 2  a (X0 )
+1 and every X0 2 S
which proves S = E
.

a (X0 ) =

© 2004 by Chapman & Hall/CRC

4.10 Exponential stability domain estimate 4.10.1 De nition of exponential stability domain estimate

Denition 4.21 (a) A set S , S
4.10.2 Estimates of the exponential stability domain of a set

Theorem 4.42 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X 0,  2 < ,  2 < and a function v : N (
A) ! < satisfying 1){5), 1) S is an open connected neighbourhood of A and subset of N (
A): A  S

S N (
A), 2) (X
A)  v(X)  (X
A) for all X 2 N (
A), 3) V (A) = S , 4) V (A) is bounded for every  2 ]0
], 5) D v(X)  ; v(X) for all X 2 N (
A). rrr Proof. Let all the conditions hold. All the conditions of Theorem 4.32 are satised. The proof of Theorem 4.32 shows that S is positively invariant. Let X 2 S so that X(t X ) 2 S for all t 2 < . Integrating D v(X)  ; v(X) for X = X(t X ) we get vX(t X )]  v(X ) exp (; t), 8t 2 < . This result combined with the condition 2) yield X(t X )
A]  (X
A) exp (; t) for all t 2 < and arbitrary X 2 S . Hence, S = Ee (A 
). 0

0

0

0

0

0

+

+

+

+

0

0

0

0

0

+

+

0

+

0

+

0

If we wish to avoid a test of the existence and C0 -continuity of system generalised motions, we may use the following result.

© 2004 by Chapman & Hall/CRC

Theorem 4.43 Let A 
+

+

+

+

0

+

0

0

0

0

+

0

4.10.3 Estimates of the exponential stability domain of X = 0

Theorem 4.44 Let the system (4.1) have generalised motions X(t X ) 2 C (I  X ), X X 0,  2 < ,  2 < and a function v : N (
O) ! < satisfying 1){5), 1) S is an open connected neighbourhood of O and subset of N (
O): O  S = S N (
O), 2) kX k  v(X)  kX k for all X 2 N (
A), 3) V = S , 4) V is bounded for every  2 ]0
], 5) D v(X)  ; v(X) for all X 2 N (
O). rrr Proof. Let A = O. Theorem 4.42 takes the form of Theorem 4.44. 0

0

0

0

0

0

+

+

+

+

If we do not require C0-continuity of system generalised motions and/or wish to avoid a test of their existence, then the following theorem is applicable.

Theorem 4.45 For a set S
+

+

+

© 2004 by Chapman & Hall/CRC

4.11 Estimates of the asymptotic stability domains on ( )( ) N

In this section we will consider Lurie systems described by (4.29) (see Section 1.1.5), dX = AX + Bf(w)
dt

(4.29a)

w = CX + Df(w)


(4.29b)

where f 2 Ni(L
M S ), i 2 f0
1g, and S will be determined by wk < 0 and wk > 0, 8k = 1
2
: : :
m, via W = (w1 w2 : : :wm )T , W = (w1 w2 : : :wm )T and S = fX : W  (I ; DL);1 CX]  W
det(I ; DL) 6= 0
8L 2 Lg. If and only if both W = ;11 and W = +11 then S =
4.11.1 De nition of estimate of the asymptotic stability domain on ( )( ) N

Denition 4.22 (a) A set S , S 
(c) Ei (L
M A S ) will be denoted by Ei (L
M S ) if and only if A = O. (d) Ei (L
M S ) will be denoted by Ei if and only if L, M and S are prespecied and xed.

4.11.2 Algebraic approach Two classes of Lurie systems (4.29), will be diered: a) the general class characterized by D 6= 0, and b) the special class specied by D = 0.

© 2004 by Chapman & Hall/CRC

In both cases -, E(L), F(L), R(L) and Q(L
M) will be used as dened in Section 4.6.5. However, T(L
M) will be dened dierently according to the case considered. In both cases K1 = 0 2 <m
m , K2 2 <m+
m , L = 0
K2], M1 2 <m;
m , M2 2 <m+
m and M = M1
M2].

The general case

In this case H = (Hij ) = H T 2 <(n+m)
(n+m) , i
j = 1
2, and

  T(L
M) = F T (L) QT (L
M)M H I E T (L)L T + E T (L)L-Q(L
M) +   + I E T (L)L H F T (L) QT (L
M)M T + QT (L
M)-LE(L):

T 2 Notice that C = (c1 c2 : : :cm )T with ck 2
Theorem 4.46 Let Assumption 4.1 hold, 1) A be nonsingular with negative real parts of all its eigenvalues, h; i ; 2) D ; CA;1B T + D ; CA;1B be negative denite, 3) R(L) be positive denite for every L 2 L, 4) T (L
M) be negative denite for every (L
M) 2 L  M, o n  5) = min min (w2
w2 ) cT R(L)];1c ;1 : L 2 L
i 2 f1
2
: : :
mg , i

i

6) set E1 (4.30),

i

i







E = \ X : X T R(L)X < : L 2 L
1

(4.30)

be a neighbourhood of X = 0. Then the set E is estimate E (L
M S ), E = E (L
M S ), of the domain D(L
M S ) of asymptotic stability of X = 0 of the system (4.29) on N (L
M S ). rrr Proof. Let f 2 N (L
M S ) and all the conditions hold. Then X = 0 is the unique equilibrium state of (4.29) in S on N (L
M S ) (Theorem 4.11). Let F be dened by (4.7) and vf 2 F . For every X 2
1

1

1

1

1

1

1

1

vf (X) = X T R(L)X and v_ f (X) = X T T (L
M)X (proof of Theorem 4.13). The conditions 3) and 4) now prove positive deniteness of vf and negative deniteness of v_ f on S . From condition 5) and (4.30) it follows that

E S: 1

© 2004 by Chapman & Hall/CRC

(4.31)

Besides,

Vf



= X

: vf (X) = (X Tf T )H(X Tf T )T

Zw

+2

0

f T (w)-dw <



S (4.32)

due to the denition of . Let X0 2 E1 be arbitrary. Then X0 2 Vf because E1 Vf in view of (4.30) and the denitions of R(L) and vf . Positive deniteness of vf and negative deniteness of v_ f on S and Vf S (4.32) imply positive invariance of Vf , which together with X0 2 Vf imply both the existence of " 2 <+ such that kX(t X0  f)k < " for all t 2 <+ and limkX(t X0  f)k : t ! +1] = 0. Hence, Vf D(L
M S ) that proves E1 D(L
M S ) due to E1 Vf . This and condition 6) show that E1 is an estimate of D(L
M S ). Let Lv denote the set of all vertices of L = 0
K], K = K2 = diagfk21 k22 : : : k2mg = diagfk1 k2 : : :km g is given, Lv = fL : L = diagf1 2 : : :m g
i 2 f0
kig
i = 1
2
: : :
mg. By following 69] we deduce from Theorem 4.46 the following criterion based on F0 = fvf : vf (X) = v(X) = X T H11X
H11 = H11 2
Theorem 4.47 Let 1) det A 6= 0, 2)

h; i ; D ; CA; B T + D ; CA; B be negative denite, 1

1

T be positive denite, 3) H11 = H11

 4) T(L) = F T (L)H11 + H11F(L) be negative denite for every L 2 Lv for F(L) = A + BL, n ; ;1c ;1 : i 2 f1
2
: : :
mgo, 5) = min min (w2i
w2i ) cTi H11 i

Then the set E0 = fX : X T H11X < g is estimate E0 (L S ), E0 = E0(L S ), of the domain D(L S ) of asymptotic stability of X = 0 of the system (4.29) on N0 (L S ).

rrr

Proof. Let all the conditions hold. Since L is compact and T(L) = F T (L)H + 11

H11F(L)] is linear in L due to linearity of F(L) = (A + BL) in L then the condition 4) guarantees negative deniteness of T (L) for every L 2 L. This fact T = 0, shows that Theorem 4.46 reduces to Theorem 4.47 for - = 0, H12 = H21 H22 = 0, which are accepted in the statement of Theorem 4.47 in view of E0 = fX : X T H11X < g.

© 2004 by Chapman & Hall/CRC

The special class of Lurie systems Let D = 0 in (4.29b) and f 2 N (L S ). Since f may not be dierentiable then we use a Lyapunov functional family in the form (4.9) that yields T(L) in the form (4.8). Notice that H = H T 2
11

preceding section. Since D = 0 then R(L) = H + L-.

Theorem 4.48 Let D = 0, 1) det A = 6 0, 2)

h; ; T ; ; i CA B + CA B be positive denite, 1

1

3) R(L) be positive denite for every L 2 L, 4) T(L) (4.8) be negative denite for every L 2 L, n o  5) = min min (w2i
w2i ) cTi R(L)];1ci ;1 : L 2 L
i 2 f1
2
: : :
mg , 6) set E0 (4.33),







E = \ X : X T R(L)X < : L 2 L
0

(4.33)

be a neighbourhood of X = 0. Then the set E0 is an estimate E0 (L S ), E0 = E0 (L S ), of the domain D(L S ) of asymptotic stability of X = 0 of the system (4.29) on N0 (L S ). rrr

Proof. Let f 2 N (L S ) and all the conditions hold. Then X = 0 is the unique equilibrium state of (4.29) in S on N (L S ) (Theorem 4.11). Let F be dened by (4.9) and vf 2 F . For every X 2
0

0

0

0



Vf = X : vf (X) = X T HX + 2

Zw 0



0

f T (w)-dw <  E0


due to the condition 5) and denitions of vf , R(L) and E0 . Hence, vf X(t X0 f)]  vf (X0 ) for all t 2 <+ and vf X(t X0 )] ! 0 as t ! +1 in view of vf > 0 and v_ f < 0 on N0 (L S ), which prove the positive invariance of Vf and Vf D(L S ). This, E0 Vf and the condition 6) prove that E0 is an estimate of D(L S ). We can avoid deniteness tests for every L 2 L if we use Theorem 4.49.

Theorem 4.49 Let D = 0, - be either non-positive or non-negative diagonal, 1) real parts of all the eigenvalues of A be negative, 2) H , (H + C T K-C) and (2K ;1 ; -CB ; B T C T -) be positive denite,

© 2004 by Chapman & Hall/CRC

h ; ; ; 3) AT H+ HA + AT C T - + C T + HB 2K ;1 ; -CB ; B T C T - ;1 AT C T i + C T + HB T be negative denite, n



;

4) = min min w2i
w2i o i = 1
2
: : :
m .

min h;cT H ; c 1

i

;1
cT ;H + C T K-C ;1 c i : i i i

Then the set E0 = fX : X T HX < g i -  0 and E0 = fX : X T (H + C T K-C)X < g i - 0 is estimate E0 (L S ), of the domain D(L S ) of asymptotic stability of X = 0 of the system (4.29) on N0 (L S ). rrr

Proof. Let all the conditions be valid. Hence, F (4.9) is a Lyapunov functional family on N (L S ) for the system (4.29) with D = 0 (proof of the su ciency part of Theorem 4.15). Let vf 2 F . The condition 4) guarantees 0

0

Vf



= X

0

: vf (X) = X T HX + 2

Zw 0

f T (w)-dw <



S:

Besides, condition 4) and the denition of E0 yield

E Vf : 0

Since, vf > 0 and v_ f < 0 on S and Vf S then Vf is positively invariantt, vf X(t X0  f)]  vf (X0 ) for all t 2 <+ , vf X(t X0 f)] ! 0 as t ! +1, which imply Vf D(L S ). This, E0 Vf and the fact that E0 is a neighbourhood of X = 0 due to positive deniteness of both H and (H + C T K-C) prove that E0 is an estimate of D(L S ).

4.11.3 Frequency domain approach

What follows generalises results by Popov 168], S,iljak 185], S,iljak and Weissenberger 186], 187], Walker and McClamroch 204] and Weissenberger 210]{215]. We allow vector nonlinearity f in what follows, which is based on the matrix Popov criterion (Theorem 4.16). Theorem 4.50 Let D = 0, 1) 2) 3) 4)

the real parts of all eigenvalues of the matrix A be negative, the pair (A
B) be controllable, the pair (A
C) be observable, - 2 <m+
m be such that K ;1 + He (I + j2!-)F(j!)] is positive denite for all ! 0,

© 2004 by Chapman & Hall/CRC

o n ; 5) = min min (w2i
w2i ) cTi H ;1 ci ;1 : i = 1
2
: : :
m . Then the set E0 = fX : X T (H + C T K-C)X < g is the estimate E0 (L S ), E0 = E0 (L S ), of the domain D(L S ) of asymptotic stability of X = 0 of the system (4.29) on N0(L S ). rrr

Proof. Let all the conditions hold. Then F (4.9) is Lyapunov functional family for the system (4.29) on N (L S ) (proof of Theorem 4.16). From this point on we 0

0

repeat the rest of the proof of Theorem 4.49.

This theorem broadens essentially the class of systems to which the Popov frequency approach can be eectively used. For example, nonlinearities can be sinusoidal and/or sigmoidal. The latter show a direct application of Theorem 4.50 to neural networks.

© 2004 by Chapman & Hall/CRC

Chapter 5

Novel development of the Lyapunov method 5.1 Introductory comment The fundamental results of the Lyapunov (second, direct) method are expressed in terms of the existence of a function v with appropriate properties. They do not clarify how to construct the function v for a given system (5.1), dX = f(X): (5.1) dt They do not provide necessary and su cient conditions for 1) an exact direct generation of the function v, and for 2) an exact direct determination of a stability domain when the system (5.1) is prespecied. In order to avoid solving the problems raised from 1) and 2) we usually a priori accept a form of the v-function and then look for conditions on the system under which the so chosen v will satisfy the corresponding fundamental theorem of the Lyapunov method. Unfortunately, the available forms of v used eectively so far are few. Their use can be inadequate for a given system, which results in too conservative, restrictive conditions on the system. It is intuitively clear that the form of the function v depends on the form of the function f dening the system (5.1). The problems of exact direct (from f) construction of the function v and of exact direct determination of a stability domain have been until recently (71], 72], 74], 75]{86].) unsolved fundamental problems of the Lyapunov stability theory. This chapter aims to present the original results on these problems, which open a new direction for the use and development of the Lyapunov method. This chapter is based on 71], 72], 74]{86].

© 2004 by Chapman & Hall/CRC

5.2 Systems with di erentiable motions 5.2.1 Smoothness property

Denition 5.1 Weak smoothness property: (i) There is an open neighbourhood S of X = 0, S 2
(a) the system (5.1) has the unique solution X(t X0 ) through X0 at t = 0 over time interval I0, and (b) the motion X(t X0 ) is dened, continuous and dierentiable in (t
X0 ) 2 I0  S . (ii) for every X0 2 (
Strong smoothness property:

(i) The system (5.1) has weak smoothness property. (ii) If the boundary @ S of S is non-empty then every motion of the system (5.1) passing through X0 2 @ S at t = 0 obeys inf kX(t X0 )k : t 2 I0 ] > 0 for every X0 2 @ S .

Specic smoothness property: There is an open neighbourhood S of X = 0 such that both (i) the function f is dierentiable on S : f(X) 2 C (1)(S ), and (ii) there is 2 <+ such that sup kfX (X)k : X 2 S ] < , where fX is the Jacobian of f .

5.2.2 Two-stage approach Distinction between problems

The two-stage approach means the following:

© 2004 by Chapman & Hall/CRC

1) At rst a system Lyapunov function v is generated exactly, at one shot, directly via ordinary dierential equations (5.1) and (5.2), dv(X) = ;q(X)
(5.2a) dt v(0) = 0
(5.2b) or equivalently via the partial dierential equation (5.3a) grad v(X)]T f(X) = ;q(X)
(5.3a) so that it obeys the boundary condition (5.3b) i.e. (5.2b)], v(0) = 0: (5.3b) Such a function v need not explicitly provide informationabout the asymptotic stability domain D of X = 0. Properties of v are governed by properties of q for a given system (5.1), i.e. for a given f. 2) At the second stage we use the same function q in a specic way to be claried later on and resolve the equations (5.1) and (5.2), or (5.3), for a new form of v that will yield exact determination of the asymptotic stability domain D. This approach was developed for the systems (5.1) with the global weak smoothness property or with the weak smoothness property or with the strong smoothness property in 71], 72], 74]{76], 78], 79]. The properties and the form of a function solution v to (5.1), (5.2), or (5.3) are governed by properties of both f and q. Once the system (5.1) is given, the function f is xed. We meet a question how to choose a suitable function q. A selection of q should meet two goals: a) it should guarantee the existence of a solution v to the equations (5.1) and (5.2), or equivalently to (5.3), and b) it should give such properties to v that we can draw conclusions about asymptotic stability and, if aimed, the asymptotic stability domain. a) The problem of the existence of a function solution v to the equations (5.1), (5.2), or (5.3), is purely mathematical problem that is independent of stability properties of the system (5.1). This is illustrated as follows:

Example 5.1 Let a rst order system (5.1) be dened by

dx = x5=3
 2 f;1
+1g: dt The zero state x = 0 is globally asymptotically stable for  = ;1 and unstable for  = +1. We shall make two dierent choices of the function q and look for a solution v to (5.2):

© 2004 by Chapman & Hall/CRC

1) q(x) = x2=3. The equation (5.3a) now takes the following form: dv 5=3 2=3 dx x = ;x : The solution to this equation is v(x) = 1 ln x: It is not dened at x  0 nevertheless what is  2 f;1
+1g. Hence, the this case. Notice solution to (5.3a) does not obey (5.3b), v(0) = ; 1  6= 0,5in = 3 that both f and q are continuous everywhere, f(x) = x and q(x) = x2=3. Besides, f is dierentiable on <1 and q on (<1 ; O). Notice that q is not Lipschitzian at x = 0. 2) q(x) = jxj. The equation (5.2a) (and (5.3a)) becomes now dv x5=3 = ;jxj: dx The solution to this equation is found as v(x) = ; 3 jxj1=3: This solution obeys (5.2b) (and (5.3b)), v(0) = 0: Certainly, f is dierentiable on <1 (it is unchanged), and q is continuous on <1 and dierentiable on (<1 ; O). In this case q is obviously Lipschitzian at x = 0. This example illustrates the challenging complexity of the following purely mathematical problem: what are the (necessary and) su cient conditions for the function q to guarantee the existence of a solution function v to the equations (5.1), (5.2), or equivalently to (5.3). A study of this problem is out of the scope of this book. The preceding analysis emphasizes the importance of determining the family of all functions q for which the equations (5.3) have a solution.

Note 5.1 The function q need not possess a (semi-)deniteness property in order

to guarantee existence of a solution function v to the equations (5.3). To illustrate this we can accept q(x) = x in Example 5.1, which yields a solution v to (5.3) as v(x) = ; 3 x1=3 2 C(<1) \ C (1) (<1 ; O)]: © 2004 by Chapman & Hall/CRC

b) Let Q be the family of all functions q guaranteeing existence of a solution function v to the equations (5.1), (5.2), or (5.3). The goal of what follows is to determine a subfamily Q of Q , Q  Q , and necessary and su cient properties of a solution function v to (5.3) for arbitrary q 2 Q in order for X = 0 to be an asymptotically stable state of the system (5.1), or for both X = 0 to be asymptotically stable and a set S to be its domain of asymptotic stability. Family Q( f) Denition 5.2 (a) A function q :
+

if and only if 1) q is dierentiable on B :q(X) 2 C (1)(B ) if B 
grad (X)]T f(X) = ;q(X) 1 ; (X)]
(5.4a) (0) = 0
(5.4b) have a solution  that is well dened in < and continuous on B . (b) A function q :
dierentiable on B . (c) Q( f) and Q1( f) will be denoted by Q(f) and Q1 (f), respectively if and only if B =
Note 5.2 Let v and  be solutions to (5.3) and (5.4), respectively. Then they obey (Vanelli and Vidyasagar 198]) (X) = 1 ; exp ;v(X)]: and v(X) = ; ln 1 ; (X)]: To verify this let  be replaced by 1 ; exp ;v(X)] in (5.4) so that d = ;(;1) exp ;v(X)] dv = ;q(X)f1 ; 1 + exp ;v(X)]g
dt dt or, dv(X) = grad v(X)]T f(X) = ;q(X) dt © 2004 by Chapman & Hall/CRC

(5.2a)

and 0 = (0) = 1 ; exp ;v(0)]: Hence, v(0) = 0:

(5.2b)

We have proved that the equations (5.4) become (5.3) when  is replaced by 1 ; exp (;v) in them. Analogously, the equations (5.3) take the form of the equations (5.4) when v is replaced by ;ln (1 ; ) in them. Altogether, the equations (5.3) and (5.4) are equivalent. The equation (5.3a) is linear in v and q, which is simpler than the equation (5.4a) that is bilinear in  and q. However, if for some X  , v(X  ) = +1 then (X  ) = 1. The latter is more suitable for computation than the former.

Results Theorem 5.1 In order for the state X = 0 of the system (5.1) with the strong smoothness property to have the domain D of asymptotic stability and for a set N , N <, to be the domain of its asymptotic stability, N = D, it is both necessary and sucient that

1) the set N is an open connected neighbourhood of X = 0 and N S , 2) f(X) = 0 for X 2 N i X = 0, and 3) (a) for arbitrarily chosen  2 <+ satisfying B  N , for arbitrary positive denite p 2 Q( f) and for  = min (
) the equations (5.5),

dv(X) = grad v(X)]T f(X) = ;w(X)
dt 8 p(X) if X 2 B
> < w(X) = > kX k  X  :  p kX k if X 2 (S ; B )
v(0) = 0


(5.5a) (5.5b) (5.5c)

have the unique solution function v with the following features:

or,

(i) v is positive denite on N , and (ii) if the boundary @ N of N is non-empty then v(X) ! +1 as X ! @N , X 2 N ,

© 2004 by Chapman & Hall/CRC

(b) for arbitrarily chosen  2 <+ satisfying B  N , for arbitrary positive denite p 2 Q1( f) and for  = min (
) the equations (5.5) have the unique solution function v with the following features: (i) v is positive denite on N and dierentiable on (N ; @ B ), and (ii) if the boundary @ N on N is non-empty then v(X) ! +1 as X ! @N , X 2 N . rrr

Proof. Necessity. Let X = 0 of the system (5.1) with the strong smoothness property have the asymptotic stability domain D. Then, Da = Dac = D = Dc Dsc Ds (Theorem 3.35, Section 3.7) and Da is a neighbourhood of X = 0 (Denition 2.4, Section 2.2.3). S is also a neighbourhood of X = 0 in view of the strong smoothness property. Hence, Da \ S = 6 . Let us prove Da S . If @ S =  then S =
0

0

0

0

0

+

0

0

0

+

0

0

0

the condition 2). Since N = D then I0 = <+ for every X0 2 N , I0 = I0 (X0 ), due to denitions of D and Da . Let  2 <+ satisfying B  N be arbitrarily chosen. Let positive denite p 2 Q( f) be arbitrarily selected. Then, the equation (5.5a) has a solution v dened on B for  given by  = min (
) > 0


because (5.5a) reduces to (5.3a) on B  for so dened . Since  < +1 due to its denition and  < +1 then B is compact. This and (a-2) of Denition 5.2 imply

jv(X)j < +1 for every X 2 B : Let X 2 N and 2 < , = (X
), obey (5.7), X(t X ) 2 B for all t 2 
+1: 0

+

(5.6)

0

0

© 2004 by Chapman & Hall/CRC

(5.7)

Such exists for X0 2 N , N = D = Da , and the denition of Da . Let (5.5a) be integrated from t 2 <+ to +1 so that vX(1 X0)] ; vX(t X0 )] = ;

Z

1

+

wX( X0 )]d for every (t
X0) 2 <+  N : (5.8) N = D and X0 2 N imply X(1 X0) = 0 (the denitions of Da and D). Let, in view of (5.5c), v(0) = 0
(5.9) so that (5.8) and (5.9) yield (5.10), t

Z


vX(t X0)] = wX( X0 )]d +

Z

t

1

+




wX( X0)]d for every (t
X0) 2 0
]  N :

(5.10) Positive deniteness of w on N , invariance of Da with respect to system motions, N = D = Da , (i) of the weak smoothness property and compactness of t
] for every t 2 0
] and 2 <+ yield

 Z
 wX( X )]d < +1 for every (t
X ) 2 0
]  B : t 0

0

(5.11)

Integrating (5.5a) now from to +1, using X(1 X  ) = 0 for X  2 B due to B  D, and utilizing (5.9) we get

 Z 1  
wX( X )]d < +1:

 jv(X  )j = 

+

(5.12)

Let X  = X(  X0 ) 2 B  for X0 2 N . This and (5.10){(5.12) prove (5.13), jvX(t X0 )]j < +1 for every (t
X0 ) 2 0
]  N : (5.13) Let t = 0 and X0 = X be set in (5.13) that becomes (5.14), jv(X)j < +1 for every X 2 N : (5.14) Invariance of D, D = Da , the strong smoothness property, B   N , dierentiability of w on B due to dierentiability of p on B and (5.5b), and (5.14) prove continuity of the function v on B , and its dierentiability on (N ; @ B ) in the case p 2 Q1( f):

8 < C(B ) if p 2 Q( f) v(X) 2 : C (B ) if p 2 Q ( f): (1)

1

(5.15)

Let X^0 2 (N ; B ) and let  = (X0 ) 2 <+ denote the last moment when kX(t X^0 )k = . Such  exists because X^0 2 N , N = D, and (i-b) of the weak © 2004 by Chapman & Hall/CRC

smoothness property. Hence, continuity of both system motions on <+  N and the function w on N , N = Da , invariance of Da with respect to system motions and compactness of t
] and 
] for every t 2 0
] imply

 Z   wX( X^ )]d < +1 for every (t
X^ ) 2 0
]  (N ; B )
 t 0

and

0

Z
  wX( X^ )]d < +1 for every X^ 2 (N ; B ):  0

0

From the denition of , (5.6) and (5.7) follows

Z 1   wX( X^ )]d < +1 for every X^ 2 (N ; B ):
+

0

0

These inequalities, B B , (5.10), continuity of the function w on N due to p 2 Q( f) and (5.4b), and continuity of motions X on <+  N prove continuity of the function v on (N ; B  ). Its dierentiability on (N ; B  ) in case p 2 Q1( f) is implied by B B , by dierentiability of the function w on N , which is ensured by p 2 Q1 ( f) and (5.5b), and by dierentiability of motions on <+  N (due to (i-b) of the weak smoothness property), v(X) 2

( C(N ; B )
p 2 Q( f)

C(N ; B ) \ C (1)(N ; B  )
p 2 Q1( f):

This result and (5.14) prove both

jv(X)j < +1 for every X 2 N and

(5.16)

8 C(N )
p 2 Q( f) < v(X) 2 : C(N ) \ C (N ; @ B )
p 2 Q ( f): (1)

1

(5.17)

(5.10), (5.16) and (5.17) prove the existence of a solution function v to the equations (5.5) for arbitrarily chosen p 2 Q( f)  Q1( f)], where  was arbitrarily selected provided B   N . Let v1 and v2 be two such solutions to (5.5). Uniqueness of the system motion X(t X0) for every X0 2 N due to (i-a) of the weak smoothness property, invariance of N = Da with respect to system motions and uniqueness of w on N due to its positive deniteness on N yield vi (X0 ) =

Z

0

1

+

wX( X0)]d for every i = 1
2 and every X0 2 N


v1 (X0 ) = v2(X0 ) for every X0 2 N : © 2004 by Chapman & Hall/CRC

This proves the uniqueness of the solution v to (5.5). Invariance of Da with respect to system motions, N = D = Da , positive deniteness of w on N and (5.18) obtained from (5.10) for t = 0 and X = X0 , v(X) =

Z1 0

wX( X)]d for every X 2 N


(5.18)

imply (5.19): v(X) > 0 for every (X 6= 0) 2 N :

(5.19)

(5.9), (5.16), (5.17) and (5.19) prove the necessity of the conditions (3a-i) and (3b-i) for p 2 Q( f) and p 2 Q1( f), respectively. Let X1
X2
: : :
Xk
: : : ^ Xk ! X^ as k ! +1, where Xk 2 N and be a sequence converging to X, ^ X 2 @ N in case @ N is non-empty. Let k = (Xk
) 2 <+ be the rst instant obeying X(t Xk ) 2 B  for all t 2  k
+1. Existence of such k is guaranteed by Xk 2 N and N = D = Da (due to denitions of Da and D). Continuity of motions X in (t
X0 ) 2 <+  N due to (i-b) of the weak smoothness property and N = N = D = Da imply (Theorem 3.34, Section 3.7) (Xk
) ! +1 as k ! +1. Let m be such a natural number that Xk 2 (N ; B ) for all k = m
m + 1
: : : Such m exists because N is open, B  N and Xk ! @ N as k ! +1. Let  be dened by (5.20):  = min w(X) : X 2 (N ; B )]: (5.20) Positive deniteness of p on B and (5.5b) imply  2 <+ . Combining (5.10) and (5.20) we get v(Xk )

Z
k 0

d +

Z

1

+


k

wX( Xk )]d:

This inequality, positive deniteness of w on B  and X(t Xk ) 2 B for all t 2  k
+1 imply v(Xk ) >  k
k = (Xk
)
Xk 2 N
k = m
m + 1
: : : Since k ! +1 as k ! +1 then (5.21) shows that

(5.21)

v(Xk ) ! +1 as k ! +1
or, in view of the denition of Xk , v(X) ! +1 as X ! @ N
X 2 N
which proves the necessity of the conditions (3a-ii) and (3b-ii) for p 2 Q( f) and p 2 Q1 ( f), respectively. Suciency. (a) Let all the conditions hold for p 2 Q( f). Then, X = 0 is asymptotically stable (Theorem 4.9, Section 4.6.3). Hence it has the domains Ds, Da and D, which are related by Da = Dac = D = Dc Dsc Ds . This follows

© 2004 by Chapman & Hall/CRC

from Theorem 3.35, Section 3.7. Besides, D is a neighbourhood of X = 0 by the denition. Hence, N \ D 6= . If both @ N and @ D are empty then D = N =
Theorem 5.2 For the state X = 0 of the system (5.1) possessing the weak smoothness property to be asymptotically stable it is both necessary and sucient that, for any 2 <+ and any function p 2 Q(  f), there exists the unique solution function v to (5.22) with (5.22a) determined along system motions, d dt v(x) = ;p(x)
v(0) = 0
which is also positive denite, and dierentiable if p 2 Q1 (  f).

(5.22a) (5.22b)

rrr

Proof. Necessity. Let the system (5.1) possess weak smoothness property. Let X = 0 be asymptotically stable. Then it has Da , Ds and D, and Da \ S 6= , Ds \ S 6=  and D \ S 6=  because Da , Ds, D and S are neighbourhoods of X = 0. Let any 2 <+ be chosen and a positive denite p 2 Q(  f) be arbitrarily selected, that is that, on an open connected neighbourhood A of X = 0, A B , it obeys 1: p(X) 2 C (1)(A)
2: p(0) = 0
and 3: p(X) > 0 for (X 6= 0) 2 A: The set L = A \ D D is also open connected neighbourhood of X = 0 (see the proof of Theorem 5.1 for such property of D). Let " satisfying B "  L be arbitrarily chosen. Then B"  D. Let  2 ]0
min ("
) fullling B  Ds (") be also arbitrarily © 2004 by Chapman & Hall/CRC

accepted. From the proofs of (5.11) and (5.15) we prove that v dened by (5.22) has the following properties since B  Ds (") B"  L, jv(X)j < +1 for every X 2 B 
(5.23a)

8 C(B )
p 2 Q(  f) < v(X) 2 : C(B )  C (B )
p 2 Q (  f): (1)

1

(5.23b)

Denitions 5.2 and 5.3, B"  A \ D, positive deniteness of p on A, X(t X0 ) 2 B" for all (t
X0) 2 <+  B due to B  Ds ("), (5.22a) used after integration from t = 0 to t = +1 together with (5.22b) and X(1 X0 ) = 0 imply v(X0 ) > 0 for every (X0 6= 0) 2 B : (5.24) Now, (5.22b){(5.24) prove positive deniteness of the function solution v to (5.22) on any B   Ds(") and its dierentiability on B . Its uniqueness is proved in the same way as in the proof of Theorem 5.1. This completes the proof of the necessity part. Suciency. Su ciency of the conditions of Theorem 5.2 for asymptotic stability

of X = 0 of the system (5.1) is due to Lyapunov 127] (Theorem 3.9, Section 3.7). This completes the proof of Theorem 5.2. From the computational point of view the condition that v(X) ! +1 as X ! @ N , X 2 N , is not suitable. It can be overcome as follows. Theorem 5.3 For the state X = 0 of the system (5.1) with the strong smoothness

property to have the domain of asymptotic stability D, and for a set N , N
d (X) = ;w(X)
dt 8 p(X)1 ; (x)]
X 2 B > <   w(X) = > kX k X :  p kX k 1 ; (X)]
X 2 (S ; B )
(0) = 0
have the unique solution function  with the following features:

© 2004 by Chapman & Hall/CRC

(5.25a) (5.25b) (5.25c)

(i)  is positive denite on N , (ii) if the boundary @ N of N is non-empty then (X) ! 1 as X ! @ N , X 2 N,

or, (b) for arbitrary  2 <+ obeying B   N and for arbitrarily chosen q 2 Q1 ( f) the equations (5.25) with (5.25a) determined along system motions and with  = min (
) have the unique solution function  with the following features: (i)  is positive denite on N and dierentiable on (N ; @ B ), and (ii) if the boundary @ N of N is non-empty then (X) ! 1 as X ! @ N , X 2 N. rrr

Proof. Since the equations (5.4) and (5.25) are equivalent (Note 5.2) then Theorem 5.3 results directly from Theorem 5.1.

Dierently from Theorem 5.1, Theorem 5.2 permits the system (5.1) to possess the weak smoothness property only. For such systems the necessary and su cient conditions for determination of both their Lyapunov functions and asymptotic stability domains are discovered as follows. Theorem 5.4 For the state X = 0 of the system (5.1) possessing the weak smooth-

ness property to have the domain of asymptotic stability D and for a subset N of S , N S , to be equal to D: N = D, it is both necessary and sucient that

1) the set N is an open connected neighbourhood of X = 0, 2) f(X) = 0 for X 2 N i X = 0, and 3) (a) for arbitrary  2 <+ satisfying B   N and for arbitrary positive denite function p 2 Q( f) the equations (5.26) with (5.26a) determined along system motions and for  = min (
),

d dt v(X) = ;w(X)
8 p(X) if X 2 B > <   w(X) = > kX k X :  p kX k if X 2 (
(5.26a) (5.26b) (5.26c)

and (ii) if the boundary @ N of N is non-empty then v(X) ! +1 as X ! @N , X 2 N ,

© 2004 by Chapman & Hall/CRC

or, (b) for arbitrary  2 <+ satisfying B   N and for arbitrary positive denite function p 2 Q1 ( f) the equations (5.26) with (5.26a) determined along system motions and for  = min (
) have the unique solution function v with the following features: (i) v is positive denite on N and dierentiable on (N ; @ B ), and (ii) if the boundary @ N on N is non-empty then v(X) ! +1 as X ! @N , X 2 N . rrr

Proof. Necessity. Let the system (5.1) possess the weak smoothness property. Let X = 0 have the asymptotic stability domain D and let N S be equal to D: N = D. From this point on we have to repeat the proof of Theorem 5.1 to show that the conditions 1){3) of Theorem 5.4 hold. In such a way we complete the proof of the necessity part. Suciency. Su ciency of the conditions for N = D results from Theorem 5.1.

If we wish to avoid computational di culties due to v(x) ! +1 as X ! @ N , X 2 N , then we can use the following theorem that follows from Theorem 5.4 and Note 5.2.

Theorem 5.5 For the state X = 0 of the system (5.1) possessing the weak smoothness property to have the domain of asymptotic stability D and for a subset N of S , N S , to be equal to D: N = D it is both necessary and sucient that 1) the set N is an open connected neighbourhood of X = 0, 2) f(X) = 0 for X 2 N i X = 0, and 3) (a) for arbitrary  2 <+ satisfying B   N and for arbitrary positive denite p 2 Q( f) the equations (5.27) with (5.27a) determined along system motions and for

 = min (
) d (X) = ;w(X)
dt 8 p(X) 1 ; (x)]
> < w(X) = > kX k  X  :  p kX k 1 ; (X)]


(5.27a) X 2 B X 2 (
(0) = 0
have the unique solution function  with the following features: (i)  is positive denite on N ,

© 2004 by Chapman & Hall/CRC

(5.27b) (5.27c)

(ii) if the boundary @ N of N is non-empty then (X) ! +1 as X ! @ N , X 2 N,

or (b) for arbitrary  2 <+ satisfying B   N and for arbitrary positive denite p 2 Q1( f), the equations (5.27) with (5.27a) determined along system motions and for  = min (
) have the unique solution function  with the following features: (i)  is positive denite on N and dierentiable on (N ; @ B), (ii) if the boundary @ N on N is non-empty then (X) ! +1 as X ! @N , X 2 N . rrr

Examples

The following examples are taken from 78]. Example 5.2 Let a rst (n = 1) order system (5.1) be described by dx = (x2 ; 1)x
f(x) = (x2 ; 1)x 2 C (1)(<1 ): (5.28) dt It possesses the strong smoothness property on <1 . Now, from f(x) = (x2 ; 1)x = 0 it results that the system (5.24) has three equilibrium states: x1e = ;1
x2e = 0
x3e = +1: They suggest to select  2 ]0
+1 and S = ] ; 1
+1 because @ S = f;1
+1g = fx1e
x3eg which shows that (ii) of the strong smoothness property holds on S = ] ; 1
+1. Hence, the system (5.28) has strong smoothness property also on S = ] ; 1
+1. Two dierent forms of the function p will be considered. a) By referring to the form of the function f (5.28) let p(x) = (1 ; x2)x2 : (5.29) The function p is dierentiable and positive denite on S = ] ; 1
+1. Now, for any  2 ]0
1, the function w is determined by (5.4) and (5.29) 8 (1 ; x2)x2
jxj  p > < (5.30) w(x) = > jxj  2 x2  2 x2
:  1 ; jxj2 jxj2 = (1 ; 2)jxj
jxj 2 
1: It is also positive denite on S = ] ; 1
+1 and dierentiable on the set B = fx : x 2 ] ; 1
1
jxj 6=  < 1g. Now, from (5.4), (5.28) and (5.30) it results that 812 > for jxj 2 0
] < x
v(x) = > 2 (1 ; 2 ) (1 ; )(1 + jxj)
1  (5.31) 2 : ln +  for j x j 2 
+1: 2 (1 + )(1 ; jxj) 2 © 2004 by Chapman & Hall/CRC

The function v determined by (5.31) is positive denite on S = ] ; 1
+1, dierentiable on the set B and v(x) ! +1 as x ! @ S , x 2 S . We conclude that all the conditions of Theorem 5.1 are satised for N = S = ] ; 1
+1. Hence, the set N is the asymptotic stability domain of x = 0 of the system (5.28) that is N = D = ] ; 1
+1. b) Let the function p be selected in another form, p(x) = x2: (5.32) Obviously, the function p is dierentiable and positive denite on S = ] ; 1
+1. The function w is now determined by arbitrarily chosen  2 ]0
1, (5.4b) and (5.32): 8 x2
for jxj 2 0
] > <   (5.33) w(x) = > jxj jxj 2 :  jxj = jxj for jxj 2 
+1: Now, (5.27a), (5.32) and (5.33) yield 81 1 > for jxj 2 0
] > < 2 ln 1 ; x2 v(x) = >  (5.34)
1 1   (1 + j x j )(1 ; ) > : 2 ln (1 ; jxj)(1 + ) + 2 ln 1 ; 2 for jxj 2 
+1: The function v dened by (5.34) obeys evidently all the conditions of the Theorem 5.1 on N = S = ] ; 1
+1. Hence, the domain D of asymptotic stability of x = 0 of the system (5.1) equals S = ] ; 1
+1: D = ] ; 1
+1:

Example 5.3 Let

dx = ;(x2 ; 1)x: (5.35) dt In this case f(x) = ;(x2 ; 1)x 2 C (1)(<1 ). The strong smoothness property is valid on <1. Equilibrium states are again x1e = ;1, x2e = 0 and x3e = +1 so that S = ] ; 1
+1 is selected. The system (5.35) has strong smoothness property also on S . Let the function p be dened by (5.29). Repeating the procedure explained in Example 5.2 under a) we get v(x) = ; 12 x2 for jxj 2 0

 2 ]0
1: Hence, the function v is not positive denite. Theorem 5.2 is not satised. Since its conditions are necessary for x = 0 to be asymptotically stable then the result is that x = 0 is not asymptotically stable. It does not have the asymptotic stability domain. This result and Theorem 3.35 (Section 3.7) show that x = 0 does not have the attraction domain, i.e. it is not attractive. In this example it is also unstable.

© 2004 by Chapman & Hall/CRC

Example 5.4 Let

" x_ ! " ;x ; x exp (kxk) ! =

1

x_ 2

1

2

2

1

;x ; x exp (kxk)

:

(5.36)

The system (5.36) possesses the weak smoothness property on <2. Let p(x) = 8kxk2, so that (5.22) yields v(x) = 4kxk2. Hence, x = 0 of the system (5.36) is asymptotically stable (Theorem 5.2).

Example 5.5 Let

" x_ ! " ;(x + x sin x )(1 + x ) ! 1

x_ 2

=

1

2

2

2

;(1 ; 2x x ) sin x 1 2

2

4 1

:

(5.37)

The system (5.37) has the weak smoothness property. Let p(x) = 2x21 + sin2 x2. Now, (5.22) yields v(x) = arctan x21 + 1 ; cos x2 . All the conditions of Theorem 5.2 are satised. Hence, x = 0 of the system (5.37) is asymptotically stable.

5.2.3 Approach via -uniquely bounded sets O

This section presents results and examples from 84].

Introductory comment

The two-stage approach requires the partial dierential equation gradv(X)]T f(X) = ;w(X) to be solved twice in order to determine the asymptotic stability domain D. It should be rst  solved  for w(X) = p(X) on B . Afterwards it should be k X k X solved for w(X) =  p kX k . The same procedure holds if w(X) = p(X) 1 ; (X)] is rst accepted to nd  from grad(X)]T f(X) = ;w(X)on B .Once it has been done we should solve the same equation for w(X) = kX k p kX X k 1 ; (X)]. In order to avoid such double solving of the partial dierential equation we need another approach.

Family Q(S  f) If we replace B by S in Denition 5.2 we get the following: Denition 5.3 (a) A function q :
1) q is dierentiable on S : q(X) 2 C (1) (S ), and

© 2004 by Chapman & Hall/CRC

2) the equations (5.1), (5.2), or equivalently (5.3), have a solution v that is well dened in < and continuous for every X 2 B  for some  2 ]0
+1,  = (q
f) or equivalently, the equations (5.4) have a solution function  that is well dened in < and continuous on B  . (b) A function q :
Results Theorem 5.6 In order for the state X = 0 of the system (5.1) with the strong smoothness property to have the domain of asymptotic stability D and for a set N , N
1) the set N is an open connected neighbourhood of X = 0 and N S , 2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrarily selected O-uniquely bounded set U , U  S , with the generating function u on S obeying u 2 Q(S  f), the equations (5.3) for q = u have the unique solution function v on N with the following properties: (i) v is positive denite on N , and (ii) if the boundary @ N of N is non-empty then v(X) ! +1 as X ! @ N , X 2 N, or 4) for arbitrarily selected O-uniquely bounded set U , U  S , with the generating function u on S obeying u 2 Q1(S  f), the equations (5.3) for q = u have the unique solution function v on N with the following properties: (i) v is dierentiable and positive denite on N , and (ii) if the boundary @ N on N is non-empty then v(X) ! +1 as X ! @ N , X 2 N. rrr

© 2004 by Chapman & Hall/CRC

Proof. Necessity. Let X = 0 of the system (5.1) with the strong smoothness property have the asymptotic stability domain D. Denitions of Da and D show that it has also the attraction domain Da , D Da . It is a neighbourhood of X = 0 due to denition of Da , and S is a neighbourhood of X = 0 in view of the smoothness property. Hence, Da \ S = 6 . Let us prove Da S . If @ S =  then S =
0

0

0

0

0

+

0

0

0

+

0

0

0

0

+

0

+

0

0

is an open neighbourhood of X = 0 (Lemma 4.10, Section 4.3.1). Property UB (Section 4.3.2), the same lemma and Denition 4.13 (Section 4.3.2) show that the function u is also positive denite on S . Since N = D S , then the function u is the dierentiable positive denite generating function on N , too. The property of u 2 Q(S  f) guarantees existence of  > 0 such that there exists a solution function v to the equations (5.3) for q = u, which is well dened in < and continuous for every X 2 B  , that is that jv(X)j < +1 for every X 2 B : (5.38) Let 2 ]0
+1 be such that U  B : (5.39) The existence of such is assured by Lemma 4.11 (Section 4.3.3). Let 2 ]0
+1, = (X0  f u ), be such that for any X0 2 N the condition (5.40) holds: (5.40) X(t X0) 2 U 
for every t 2 
+1: Such exists in view of denitions of Da and D, Da = D, N = D and X0 2 N . Notice that X0 2 N implies also X(+1 X0) = 0: (5.41) © 2004 by Chapman & Hall/CRC

After integrating (5.3a) for q = u from t 2 <+ to +1 we derive (5.42): vX(+1 X0)] ; vX(t X0 )] = ;

Z

1

+

t

uX( X0 )]d for every (t X0 ) 2 <+  N : (5.42)

Since u 2 Q(S  f) then (5.43) holds: v(0) = 0: Now, (5.41){(5.43) yield (5.44): vX(t X0 )] =

Z

1

+

t

(5.43)

uX( X0 )]d for every (t X0 ) 2 <+  N :

(5.44)

This can be rewritten in the following form:

Z


vX(t X0)] = uX( X0 )]d + t

Z

1

+




uX( X0 )]d for every (t X0 ) 2 <+  N :

(5.45) Positive invariance of D with respect to system motions, N = D, continuity of the motions X due to the smoothness property, continuity of u on N in view of its dierentiability on N , the denition of (5.40), (5.38), and compactness of  ; t] for any t 2 <+ prove (5.46):

Z
  uX( X )]d < +1 for every (t X ) 2 <  N : t 0

0

+

(5.46)

From (5.38){(5.40) it follows that (5.47)

Z  t

1

+

 uX( X )]d < +1 for every X 2 N : 0

0

(5.47)

Altogether (5.45){(5.47) prove boundedness of vX(t X0 )] expressed by (5.48): (5.48) jvX(t X0 )]j < +1 for every (t X0) 2 <+  N : Hence, by setting t = 0 and X0 = X in (5.48) we derive (5.49): jv(X)j < +1 for every X 2 N : (5.49) Dierentiability of the motion X in X0 2 N , dierentiability of u in X 2 N , positive invariance of N = D with respect to system motions, (5.44) and (5.48) prove continuity of v in X 2 N (5.50): v(X) 2 C(N ): (5.50) @v @v : : : @v )T and u Let now u 2 Q1(S  f) be arbitrarily chosen. Let vX = ( @x X 1 @x Z2
@xn be analogously dened. From (5.45) it follows that v(X) = uX( X0 )]d + 0

© 2004 by Chapman & Hall/CRC

^ 2 C (1)(B  ), ^ X^ = X(  X0). Since X^ 2 B then u 2 Q1 (S  f) implies v(X) v(X), which together with b) of the weak smoothness property implies vX(  X)] 2 C (1), so that there exists (X) 2 ]0
+1, 8X 2 N , such that kvX X(  X0 )]k < (X), 8X 2 N . Now, a-1) of Denition 5.3, b) of the weak smoothness property and compactness of 0
!] imply existence of (X) 2 ]0
+1, 8X 2 N , such that !!Z
!! uX X( X)]d!!! < (X), 8X 2 N . Altogether, kvX (X)k < (X) + (X) < 0 +1, 8X 2 N . Therefore, v(X) 2 C (1)(N ): (5.51) Positive invariance of N with respect to system motions, positive deniteness of u on N and (5.44) imply (5.52): v(X) > 0 for all (X 6= 0) 2 N : (5.52) Now, (5.43), (5.50) and (5.52) prove the necessity of the condition 3i), and (5.43), (5.51) and (5.52) prove the necessity of the condition 4i). Let v1 and v2 be two solutions to (5.3). Hence, v1 (X0 ) ; v2(X0 ) =

Z1 0

fuX ( X )] ; uX ( X )]gd for every X 2 N : 1

0

2

0

0

Since u(X) is uniquely determined by X 2 N due to (a) of Property UB and Denition 4.13 (Section 4.3.2), and the motion of the system is unique through X0 , then X 1 ( X0 )  X 2( X0 ) and uX 1( X0 )]  uX 2 ( X0 )] so that v1 (X0 ) ; v2(X0 ) = 0 for every X0 2 N : This proves the uniqueness of the solution v to (5.3) and completes the proof of 3i) and 4i). ~ Xk ! X~ as k ! +1, Let X1
X2
: : :
Xk
: : : be a sequence converging to X, where Xk 2 N for all k = 1
2
: : : @ N be non-empty and X~ 2 @ N . Let 2 ]0
+1 be arbitrarily chosen so that U = fX : u(X) < g U . Such exists because the set U is O-uniquely bounded and the function u is its generating function on N (Denition 4.12, Property UB, Lemma 4.10, Denition 4.11 and Denition 4.13 (Sections 4.2 and 4.3)). The set U is connected open neighbourhood of X = 0 (Lemma 4.11, Section 4.3.3). Let Tk , Tk = T (Xk
) 2 0
+1 be the rst instant obeying (5.53): (5.53) X(t Xk ) 2 U  for all t 2 Tk
+1: The existence of such Tk is guaranteed by Xk 2 N and N = D (denition of Da and denition of D). Continuity of the motions X in (t
X0 ) 2 <+  N due to the strong smoothness property and N = N = D (Theorem 3.35, Section 3.7) and D S imply Tk ! +1 as k ! +1 (Theorem 5.34, Section 3.7). Let m be such a natural number that Xk 2 (N ; U  ) for all k = m
m + 1
: : :. Such m exists because N is open, U   N and Xk ! @ N as k ! +1. Let ^ be dened by (5.54): ^ = min u(X) : X 2 (N ; U )]: (5.54) © 2004 by Chapman & Hall/CRC

The O-unique boundedness of the set U and the fact that the function u is its generating function on N (Lemma 4.11, Section 4.3.3) guarantee 4 and 5 of Property UB and Lemma 4.10 (Section 4.3.2)] that ^ dened by (5.54) satises (5.55): ^ = 2 ]0
+1: (5.55) From (5.45) we get (5.56) after replacing by Tk : vX(t Xk )] =

Z Tk t

uX( Xk )]d +

Z

1

+

Tk

uX( Xk )]d for (t Xk ) 2 <+  N


and for k = m
m + 1
: : : Setting t = 0 in (5.56) and using (5.54) and (5.55) we derive v(Xk )

Z Tk 0

d +

Z

1

+

Tk

(5.56)

uX( Xk )]d for Xk 2 N and all k = m
m + 1
: : :

(5.57) Positive invariance of N = D with respect to system motions, positive deniteness of u on N and (5.57) imply v(Xk ) Tk
for Xk 2 N and all k = m
m + 1
: : : Since Tk ! +1 as k ! +1 then v(Xk ) ! +1 as Xk ! @ N due to k ! +1
Xk 2 N
which proves necessity of the condition (3-ii). Suciency. Let the conditions of the theorem to be proved hold. The generat-

ing function u of the selected O-uniquely bounded set U is positive denite on S (Lemma 4.11, Section 4.3.3). Since the function v is positive denite solution to (5.3) on N , and N is an open connected neighbourhood of X = 0, it follows that X = 0 is asymptotically stable (Theorem 4.9, Section 4.6.3). Hence, X = 0 has the asymptotic stability domain. Since the function v is the solution to (5.3) obtained for positive denite u 2 Q(S  f) then v(X) 2 C(D) and v(X) ! +1 as X ! @ D in case @ D 6=  (see the proof of the necessity part). These facts, positive deniteness of v on N , v(X) ! +1 as X ! @ N if @ N 6= , and the fact that both D and N are connected neighbourhoods of X = 0, prove D = N (for details see the proof of su ciency part of Theorem 5.6). In view of equivalence of equations (5.3) and (5.4) (Note 5.2) Theorem 5.6 can be set in the following form: Theorem 5.7 In order for the state X = 0 of the system (5.1) with the strong

smoothness property to have the domain of asymptotic stability D and for a set N , N
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1) the set N is an open connected neighbourhood of X = 0 and N S , 2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrarily selected O-uniquely bounded set U , U  S , with the generating function u on S obeying u 2 Q(S  f), the equations (5.4) for q = u have the unique solution function  on N with the following properties: (i)  is positive denite on N , (ii) if the boundary @ N of N is non-empty then (X) ! 1 as X ! @ N , X 2 N, or 4) for arbitrarily selected O-uniquely bounded set U , U  S , with the dierentiable generating function u on S obeying u 2 Q1(S  f), the equations (5.4) with q = u have the unique solution function  on N with the following properties: (i)  is dierentiable and positive denite on N , and (ii) if the boundary @ N of N is non-empty then (X) ! 1 as X ! @ N , X 2 N. rrr

However, the conditions slightly change if the system (5.1) possesses the weak smoothness property rather than the strong smoothness property.

Theorem 5.8 For the state X = 0 of the system (5.1) possessing the weak smoothness property to have the domain of asymptotic stability D, and for a subset N of S , N S to be equal to D: N = D, it is both necessary and sucient that 1) the set N is an open connected neighbourhood of X = 0,

2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrarily selected O-uniquely bounded set U , U  S , with the generating function u obeying u 2 Q(
© 2004 by Chapman & Hall/CRC

4) for arbitrarily selected O-uniquely bounded set U , U  S , with the generating function u obeying u 2 Q1 (
Proof. Necessity. Let the system (5.1) possess the weak smoothness property. Let X = 0 have the asymptotic stability domain D and let N S be equal to D: N = D. Let an O-uniquely bounded set U , U  S , with generating function u obeying u 2 Q(
conditions 1){3), or 1), 2), 4), hold. Then, X = 0 of the system (5.1) is asymptotically stable (Theorem 4.9, Section 4.6.3). Therefore, X = 0 has the domain D of asymptotic stability. Let X0 2 (
ness property to have the domain of asymptotic stability D and for a subset N of S , N S , to be equal to D: N = D, it is both necessary and sucient that

1) the set N is an open connected neighbourhood of X = 0, 2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrarily selected O-uniquely bounded set U , U  S , with the generating function u obeying u 2 Q(
© 2004 by Chapman & Hall/CRC

(i)  is positive denite on N , and (ii) if the boundary @ N of N is non-empty then (X) ! 1 as X ! @ N , X 2 N, or 4) for arbitrarily selected O-uniquely bounded set U , U  S , with the generating function u on
Examples Example 5.6 Let n = 1 and f(X) = ;sin X so that (5.1) becomes (5.58), dX = ;sin X: (5.58) dt The function f is dierentiable on <1 . Hence, the system (5.58) possesses the strong smoothness property on <1 . The equilibrium states are determined  by1 Xek = k, k = 0
1
2
: : : This 2 suggests S = ] ; 
 and U = X : sin 2 X <  for  2 ]0
1. The system has the  1 strong  smoothness property also on S . The function u dened by u(X) = 2 sin 2 X is the dierentiable generating function on S of the O-uniquely bounded set U . Besides, the function u is positive denite on S . The denition of the function u, (5.3a) and (5.58) yield

  dv = 1 tan 1 X
dX 2 2 which has the solution obeying v(0) = 0 (5.3b) determined by 1  v(X) = ; ln cos 2 X : Evidently, u 2 Q(] ; 
 ;sin( )). The function v is dierentiable and positive denite on S and v(X) ! +1 as X ! @ S = f;
g, X 2 S . All the conditions of Theorem 5.6 are satised for N = S = ];
. Hence, the set D = N = S = ];
 is the domain of asymptotic stability of X = 0 of the system (5.58). Notice that © 2004 by Chapman & Hall/CRC

f 2 (X) is dierentiable and positive denite on N = S = ] ; 
 but u(X) 6= f 2 (X) in this example because f 2 (X) = sin2 X cannot be the generating functionh on hS of any O-uniquely bounded set. It is not strictly increasing in X for jX j 2 2
 . Notice also that Xe = ; and Xe =  are unstable and not attractive due to the domain of asymptotic stability of X = 0: D = ] ; 
. If we use Theorem 5.7 then the equations (5.4) take the following form:   d = 1 tan 1 X dx 1; 2 2 and (0) = 0: 1  The solution (X) = 1 ; cos 2 X obeys all the conditions of Theorem 5.7. Motions of system (5.58) for jX0 j <  are dened by  1 
X(t X0 ) = 2 arctan exp (;t) tan 2 X0
jX0 j < : It is now simple to verify the above result.

Example 5.7 Let f(X) = sin X and dX = sin X: (5.59) dt This system possesses the strong smoothness property on S = <1 . Its equilibrium statesare X ek = k, k = 0
1
2
: : : Hence, N =];
 is accepted and U = fX : 1 sin2 2 X < g,  2 ]0
1. By repeating the procedure explained in Example 5.6 we derive   v(X) = ln cos 12 X : The function v is negative denite on N . Thus, it does not satisfy either Theorem 5.6 or Theorem 5.7, which shows that X = 0 of the system (5.59) does not have the asymptotic stability domain D <1, or equivalently that it is not asymptotically stable. In fact, it is unstable and not attractive.

Example 5.8 Let

dX = ;tan X: dt

(5.60)

i h The system possesses the strong smoothness property on S = ; 2
2 but not on <1 . The equilibrium state X = 0 is unique in S . This and the form of jf(X)j = jtan X j suggest U = fX : tan2X < 1g  S . The generating function on S of the © 2004 by Chapman & Hall/CRC

O-uniquely bounded set U is u(X) = tan X. It is dierentiable on S . Now, (5.3a), 2

(5.60) and the denition of u yield dv = tan X: dX Its solution satisfying v(0) = 0 (5.3b) is found as v(X) = ; ln cos X: (5.61) h  i This shows that u 2 Q ; 2
2  ;tan( ) . The function v (5.61) is dierentiable and positive denite on S and n o v(X) ! +1 as X ! @ S
X 2 S
@ S = ; 2
2 : i h All the conditions of Theorem 5.6 are satised for N = S . Hence, D = ; 2
2 is the domain of asymptotic stability of X = 0 of the system (5.60). In this example u(X) = f 2 (X). Motions of the system are dened by: 8 X for X = k
> 0 < 0
2k ; 1 2k + 1 X(t X0) = > : arcsin (exp (;t) sin X0 )
X0 2 2 
2 
k = 0
1
: : : i h Using this it is easy to verify D = ; 2
2 . Example 5.9 The function f dened by f(X) = (X 2 ; 4)X is dierentiable on <1. Therefore, the system (5.62), dX = (X 2 ; 4)X
(5.62) dt possesses the strong smoothness property on <1. The equilibrium states are: Xe1 = ;2, Xe2 = 0 and Xe3 = 2. They suggest S = ] ; 2
2. Hence, the system (5.62) has the strong smoothness property also on S . The form of the function f suggest U = fX : X 2 < g for  2 ]0
2 so that the generating function on S of the O-uniquely bounded set U is u(X) = X 2 . It is dierentiable and positive denite on S . The denition of the function u, (5.3a) and (5.62) result in X dv (5.63) dX = 4 ; X 2 : The solution of (5.3) obeying v(0) = 0 (5.3b) is derived as
2 1 4 ; X v(X) = ; 2 ln 4 : (5.64) Hence, u 2 Q(] ; 2
2 (X 2 ; 4)X). The function v (5.64) is positive denite and dierentiable on S = ] ; 2
2 and v(X) ! +1 as X ! @ S = f;2
2g, X 2 S . All © 2004 by Chapman & Hall/CRC

the conditions of Theorem 5.6 are satised for N = S = ];2
2. The set N = ];2
2 is the domain of asymptotic stability of X = 0 of the system (5.63) (Theorem 5.6). Example 5.10 Let dX = ;X(1 + X 4 ): (5.65) dt Since the function f, f(X) = ;X(1 + X 4 ), is dierentiable on <1 then the system (5.65) has the strong smoothness property on S = <1 . It has the unique equilibrium state Xe = 0. Let U = fX : X 2 < 1g. The function u, u = X 2 , is the generating function on (S = <1 ) of the O-uniquely bounded set U . The solution of the equations (5.3) in this case is v(X) = 12 arctan X 2: (5.66) Hence, u 2 Q(<1  f). The function v obeys all the conditions of Theorem 5.6 on N = S = <1 . Hence, the asymptotic stability domain D of X = 0 of the system (5.65) is the whole state space, D = <1
which shows that X = 0 is globally asymptotically stable. In this connection it is emphasized that the system Lyapunov function v (5.66) is not radially unbounded because v(X) ! 4 as jX j ! +1: This result does not contradict the well-known Barbashin{Krasovskii criterion for the global asymptotic stability 6], which requires existence of radially unbounded globally positive denite function v with globally negative denite derivative v_ . However, it does not require that every function v proving the global asymptotic stability is radially unbounded. In order to show that Theorem 5.6 leads to a radially unbounded Lyapunov function under an appropriate choice of U , let U = fX : 4X 4 < 1g. The generating function of O-uniquely bounded set U is u1(X) = 4X 4 , which with the equations (5.3) implies ; v1 (X) = ln 1 + X 4 : Hence, u1 2 Q(<1  f). This function v1 obeys also all the conditions of Theorem 5.6 on N = S = <1. In addition it is radially unbounded, that is v1 (X) ! +1 as jX j ! +1: Radial unboundedness of v is not a necessary condition for global asymptotic stability in case v is constructed via Theorem 5.6 { Theorem 5.9. Example 5.11 A second order system described by (5.67), dX1 = ; X1 (1 ; X12 )2 (1 ; X22 )
dt 1 + X22 (5.67) dX2 = ; X2 (1 ; X12 )(1 ; X22 )2
dt 1 + X12 © 2004 by Chapman & Hall/CRC

has the strong smoothness property on <2 because f : <2 ! <2,

2 66 ; X f(X) = 66 4 ;X

1

;1 ; X ;1 ; X 1+X ;1 ; X ;1 ; X 2 2 1

2 2

2 2

2

2 2 2

2 1

3 77 77 2 C (< ): 5 (1)

(5.68)

2

1 + X12 Obviously the set Se of the equilibrium states is determined by

Se = fX : (X = 0) or (X = 1
X 2 <) or (X 2 <
X = 1)g: Let S = fX : jX j < 1
jX j < 1g. Then, X = 0 is the unique equilibrium state of the system (5.67) in S , and the system possesses the strong smoothness property also on S . By referring to (5.67) and the set S we accept U = B = fX : kX k < 1g  S . The set U is O-uniquely bounded with the dierentiable generating function u on S : u(X) = kX k . Such a choice of the function u and (5.3) leads to 1

1

2

1

2

2

2

1

2

2 + X22 : v(X) = 12 (1 ;XX12 )(1 ; X 2) 1

(5.69)

2

Hence, u 2 Q(S  f). The function v is dierentiable and positive denite on S and v(X) ! +1 as X ! @ S
@ S = X : (jX1 j = 1
jX2j  1) (5.70)



or (jX1 j  1
jX2j = 1) : The set N = S and the function v (5.69) obey all the conditions of Theorem 5.6, which shows that D = fX : jX1j < 1
jX2 j < 1g:

Example 5.12 Let X_ = ;X (4 ; X )(1 + X )


X_ 2 = ;X2 (4 ; X22 )(1 + X12 ): (5.71) The system (5.70) satises the requirements for the strong smoothness property on <2. The set Se of the equilibrium states is found in the following form Se = fX : X1 2 f;2
0
2g and X2 2 f;2
0
2gg: Let S = fX : jX1 j < 2
jX2j < 2g. The system (S ) possesses the strong smoothness property also on S . Let U = fX : 2X12 (1 + X22 ) + X22 (1 + X12 )] < 2g  S be selected. It is O-uniquely bounded set with the dierentiable generating function u on S : u(X) = 2X12 (1 + X22 ) + X22 (1 + X12 )]. Such a form of the function u and (5.3) imply 2 )(4 ; X22 )
: v(X) = ; ln (4 ; X116 (5.72) 1

1

2 1

© 2004 by Chapman & Hall/CRC

2 2

Hence, u 2 Q(S  f). The function v (5.71) is dierentiable and positive denite on S . Besides, v(X) ! +1 as X ! @ S
@ S = X : (jX1 j = 2
jX2j  2)



or (jX1 j  2
jX2j = 2) : The function v (5.71) and the set N = S obey all the conditions of Theorem 5.6 which veries that D = fX : jX1j < 2 and jX2 j < 2g:

Example 5.13 The system (5.72), dX1 = ;sin X cos3 X cos2 X
dX2 = ;sin X cos2 X cos3 X
(5.73) 1 1 2 2 1 2 dt dt obeys all the requirements for the strong smoothness property on <2. The set Se of all the equilibrium states is found in the form:   2k + 1 Se = fX : Xi 2 fk : k = 0
1
: : :g  2  : k = 0
1
: : :
i = 1
2g: Let S = fX : jX1j < 2
jX2j < 2 g. The system (5.72) has the strong smoothness property on S . Let U = fX : 2sin2 X1 (1 + sin2 X2 ) + sin2 X2 (1 + sin2 X1 )] < 2g. It is O-uniquely bounded set with the dierentiable generating function u on S : u(X) = 2sin2X1 (1 + sin2 X2 ) + sin2 X2 (1 + sin2 X1 )]. Hence, the function v (5.3) is found in the following form: 2 X1 + sin2 X2 : (5.74) v(X) = sin cos2 X1 cos2 X2 Therefore, u 2 Q(S  f). The function v (5.73) is dierentiable and positive denite on S . Furthermore, n   v(X) ! +1 as X ! @ S
@ S = X : jX1j = 2
jX2 j  2  o or jX1 j  2
jX2j = 2 : The function v and the set N = S satisfy all the conditions of Theorem 5.6. The domain D of asymptotic stability of X = 0 of the system (5.72) is the set S : n o D = X : jX1 j < 2
jX2j < 2 :

Comment 5.1 The classical forms of a Lyapunov function are  quadratic form v(X) = X T HX


© 2004 by Chapman & Hall/CRC

 Lurie form (Section 4.11.2, (5.32)) vf (X) = (X T f T )H(X T f T )T +

Zw 0

f T (w)-dw


and  the natural form (Section 4.4.1, Example 4.16)

v(X) = bT jX j: However, Lyapunov functions constructed directly very often do not have a classical form as shown by (5.34), by

 v(X) = ; ln cos 12 X



 

and (X) = 1 ; cos 1 X


2

of Example 5.6, by (5.64), by (5.66), by (5.69), by (5.71) and by (5.73).

5.2.4 General one-shot approach

What follows is deduced from 81] by accepting I = f0g, and from 83].

Introductory comments Like the O-uniquely bounded set approach, the general one-shot approach is aimed to provide the necessary and su cient conditions for a system Lyapunov function construction with a simultaneous exact determination of the asymptotic stability domain by solving v_ = ;p, or equivalently _ = ;(1 ; )p, only once. Unlike the former the latter does not require use of an O-uniquely bounded set that gives the following property to the generating function u: min u(X) : X 2 (S ; B )]  > 0
for every  2 <+ for which S is neighbourhood of B . This property of u is used to prove that v(x) ! +1 (or equivalently (X) ! 1) as X ! @ N , X 2 N , in case @ N is nonempty. This property of u will be used in what follows, too.

Family Q(S  f)

In the one-shot approach the family Q(S
f) is determined dierently than in the preceding approaches.

Denition 5.4 (a) Q(S  f) is the family of all the functions q :
1) q is continuous on S ,

© 2004 by Chapman & Hall/CRC

2) for any  > 0 such that S is a neighbourhood of B = fX : kX k < g, there is  > 0,  = (
q
S ), obeying inf q(X) : X 2 (S ; B ] =  , 3) there is  2 ]0
+1,  = (q f), such that there exists a solution v :
Results Theorem 5.10 In order for the state X = 0 of the system (5.1) with the strong smoothness property to have the domain of asymptotic stability D and for a set N to be the domain of its asymptotic stability, N = D it is both necessary and sucient that

1) the set N is an open connected neighbourhood of X = 0 and N S , 2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrary positive denite q 2 Q(S  f), the equations (5.3) with (5.3a) determined along system motions have the unique solution function v with the following features: (i) v is positive denite on N , and (ii) if the boundary @ N of N is non-empty then v(X) ! +1 as X ! @ N , X 2 N, or 4) for arbitrary positive denite q 2 Q1(S  f), the equations (5.3) with (5.3a) determined along system motions have the unique solution function v with the following features: (i) v is positive denite and dierentiable on N , and (ii) if the boundary @ N of N is non-empty then v(X) ! +1 as X ! @ N , X 2 N. rrr

Proof. Necessity. Let X = 0 of the system (5.1) with the strong smoothness property have the asymptotic stability domain D. Hence, it has also the attraction domain Da , D Da . It is a neighbourhood of X = 0 in view of the strong smoothness property. Hence, Da \ S = 6 . Let us prove Da S . If @ S =  then S =
© 2004 by Chapman & Hall/CRC

and X0 2 (
Z

1

+

t

qX( X0 )]d for every (t X0 ) 2 <+  N : (5.77)

N = D and X 2 N imply X(1 X ) = 0. Let, in view of (5.2b), 0

0

v(0) = 0
so that (5.76) and (5.77) yield (5.78), vX(t X0)] =

Z
t

qX( X0 )]d +

Z




(5.78)

1

+

qX( X0)]d for every (t X0 ) 2 0
]  N :

(5.79) Positive deniteness of q on N , invariance of Da with respect to system motions, N = D = Da , B N \ B , (i) of the weak smoothness property, compactness of t
] for every t 2 0
] and 2 <+ yield

Z
  qX( X )]d < +1 for every (t
X ) 2 0
]  N
t 0

© 2004 by Chapman & Hall/CRC

0

which together with  > 0 (5.74), (5.75) and (5.78) proves jvX(t X0 )]j < +1 for every (t
X0) 2 <+  N :

(5.80)

Hence, for (t = 0) 2 <+ ,

jv(X )j < +1 for every X 2 N : (5.81) The invariance of N = D = Da , the weak smoothness property, N S , continuity of q on S due to q 2 Q(S  f) and (5.80) prove continuity of the function v on N : v(X) 2 C(N ): (5.82) 0

0

Now, (5.76), (5.77) and (5.80) complete the proof of the existence of a solution function v to (5.2) obeying (5.81) for every q 2 Q(S  f). Let now q 2 Q(1)(S  f) be arbitrarily chosen. Let vX = (@v=@XR1
: : :@v=@X2 )T and qX be analogously dened. ^ X^ = X(  X0 ). Since From (5.78) it follows that v(X) = 0 qX( X)]d + v(X), X^ 2 B  then q 2 Q(1)(S  f) implies v(X) 2 C (1) (B ) which together with b) of the weak smoothness property implies vX(  X0 )] 2 C (1)(N ) so that there is (X) 2 ]0
+1, 8X 2 N , such that kvX X(  X0 )]k < (X), 8X 2 N . Now, 1) of Denition 5.4, b) of the weak smoothness property and R compactness of 0
] imply existence of (X) 2 ]0
+1, 8X 2 N , such that k 0
qX X( X0 )]dk < (X), 8X 2 N . Altogether, kvX (X)k < (X) + (X) < +1, 8X 2 N . Hence, v(X) 2 C (1)(N ): (5.83) Now, (5.76), (5.77) and (5.80) prove the existence of a solution function v to (5.3) satisfying (5.82) for every q 2 Q(1)(S  f). Let v1 and v2 be two solutions to (5.3). Uniqueness of the system motion X(t X0) for every X0 2 N , invariance of N = Da , N S , and uniqueness of q on N due to its positive deniteness on N yield vi (X0 ) =

Z

0

+

1

wX( X0 )]d for every i = 1
2 and every X0 2 N
(5.84)

which implies v1(X0 ) = v2 (X0 ) for every X0 2 N : (5.85) This proves the uniqueness of the solution v to (5.2). The invariance of Da , N = Da , and positive deniteness of q on N and (5.78) imply v(X) > 0 for all (X 6= 0) 2 N :

(5.86)

(5.77), (5.81) and (5.85) prove the necessity of the condition 3 (i) and (5.77), (5.82) and (5.85) prove the necessity of the condition 4 (i). Let @ N be non-empty. Let ~ Xk ! X~ as k ! +1, where X1
X2
: : :
Xk
: : : be a sequence converging to X, ~ Xk 2 N and X 2 @ N . Let k = (Xk
) 2 0
+1 be the rst instant obeying X(t Xk ) 2 B for all t 2  (Xk
)
+1 and for  as dened above. The existence of such (Xk
) is guaranteed by Xk 2 N and N = D. Continuity of motions X © 2004 by Chapman & Hall/CRC

in (t
X ) 2 <  N due to (i-b) of the weak smoothness property, N S , and N = N = D imply (Xk
) ! +1 as k ! +1. Let m be such a natural number that Xk 2 (N ; B  ) for all k = m
m + 1
: : : Such m exists because N is open, B   N and Xk ! @ N as k ! +1. Let be dened by (5.86), and let  =  in 0

+

Denition 5.4:

= min q(X) : X 2 (N ; B ]:

(5.87)

Positive deniteness of q on S and N S imply 2 ]0
+1 due to 2) of Denition 5.4. Combining (5.78) and (5.86) we prove v(Xk )

Z
k 0

dt +

Z

1

+


k

qX(t Xk )]dt:

(5.88)

This inequality, positive deniteness of q on S and X(t Xk ) 2 B  for all t 2  k
+1 imply v(Xk ) k
k = (Xk
)
k = m
m + 1
: : : Since k ! +1 as k ! +1 then (5.88) shows that v(Xk ) ! +1 as k ! +1:

(5.89) (5.90)

Now, (5.89) and the denition of Xk , yield v(X) ! +1 as X ! @ N
X 2 N
which proves necessity of the condition 3 (ii) and of the condition 4 (ii). Suciency. Let all the conditions 1){3) of the theorem hold. Hence, all the condi-

tions of Theorem 4.26 (Section 4.9) due to Theorem 4.9 (Section 4.6.3) are satised, which proves that the set N is the domain of attraction of X = 0, N = Da . Since Da = D then N = D, i.e. the set N is the domain of the asymptotic stability of X = 0 of the system (5.1). Equivalency of (5.3) and (5.4), and the preceding theorem yield

Theorem 5.11 In order for the state X = 0 of the system (5.1) with the strong smoothness property to have the domain of asymptotic stability D and for a set N to be the domain of its asymptotic stability, N = D, it is both necessary and sucient that

1) the set N is an open connected neighbourhood of X = 0 and N S , 2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrary positive denite q 2 Q(S  f), the equations (5.4) have the unique solution function  with the following features:

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(i)  is positive denite on N , and (ii) if the boundary @ N of N is non-empty then (X) ! 1 as X ! @ N , X 2 N, or 4) for arbitrary positive denite q 2 Q1(S  f), the equations (5.4) have the unique solution function  with the following features: (i)  is positive denite and dierentiable on N , and (ii) if the boundary @ N of N is non-empty then (X) ! 1 as X ! @ N , X 2 N. rrr

If the system (5.1) possesses only the weak smoothness property then the conditions slightly change.

Theorem 5.12 For the state X = 0 of the system (5.1) possessing the weak smoothness property to have the domain of asymptotic stability D and for a subset N of S to be equal to D: N = D, it is both necessary and sucient that 1) the set N is an open connected neighbourhood of X = 0, 2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrary positive denite function q 2 Q(
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Proof. Necessity. Let the system (5.1) possess the weak smoothness property. Let X = 0 have the asymptotic stability domain D and let N S be equal to D: N = D. From this point on we have to repeat the proof of Theorem 5.10 to show that the conditions 1){3) or 1), 2) and 4) of Theorem 5.11 hold. In such a way we complete the proof of the necessity part. Suciency. Let the system (5.1) possess the weak smoothness property and the

conditions 1){3) or 1), 2) and 4) be valid. Then, X = 0 of the system (5.1) is asymptotically stable. Therefore, X = 0 has the domain D of asymptotic stability. Let X0 2 (
smoothness property to have the domain of asymptotic stability D and for a subset N of S to be equal to D: N = D, it is both necessary and sucient that 1) the set N is an open connected neighbourhood of X = 0, 2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrary positive denite function q 2 Q(
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Examples Example 5.14 Let f be dened by " ;X (1 ; X ; X ) ! f(X) = 2 C (< )
;X (1 ; X ; X ) 1

2 1

2 2 2

2

2 1

2 2 2

(1)

2

(5.91)

so that the system (5.1), (5.90) has the unique solution X(t X0 ) through every X0 2 <2, which is dened, continuous and dierentiable in (t
X0 ) 2 I0  <2. Hence, the system possesses the weak smoothness property (on <2). The set Se of all the equilibrium states is easily found as

Se = fX : (X = 0) or (X + X = 1)g: 2 1

Hence,

kX(t Xe )k = kXek for all t 2 < This suggests to accept the set S as

+

2 2

and every Xe 2 Se :

S = fX : X + X < 1g = B  < : 2 1

2 2

(5.92)

1

2

(5.93)

Now, (5.91) and (5.92) imply

inf kX(t X0 )k : t 2 <+ ] = kX0 k = 1 > 0 for every X0 2 @ S : This, (5.92) and the weak smoothness property (on <2 ) show that the system possesses also the strong smoothness property. Let the function q be determined by q(X) = (X12 + X22 ) 2 C (1) (<2): (5.94) Hence, it is dierentiable and positive denite on S . For any  2 ]0
1 there is  > 0 obeying  = inf q(X) : X 2 (B1 ; B )] = = inf (X12 + X22 ) : X 2 fX : 0 <   (X12 + X22 ) < 1g] =  > 0: In this case  = . Hence, 1) and 2) of Denition 5.4 are satised. The equations (5.2) with q dened by (5.93) have a solution 2 2 v(X) = 12 1 ;X1X+2 ;X2X 2
1

2

(5.95)

which is dened and continuously dierentiable on B for any  2 ]0
1. Hence, the condition 4) of Denition 5.4 is also satised, which completely veries q 2 Q1 (S  f). Let us verify now the conditions of the Theorem 5.10.

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1) the set S = B1 (5.92) is open connected neighbourhood of X = 0, 2) f(X) = 0 for X 2 S i X = 0, 3) for the function q (5.93) we concluded that it belongs to Q1 (S  f) and leads to the solution v of (5.2) in the form (5.94). The function v (5.94) obeys (i) and (ii), (i) v is positive denite and dierentiable on S , (ii) v(X) ! +1 as X ! @ S i.e. as (X12 + X22 ) ! 1]. The conditions 1), 2) and 4) of Theorem 5.1 are satised for N = S = B1 : Hence, the set N = B1 is the domain D of asymptotic stability of X = 0 of the system (5.1), (5.90):

D = B = fX : X + X < 1g: 2 1

1

2 2

Example 5.15 The second order system (5.95),

" # dX = (10 ; X 2 + 2X X ; 4X 2 ) ;2X1 + 4X2 = f(X)
(5.96) 1 2 1 2 dt X1 ; 4X2 has the unique solution X(t X0 ) through every X0 2 <2 , which is dened, continuous and dierentiable in (t
X0) 2 I0  <2. Therefore, the system possesses the weak smoothness property on <2 . It has the unique equilibrium state X = 0 in the set S = fX : X12 ; 2X1 X2 + 4X22 < 10g. Other equilibrium states compose the boundary @ S = fX : X12 ; 2X1 X2 + 4X22 = 10g of the set S . Therefore, N S . Notice that the system has also the strong smoothness property on S . We may apply any of the theorems. Let the function q be accepted in the form (5.96): q(X) = X T GX


G=

" 6 ;14 # ;14 40

= GT > 0:

(5.97)

The function q (5.96) obeys 1) and 2) of Denition 5.4. The solution v of (5.2) for such a choice of the function q is

" 1 ;1 # 10 v(X) = ln 10 ; X T HX
H = = H T > 0: (5.98) ;1 4 The function v is continuously dierentiable on S , which shows that the function q obeys 3) and 4) of Denition 5.1. Moreover, the function v is positive denite on © 2004 by Chapman & Hall/CRC

S and v(X) ! +1 as X ! @ S , X 2 S . All conditions of Theorem 5.10 through Theorem 5.13 corresponding to q 2 Q (S
f) have been veried. Any of them shows 1

that X = 0 of the system (5.95) is asymptotically stable. They clarify that the set N = S is the asymptotic stability domain of X = 0 of the system,

D = fX : X ; 2X X + 4X < 10g: 2 1

1

2

2 2

5.2.5 Exponential stability What follows is based on 85].

Introductory comment

In this section the following problem is addressed and solved.

Problem. What are the necessary and sucient conditions for an exact algorithmic

one-shot construction of a Lyapunov function v for the system (5.1) with the specic smoothness property such that it and its total time derivative v_ along system motions prove the following estimate:

kx(t x )k  kx k exp (;t)
for every (t
x ) 2 <  B
0

0

0

+

(5.99)

for some numbers  2 1
+1 and  2 ]0
+1, and for obeying (5.99),

0 <  ;1$


(5.100)

where $ > 0 satises B S .

Problem solution

In this framework the system will be assumed to possess the specic smoothness property (Denition 5.1). The complete solution to the problem reads as follows:

Theorem 5.14 Let the system (5.1) possess the specic smoothness property and

$ > 0 obey B S . In order for system solutions to obey (5.98) for some numbers  2 <+ ,  1,  2 <+ and given by (5.99) it is necessary and sucient that for arbitrarily chosen positive denite dierentiable function p obeying (5.100), 1 kX kk  p(X)  2kX kk
8X 2 B
© 2004 by Chapman & Hall/CRC

(5.101)

for some numbers i 2 <+ , i = 1
2, and some natural number k there exists a unique solution function v to (5.101) with (5.101a) determined along system motions,

dv(X) = ;p(X)
8X 2 B
 dt v(0) = 0


(5.102a) (5.101b)

which has the following properties: (i) v is dierentiable on B : f(X) 2 C (1)(B ), and (ii) it satises

1 kX kk  v(X)  2kX kk
8X 2 B
for some numbers i 2 <+ , i = 1
2.

(5.102)

rrr

Proof. Let the system (5.1) possess the specic smoothness property and $ > 0 obey B S . 

Necessity. The proof of the necessity part is a modication of the proof of Theo-

rem 11.1 by Krasovskii 116]. Let the estimate (5.98) hold for some numbers  2 <+ ,  1,  2 <+ , for obeying (5.99). Let us choose arbitrary dierentiable positive denite function p on B obeying (5.100) for some positive numbers i, i = 1
2, and some natural number k. Let a function v be determined by (5.101). Then, vX(t X0 )] ; v(X0 ) = ;

Zt 0

pX(  X0)]d
8(t
X0) 2 <+  B :

This, (5.98) and (5.101b) imply (5.103): v(X0 ) =

Z

1

+ 0

pX(  X0 )]d
8X0 2 B :

(5.103)

The specic smoothness property (Denition 5.1) guarantees the existence and uniqueness of motions of the system (5.1) on <+ for every X0 2 B . Besides, it ensures validity of (5.104) (inequality (4.11) by Krasovskii 116]), kX(t X0 )k kX0 k exp (;n2 t)
8(t
X0) 2 <+  B : (5.104) Let 1 = (k n2 );1 1 2 <+ and 2 = k (k);1 2 2 <+ . Now, (5.98), (5.99), (5.103) and (5.104) imply (5.105): 1kX0 kk  v(X0 )  2kX0 kk
8X0 2 B : (5.105) By substituting X0 with X, X 2 B , in (5.105) we prove (5.102). Besides, (5.103) and (5.105) for X0 = X prove that v is dened on B . Continuity and dierentiability of X(t X0) on <+  B due to the specic smoothness property, together with © 2004 by Chapman & Hall/CRC

(5.101a) and (5.103) prove continuity and continuous dierentiability of the function v on B (for all the details see the proof of Theorem 11.1 by Krasovskii 116]). Furthermore, uniqueness of the motions X of the system (5.1), and (5.103) prove the uniqueness of the solution function v to (5.101), which completes the proof of the necessity part. Suciency. Let all the conditions of the theorem to be proved hold. From (5.100), (5.101a) and (5.102) we derive: v_ X(t X0 )]  ;12;1 vX(t X0 )]
8t
X(t X0 )] 2 <+  B : A solution to this inequality satises the following: vX(t X0 )]  v(X0 ) exp (;1 2;1 t)
8t
X(t X0)] 2 <+  B : Combining this with (5.102) we derive (5.106a) kX(t X0)k  kX0 k exp (;t)
8(t
X0 ) 2 <+  B
and  = (1;1 2)1=k 2 1
+1
 = 1 (k2);1 2 <+ : (5.106b) Let now be chosen as 0 <  ;1$
(5.106c) so that kX0 k < implies kX0 k < $ that together with (5.106a) proves (5.98), and (5.106c) proves (5.99).

Example 5.16 Let a third order system be described by

0 ;x ; 2x + 6x x ; 10x x dX = B B ;10x x ; x ; 2x + 6x x dt B @ 3 1

1

2 1 2

2 1 2

2

2 1 3

3 2

2 2 3

6x21x3 ; 10x22x3 ; x3 ; 2x33

1 CC CA = f(X):

The function f is dierentiable on <3 : f(X) 2 C (1) (<3). Its Jacobian fX has bounded norm on B for any $ 2 <+ , that is for any $ 2 <+ there is 2 <+ , = ($) such that sup kfX (X)k : X 2 B ] < ($) < +1: Hence, the system possesses the specic smoothness property (Denition 5.1) with S = B . Let at rst p(X) = 2kX k2 (1 + 2kX k2): The function p obeys (5.100) for k = 2
1 = 2
2 = 2(1 + 2$2):

© 2004 by Chapman & Hall/CRC

For such a function p, the solution v to (5.101) is v(X) = kX k2 : Hence, the function v obeys Theorem 5.14 for 1 = 2 = 1
so that (5.98) is satised for  = 1
 = 4(1 +12$2)
and for 2 ]0
$]. Let the second choice of p be p(X) = 2kX k2 (1 + 2kX k2 )2 : so that (5.100) holds for k = 2
1 = 2
2 = 2(1 + 2$2)2 : The solution v to (5.101) is found as v(X) = kX k2 (1 + kX k2 ): It obeys (5.102) for 1 = 1
2 = 1 + $2: In this case (5.98) is valid for




p  = 1 + $2 2 1
+1
 = 1 +1$2



$ . and for 2 0
p 1 + $2 If we select

k2
p(X) = 1 +2k2XkX k2 © 2004 by Chapman & Hall/CRC

which yields k = 1 = 2 = 2
then the solution v to (5.101) is 2 v(X) = 1 +kX2kkX k2 :

It satises (5.102) for 1 = 1 +12$2
2 = 1
so that (5.98) holds for  = (1 + 2$2) 2 1
+1
 = 1
= 1 +$2$2 : This example shows dependence of v, ,  and on the choice of the function p.

5.3 Systems with continuous motions (generalised motions) 5.3.1 Smoothness property

Denition 5.5 Weak smoothness property: (i) There is an open neighbourhood S of X = 0, S
(a) the system (5.1) has the unique solution X(t X0 ) through X0 at t = 0 over a time interval I0, and (b) the motion X(t X0) is dened and continuous in (t
X0 ) 2 I0  S . (ii) For every X0 2 (
Strong smoothness property:

(i) The system (5.1) has the weak smoothness property. (ii) If the boundary @ S of S is non-empty then every motion of the system (5.1) passing through X0 2 @ S at t = 0 obeys inf kX(t X0 )k : t 2 I0 ] > 0 for every X0 2 @ S .

Throughout this section (5.3) a smoothness property will be used in the sense of Denition 5.5 (rather than in the sense of Denition 5.1).

© 2004 by Chapman & Hall/CRC

5.3.2 Approach via -uniquely bounded sets O

This section exposes results and examples from 80]. In what follows we shall make use of the following denition.

Denition 5.6 A function u :
D v(X) = ;u(X)
+

v(0) = 0


(5.107a) (5.107b)

have a solution v that is well dened in < and continuous for every X 2 B  for some  2 ]0
+1,  = (u
f).

Theorem 5.15 In order for the state X = 0 of the system (5.1) with the strong smoothness property to have the domain of asymptotic stability D and for a set N , N
1) the set N is an open connected neighbourhood of X = 0 and N S , 2) f(X) = 0 for X 2 N i X = 0, and either 3) for arbitrarily selected O-uniquely bounded set U , U S , with the generating function u on S obeying u 2 E (S  f), the equations (5.107) have the unique solution function v on N with the following properties: (i) v is positive denite on N , and (ii) if the boundary @ N of N is non-empty then v(X) ! +1 as X ! @ N , X 2 N, or 4) for arbitrarily selected O-uniquely bounded set U , U S , with the generating function u on S obeying u 2 E (S  f), the equations (5.108) with (5.108a) taken along system motions,

D (X) = ;u(X) 1 ; (X)]
+

(0) = 0


(5.108a) (5.108b)

have the unique solution function  on N with the following properties:

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(i)  is positive denite on N , and (ii) if the boundary @ N of N is non-empty then (X) ! 1 as X ! @ N , X 2 N. rrr

Proof. 1){3) of the statement are proved in the same way as 1){3) of Theorem 5.6. Equivalency of (5.107) and (5.108) then proves 4).

The conditions slightly change if the system (5.1) possesses the weak smoothness property rather than the strong smoothness property.

Theorem 5.16 For the state X = 0 of the system (5.1) possessing the weak smoothness property to have the domain of asymptotic stability D and for a subset N of S , N S , to be equal to D: N = D, it is both necessary and sucient that 1) the set N is an open connected neighbourhood of X = 0, 2) f(X) = 0 for X 2 N i X = 0, and either 3) for arbitrarily selected O-uniquely bounded set U , U S , with the generating function u on
Proof. This theorem is proved in the same way as Theorem 5.8 and Theorem 5.9. Example 5.17 Let n = 1, dx = ;x + h(x)
h(x) = dt

© 2004 by Chapman & Hall/CRC

( xjxj for jxj 2 0
1]


xpx for jxj 2 1
+1:

The system possesses the strong smoothness property because f(x) = ;x + h(x) is Lipschitzian on <1. The equilibrium states are xe1 = ;1, xe2 = 0 and xe3 = +1. They suggest S = ] ; 1
+1 and U = fx : jxj < g = ] ; 
+ for  2 ]0
1. The generating function u on N , u(x) = jxj, of the O-uniquely bounded set U and (5.2) yield D+ v(x) = ;jxj
x 2 S : Its solution v is v(x) = ; ln (1 ; jxj)
x 2 S : The function v(x) and the set N = S = ] ; 1
+1 obey all the requirements of Theorem 5.15, that is that 1) N = ] ; 1
+1 is an open connected neighbourhood of X = 0 and N = S , 2) f(x) = ;x + h(x) = 0 for x 2 N i x = 0, 3) (i) v(x) = 0 for x 2 N i x = 0, v(x) 2 C(N ), v(x) > 0 for every (x 6= 0) 2 N , which prove positive deniteness of v on N , (ii) v(x) ! +1 as x ! @ N = f;1
+1g, x 2 N . Hence, N = ] ; 1
+1 is the domain D of asymptotic stability of x = 0, D = ] ; 1
+1: Notice that jf(x)j = jxj(1 ; jxj), x 2 N , is not a generating function on N of any O-uniquely bounded set because it is not radially increasing on N .

Example 5.18 Let the function h be dened as in Example 5.17 and

dx = x ; h(x): dt It is clear that the system possesses the strong smoothness property on <1 and has the equilibrium states xe1 = ;1, xe2 = 0 and xe3 = +1 (see Example 5.17). Let again U = fx : jxj < g = ] ; 
+ for  2 ]0
+1 so that u(x) = jxj. From (5.107a) and u(x) = jxj we get D+ v(x) = ;jxj
x 2 N : Integrating this equation along motions of the system we derive v(x) = ln (1 ; jxj)
x 2 N
which is negative denite on N and, thus, does not satisfy the necessary and sufcient conditions for asymptotic stability of x = 0 of the system. Hence, x = 0 of the system is not asymptotically stable and does not have the asymptotic stability domain.

© 2004 by Chapman & Hall/CRC

5.3.3 General one-shot approach

In what follows we shall extend the general one-shot approach to systems with continuous (generalised) rather than with dierentiable motions 77], 82]. In this framework we shall use the family E (S  f) dened as follows: Denition 5.7 E (S  f) is the family of all the functions e, e :
1) e is continuous on S : e(X) 2 C(S ), 2) for any  > 0 for which the set S is neighbourhood of B there is  > 0,  = ( e S ), which obeys

inf e(X) : X 2 (S ; B )] = 
3) there is  2 <+ ,  = (e f), such that there exists a solution v to the equations (5.109) with (5.109a) taken along motions of the system (5.1),

D v(X) = ;e(X)
+

v(0) = 0


(5.109a) (5.109b)

which is dened in < and continuous on B , or equivalently, there exists a solution  to the equations (5.110), with (5.110a) taken along motions of the system (5.1),

D (X) = ;e(X) 1 ; (X)]
+

(0) = 0


(5.110a) (5.110b)

which is dened in < and continuous on B .

Theorem 5.17 In order for the state X = 0 of the system (5.1) with the strong smoothness property to have the domain of asymptotic stability D and for a set N , N
1) the set N is an open connected neighbourhood of X = 0 and N S , 2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrarily selected positive denite function e obeying e 2 E (S  f), the equations (5.109) equivalently (5.110)] have the unique solution function v equivalently  ] on N with the following properties: (i) v equivalently  ] is positive denite on N , and (ii) if the boundary @ N of N is non-empty then v(X) ! +1 equivalently (X) ! 1], respectively, as X ! @ N , X 2 N . rrr

© 2004 by Chapman & Hall/CRC

Proof. This theorem is proved in the same way 77], 82] as Theorem 5.10. Theorem 5.18 For the state X = 0 of the system (5.1) with the weak smoothness property to have the domain of asymptotic stability D and for a subset N of S , N S , to be equal to D: N = D, it is both necessary and sucient that 1) the set N is an open connected neighbourhood of X = 0, 2) f(X) = 0 for X 2 N i X = 0, and 3) for arbitrarily selected function e obeying e 2 E (S  f) the equations (5.109) equivalently (5.110)] have the unique solution function v equivalently  ] on N with the following properties: (i) v equivalently  ] is positive denite on N , and (ii) if the boundary @ N of N is non-empty then v(X) ! +1 equivalently (X) ! 1], respectively, as X ! @ N , X 2 N . rrr

Proof. The proof is carried out in the same way as that of Theorem 5.12 77], 82]. Example 5.19 Let the second order system be given by dx = ; (10 ; jx j) (6 ; jx j) x = f (X)
dt 30 + 2jx j dx = ; (10 ; jx j)(6 ; jx j) x = f (X)
dt 12 + 3jx j 1

1

2

2

2

2

1

2

1

1

1

2

2

2

f = (f1 f2 )T : The system has a unique (generalised) motion X(t X0 ) for every X0 2 <2 , which is continuous in t 2 <+ . The set Se of the equilibrium states is found as Se = X : X 2 <2
(X = 0) or (jx1j = 10 and x2 2 <) or (x1 2 < and jx2j = 6)g : The set Se suggest the set S as

S = X : X 2 <2
jx1j < 10 and jx2j < 6 : Since the set Se is positive invariant and @ S  Se then it is easy to verify that the system possesses the strong smoothness property with S determined above. Let e(X) = jx1j + jx2j
which yields the solution to (5.109) as x2j : v(X) = (10 ;3jxjx1j j+)(62j; jx2j) 1 © 2004 by Chapman & Hall/CRC

The function v is well dened on B for  2 ]0
5:99. Since inf e(X) : X 2 (S ; B )] =  > 0
for any  2 ]0
5:99
then we conclude that the positive denite function e obeys e 2 E (S  f). The function v satises all the conditions of Theorem 5.17 on S . Since X = 0 is a unique equilibrium state of the system in S , then we may conclude that the set S is the domain D of asymptotic stability of X = 0, N = S = D = fX : X 2 <2
jx1j < 10 and jx2j < 6g:

Example 5.20 Let the system (5.1) take the following specic form: dx1 = ;x (1 + x2) cos2 jx j = f (X)
1 1 1 2 dt dx2 = ;x (1 + x2)(1 + jx j) = f (X)
2 1 2 2 dt f = (f1 f2 )T : The system possesses the weak smoothness property on <2 : S = <2 . Let e(X) = jx1j(1 + x22) + jx2j(1 + jx1j): It satises 1) and 2) of Denition 5.7 on <2 . Furthermore, the solution function v to (5.109) is obtained in the form of v(X) = tan jx1j + arctan jx2j
which completes the proof of e 2 E (<2  f). The function v is positive denite on N , n o N = X : X 2 <2
jx1j < 2 and x2 2 < : The functions e and v, and the set N full all the conditions of Theorem 5.18. Hence, X = 0 of the system has the domain D of asymptotic stability given by D = N , that is n o D = X : X 2 <2
jx1j < 2 and x2 2 < :

Example 5.21 Let

dx1 = ;x (1 + x2 )(2 + sin x ) = f (X)
1 2 1 1 dt dx2 = ;x3 exp jx j (4 + cos x ) = f (X)
2 1 2 2 dt f = (f1 f2 )T :

© 2004 by Chapman & Hall/CRC

This system has the weak smoothness property on <2 . The function e, e(X) = jx1j(2 + sin x2 ) + jx2j3(4 + cos x1 )
obeys e 2 E (<2 f) because it yields the solution v to (5.109) in the following form: v(X) = arctan jx1j + 1 ; exp (;jx2j): The functions v and e, and the set N = <2 satisfy all the conditions of Theorem 5.18. Therefore, X = 0 of the system has the domain of asymptotic stability, D = <2 : This means that x = 0 is globally asymptotically stable, which is proved by using non-radially unbounded function v. This is possible because v is solution to (5.109) for e 2 E (<2 f) and X = 0 is the unique equilibrium state in <2 . The obtained result does not contradict the Barbashin{Krasovskii theorem 6] on asymptotic stability in the whole, which requires only the existence of a globally positive denite radially unbounded function with globally negative denite derivative along system motions. In order to show that such a function v can be generated by using Theorem 5.18 and (5.109) let the function e have the form either under (i) or (ii), (i) e(X) = 2x21(1 + x21)(2 + sin x2 ) + 2x42 exp jx2j (4 + cos x1 ), which yields the solution v to (5.109) as v(X) = kX k2
(ii) e(X) = jx1j(1 + x21)(2 + sin x2 ) + jx2j3 exp jx2j (4 + cos x1 ), which leads via (5.109) to v(X) = jx1j + jx2j: In both cases the function v is globally positive denite and radially unbounded.

5.4 Conclusion The new methodology based on the Lyapunov (direct) method 127] presented in this chapter for nonlinear systems described by (5.1) and with a corresponding smoothness property, is characterized by the following in the case where X = 0 is unique equilibrium state in its neighbourhood N : 1. A function q can be arbitrarily chosen from an appropriate functional family so that q is positive denite. 2. Either properties of a solution v to v_ = ;q with v(0) = 0, or properties of a solution  to _ = ;q(1 ; ) with (0) = 0 are to be tested. If and only if, for a chosen positive denite q from the functional family determined in the chapter for dierent cases, the solution v (or ) is positive denite then X = 0 is asymptotically stable. Hence, if v (or ) is not positive denite then X = 0 is not asymptotically stable. The procedure ends in either case after testing properties of v (or ) only once.

© 2004 by Chapman & Hall/CRC

3. If we wish to verify whether an open connected neighbourhood N of X = 0 is the domain of asymptotic stability of X = 0 then, in addition to positive deniteness of v (or ) on N , it is both necessary and su cient that v(X) ! 1 or (X) ! 1] as X ! @ N , X 2 N . In this way we achieve both a) an exact construction of a system Lyapunov function, and b) an exact determination of the asymptotic stability domain of X = 0. The same holds in the case where we test asymptotic stability of a set 81]. The methodology presented herein is inverse to Lyapunov's methodology for nonlinear systems 127] that starts with a choice of v (or ) and verication of properties of v_ (or _ ). If v_ (or ) _ does not possess the requested properties, the v (or ) should be reselected without any guideline how to reselect them. Therefore, Lyapunov's original methodology allows the procedure to be endless. It should be noticed that Lyapunov's original methodology for nonlinear systems is inverse to his own methodology for time-invariant linear systems 127]. However, the methodology presented in this chapter for nonlinear systems is the same as Lyapunov's methodology for time-invariant linear systems. This ensures consistency of the methodology based on the Lyapunov method and makes it complete. It should be also noticed that the methodology presented herein separates the stability problem reduced to the problem of testing the properties of v (or ) from the purely mathematical problems of the existence of the solution v to v_ = ;q with v(0) = 0 or  to _ = ;q(1 ; ) with (0) = 0] and of solving these equations. These mathematicalproblems are not considered in this chapter but in the preceding chapter (Section 4.9.4) and deserve further mathematical research.

© 2004 by Chapman & Hall/CRC

Chapter 6

Foundations of practical stability domains 6.1 Introductory comment A practical stability analysis is based on the Lyapunov method in a broad sense. Qualitative properties of the system (generalised) motions are studied via behaviour of a continuous function v along the motions by using its total time derivative rather than to determine the motions themselves. This is the basis of the Lyapunov method in a broad sense. Its closer sense requires the function v to be positive denite, and generally its total time derivative to possess a (semi-)deniteness property. Such demands are not imposed on the function v in the framework of practical stability. We use instead the extremal (minimal or maximal) values of the function v on appropriate sets or on their boundaries. Then, by using the total time derivative of v, we are able to compare instantaneous values of v along motions with its extremal values of importance over an accepted (nite or innite) time interval. What follows is further development of the methodology established by Weiss and Infante 208], Michel 144]{146] and Gruji&c 60]{62], and opening of new practical stability study compared with that of 11], 29], 43], 44], 52], 60]{63], 91], 93], 99], 105], 109]{111], 121], 133], 144]{146], 150], 152], 156], 179], 194], 205], 207]{209]. The novelty is related to conditions for a set to be an estimate of the domain of a requested practical stability property.

6.2 System aggregation function and sets 6.2.1 System description and sets

We follow Sections 1.2 and 2.3 so that systems to be studied are generally in forced regimes and described by (6.1), dX = f(X
i)
f :
Our interest is in system (generalised) motions X(t X0 i) over a (nite or innite) time interval <
= 0
 <+ with respect to some of, or all of the following sets possessing a non-empty interior: XA is the set of admitted system states over <
, XF is the set of admitted system states over T
s =  s
 <
, where s 2 <
so that s = 0 implies XF = XA and s = means XF =
6.2.2 De nition of estimates of practical stability domains of systems

Denition 6.1 (a) A set S is an estimate of the domain Dps (
XA
I ) of practical stability of the system (6.1) with respect to f
XA
Ig, which is denoted by Eps (
XA
I ), S = Eps (
XA
I ), if and only if (i) the interior S of S is non-empty and S is connected, (ii) S is subset of Dps(
XA
I ): S Dps (
XA
I ). (b) A set S is an estimate of the domain Dpc (
s
XF
I ) of practical contraction with the settling time s of the system (6.1) with respect to f
XF
Ig, which is denoted by Epc (
s
XF
I ), S = Epc (
s
XF
I ), if and only if (i) the interior S of S is non-empty and S is connected, (ii) S is subset of Dpc (
s
XF
I ): S Dpc (
s
XF
I ). (c) A set S is an estimate of the domain Dp (
s
XA
XF
I ) of practical stability with the settling time s of the system (6.1) with respect to f
XA
XF
Ig, which is denoted by Ep (
s
XA
XF
I ), S = Ep (
s
XA
XF
I ), if and only if

(i) the interior S of S is non-empty and S is connected, (ii) S is subset of Dp (
s
XA
XF
I ).

(d) The arguments
s
XA
XF and I may be omitted if and only if they are prespecied and xed.

Note 6.1 If Eps and Epc are estimates of Dps and Dpc, respectively, then for S = Eps \ Epc to be an estimate Ep of Dp it is both necessary and su cient that S is connected with non-empty interior S. © 2004 by Chapman & Hall/CRC

6.2.3 System aggregation function extrema and sets Let Z XA denote a nite set or Z = .

Denition 6.2 A function v : <
1) v(t
X) 2 C<+  (
Note 6.2 A Lyapunov function of a system is its aggregation function, but vice versa need not be true.

Note 6.3 Since v can depend on time t we present the denition of D v(t
X) as   D v(t
X) = lim sup vt + 
X(t +  t
X i)] ; v(t
X) :  ! 0 : +

+

+



If v(t
X) 2 C (11)(<+ 








vM ( ) (t) = sup v(t
X) : x 2 X( )
vM ( ) (t) = sup v(t
X) : x 2 X ( )
  vm( ) (t) = inf v(t
X) : x 2 X( )
vm( )(t) = inf v(t
X) : x 2 X ( )


  = sup D v(t
X
i) : (t
X
i) 2 <
 X  I :

vm@ ( ) (t) = inf v(t
X) : x 2 @ X( )
v_ M ( )


+

()

Example 6.1 Let XA = fX : kX k < eg = Be, XF = fX : kX k < 1g = B and v(t
X) = 2 ln (1 + t)kX k] 2 C<  (
+

© 2004 by Chapman & Hall/CRC

and vMF (t) = sup v(t
X) : x 2 XF ] = sup f2 ln (1 + t)kX k] : X 2 B1 g = 2 ln (1 + t)
vMf (t) = sup v(t
X) : x 2 X f ] = sup f2 ln (1 + t)kX k] : X 2 B1 g = 2 ln (1 + t)


vmF (t) = inf v(t
X) : x 2 XF ] = inf f2 ln (1 + t)kX k] : X 2 B1 g = ;1
vmF (t) = inf v(t
X) : x 2 X F ] = inf f2 ln (1 + t)kX k] : X 2 B1 g = ;1
@ (t) = inf v(t
X) : x 2 @ XF ]=inf f2 ln (1+t)kX k] : X 2 @ B1 g = 2 ln (1 + t): vmF

6.3 Estimate of the system practical stability domain

A set Z , Z 
;

@ (t) < v(t
X), 8(t
X) 2 <

Proof. Let all the conditions hold. From the conditions under 2) we easily derive @ (t) ; v (t)
8(t
X ) 2 <
 S
v(X ) = v(0
X ): vt
X(t X  i)] < v(X ) + vmA MS 0

0

0

By the denition of v-extrema: v(X0 )  vM S
8X0 2 S
vM S = vM S (0): The preceding inequalities yield @ (t)
8(t
X0 ) 2 <
 S : vt
X(t X0  i)] < vmA © 2004 by Chapman & Hall/CRC

0

0

From X(t X0  i) 2 C(<
 XA  I ), connectedness of XA , S XA and X0 2 S it follows that X(t X0  i) can leave X A only by passing through its boundary @ XA . Let be assumed that there is (X0
i
) 2 S  I  <
such that X(  X0 i) 2 @ XA . From the last inequality we deduce @ ( ) v
X(  X0 i)] < vmA

that implies X(  X0  i) 62 @ XA . This contradicts the denition of (X0
i
) 2 S  I  <
. Hence, such (X0
i
) 2 S  I  <
does not exist, which proves X(t X0  i) 2 XA for all t 2 <
and every (X0
i) 2 S  I : This, connectedness of S , S 6=  and S XA prove

S = Eps (
XA
I ): Let

i

h

v_ MAS = sup D+ v(t
X
i) : (t
X
i) 2 <
 (X A ; S)  I :

Theorem 6.2 Let the system (6.1) have generalised motions X(t X  i) 2 Ct(<
 X A  I ). In order for a compact set S , S XA , to be the estimate Eps (
XA
I ) of the domain Dps (
XA
I ) of practical stability of the system (6.1) with respect to f
XA
Ig: S = Eps(
XA
I ), XA = X A it is sucient that 1) S is connected and with non-empty interior: S = 6 , 0

and 2) there are at most nite subset Z of S and a function v : <
 X A ! (<  f;1g) such that

@ (t) < v(t
X), 8(t
X) 2 <
;
Proof. Let all the conditions hold. Let (i
X ) 2 I  S be such that there are t 2 ]0
 and  2 ]t
], t = t (X  I
S ),  = (X  I
S  t ), which obey X(t X  i) 2 (XA ; S), 8t 2 t
, and  determines the largest interval t
. 0

1

1

1

0

1

0

0

1

1

1

From the denition of v_ MAS follows

D v(t
X
i)  v_MAS
8(t
X
i) 2 ]t
(X A ; S)  I : +

1

Integrating this inequality we get vX(t X0  i)] ; v(X0 )  t_vMAS
8t 2 ]t1

which with the conditions under 2) of the theorem and the denition of t1 yields @ (t) + v(X0 ) ; v (0)
8(t
X0
i) 2 ]t1
S  I : vX(t X0  i)]  vmA MS

© 2004 by Chapman & Hall/CRC

x3

αb-13 -1

−αb 1 -1

-1

−αb 2

αb 2

-1

x2

XA

αb 1

-1

−αb 3 x1

Figure 6.1: The set XA = fX : bT jX j  g, (6.2), in the case n = 3.

By the denition vM S (0) v(X0 )
8X0 2 S : The preceding two inequalities combined imply @ (t)
8(t
X0
i) 2 ]t1
S  I : vX(t X0  i)]  vmA This, X(t X0  i) 2 Ct(<
 XA  I ), 2a) and the denitions of t1 and  prove X(t X0  i) 2 XA
8(t
X0
i) 2 <
 S  I
which together with the condition 1) proves that S = Eps (
XA
I ) (Denition 6.1a). Let (Fig. 6.1)

XA = fX : bT jX j =

n X

bijxij  g
 2 <+
b > 0


(6.2)

f(0) = 0 then the following transformation by Michel et al. 148] is valid: f(X
i) = H(X
i)X


(6.3)

i=1

where b = (b1 b2 : : : bn)T . If

(6.4a)

H( ) :
8 fi (X
i)xj 9 <
kX k 6= 0
= hij (X
i) = : kX k
0
kX k = 0  2

© 2004 by Chapman & Hall/CRC

(6.4b)

so that the system (6.1), (6.3) can be set in the form (6.5), dX = H(X
i)X: dt Let A = (aij ) 2
(6.5) (6.6a)

aij = ij sup hii(X
i) : (X
i) 2 (XA ; S)  I ] + + (1 ; ij ) max f0
sup hij (X
i) sign xj : (X
i) 2 (XA ; S)  I ]g: (6.6b) Theorem 6.3 Let the system (6.5) have generalised motions X(t X0 i) 2 Ct(<
 XA  I ). In order for the set S = fX : bT jX j  g, 2 ]0
1], to be estimate Eps (
XA
I ) of the domain Dps (
XA
I ) of practical stability of the system (6.5) with respect to f
XA (6:2)
Ig: S = Eps(
XA
I ), it is sucient that t  (1 ; )
8t 2 <

where

=

n X





max 0
 bck
k k=0

and c = (c1 c2 : : :cn )T is determined by c = AT b:

rrr

Proof. Under the conditions of the theorem statement it follows that 2a) of Theo@ =  for v(X) = bT jX j, and rem 6.2 is valid, vM S =  = , vMA _ D v(X
i) = bT D jX j = bT (X
X)H(X
i)X  bT AjX j = cT jX j 
+

+

8(X
i) 2 X A  I
so that v_ MAS = : Hence, the condition t  (1 ; ), 8t 2 <
, can be set in the following form: @ ; t_v vM S  vMA MAS
8t 2 <

which shows that the condition 2b) of Theorem 6.2 holds. The set S = fX : bT jX j  g obviously is connected with non-empty interior because  2 <+ and b > 0 so that the condition 1) of the Theorem 6.2 is also satised. All conditions of the Theorem 6.2 are valid, which proves the statement of Theorem 6.3.

© 2004 by Chapman & Hall/CRC

x3 -2 αp33

-2

−αp11 -2

αp22

-2 −αp22

x2

-2

αp11

-2 −αp33

x1

XA

Figure 6.2: The set XA = fX : X T PX  g, (6.7), in the case n = 3.

Comment 6.1 Stability of the matrix A is NOT required in Theorem 6.3. This is due to possible nite : < 1, and represents an essential dierence from Lyapunov's conditions.

Let (Fig. 6.2)

XA = fX : X T P X  g
 2 <
P = P T = (pij ) > 0

(6.7)

+

and K = (kij ) 2
(6.8a)

kij = ij sup wii(X
i) : (X
i) 2 (XA ; S)  I ] +

+ (1 ; ij ) maxf0
sup wij (X
i) sign(xi xj ) : (X
i) 2 (XA ; S)  I ]g
(6.8b) W ( ) = wij ( )] = H T ( )P + P H( ):

(6.8c)

Theorem 6.4 Let the system (6.5) have generalised motions X(t X  i) 2 Ct(<
 XA  I ). In order for the set S = fX : X T PX  g, 2 ]0
1] to be estimate Eps (
XA
I ) of the domain Dps (
XA
I ) of practical stability of the system (6.5) with respect to f
XA (6:7)
Ig: S = Eps(
XA
I ), it is sucient that -t  (1 ; )
8t 2 <

0

where

8 (K) > < - = > (P) : 0

9 > = > (K)  0: 

if (K) 0
if

rrr © 2004 by Chapman & Hall/CRC

x3

αb3 0

αb2

x2

αb1

XA x1









Figure 6.3: The set XA = X : max jxb i j : i = 1
2
: : :
n   , (6.9), in case n = 3. i

Proof. The set XA (6.7) obeys the condition 1) of Theorem 6.2 due to  2 < and P

= PT

> 0. Let

v(X) = X T P X

so that 2a) of Theorem 6.2 holds,

+

@ = vM S =  = 
vMA

and





v_ (X
i) = X T P H(X
i) + H T (X
i)P X = = X T W (X
i)X  jX jT K jX j
8(X
i) 2 XA  I : This result, the fact that kX k2  = (P ) for all X 2 XA and the denition of -, yield v(X
_ i)  -
8(X
i) 2 XA  I
v(X
_ i)  -
8(X
i) 2 S  I : Hence, v_ MAS = -: @ =  together with the condition -t  (1 ; ), This result, vM S =  and vMA 8t 2 <
, yield @ ; t_v vM S  vMA MAS
8t 2 <
:

All the conditions of Theorem 6.2 hold, which completes the proof of Theorem 6.4. Let (Fig. 6.3)

    XA = X : max jxb ij : i = 1
2
: : :
n  
 2 <
b > 0: i

© 2004 by Chapman & Hall/CRC

+

(6.9)

We will look for an estimate of the practical stability domain in the same form. Therefore,     S = X : max jxbij : i = 1
2
: : :
n  
i 2 ]0
1]
i = 1
2
: : :
n
(6.10) i i is accepted for such a tentative estimate. Let c = Ab
c = (c1 c2 : : : cn )T : (6.11)

Theorem 6.5 Let the system (6.5) have generalised motions X(t X  i) 2 Ct(<
 Xa  I ). In order for the set S , (6.10), to be estimate Eps(
XA
I ) of the domain Dps (
XA
I ) of practical stability of the system (6.5) with respect to f
XA (6:9)
Ig: 0

it is sucient that

ci  (1 ; i)bi
i = 1
2
: : :
n:

rrr

Proof. Let the condition of the theorem hold. The set S (6.10) is compact and connected. This and X(t X  i) 2 Ct(<
 Xa  I ) show that any motion starting from the set S (6.10), which leaves the set S after some time t 2 ]0
 must be at the boundary @ S of the set S at some time t 2 <
. Let t 2 0
 and  2 ]0
] 0

1

obey

1

X(t X0  i) 2 (XA ; S)
8t 2 t1

where  determines the largest interval t1
. Now we estimate the upper bound of D+ jxij along motions of the system (6.5) by using (6.6),

D jxij  +

n X j =1

and make use of n X j =1

aij jxj j  

by combining them,

D jxij   +

aij jxj j
8X 2 (XA ; S )
i = 1
2
: : :
n: n X j =1

n X j =1

aij bj
8X 2 (XA ; S )
i = 1
2
: : :
n


aij bj
8X 2 (XA ; S )
i = 1
2
: : :
n:

After integrating this inequality from t1 to t 2 ]t1
 we derive jxi (t X0 i)j ; jxi (t1  X0 i)j  (t ; t1 )ci
8(t
X0
i) 2 t1
S  I
i = 1
2
: : :
n
where ci is the i'th element of c = Ab.

© 2004 by Chapman & Hall/CRC

(6.12)

If ci  0 then (6.12) yield jxi (t X0 i)j  jxi (t1 X0  i)j
8(t
X0
i) 2 t1
S  I
ci  0: (6.13a) Hence, the assumed t1 does not exist. If ci 0 then (t ; ti)ci < so that from (6.12) follows: (6.13b) jxi (t X0 i)j  jxi (t1  X0 i)j +  ci = ( i bi + ci)
ci 0
where the equality is due to jX(t1  X0 i)j 2 @ S that yields jxi (t X0 i)j =  i bi. If t1 as dened above does not exist then X0 2 S , X(t X0 i) 2 Ct(<
 Xa  I ), connectedness and compactness of S imply X(t X0  i) 2 S , 8t 2 <
. In that case X(t X0 i) 2 XA , 8t 2 <
, due to S XA. Hence, we continue considering (6.13b). Since ( ibi + ci)  bi ,8i = 1
2
: : :
n, due to the condition ci  (1 ; i )bi and  2 <+ , then (6.13b) yields jxi (t X0 i)j  
c 0: i bi This together with connectedness and compactness of XA and X(t X0  i) 2 Ct(<
 Xa  I ), and (6.14) ensure that X(t X0  I ) 2 XA also in the case when there exist t1 and  as dened above. Altogether, X(t X0  i) 2 XA
8(t
X0
i) 2 <
 S  I
which proves the theorem in view of Denition 6.1(a). It is interesting to note that the set XA is O-uniquely bounded in all three cases with the generating function u( ) determined by a) u(X) = bT jX j in case XA is dened by (6.2), b) u(X) = X T P X in case XA is dened by (6.7),   c) u(X) = max jxb ij : i = 1
2
: : :
n in case XA is dened by (6.9). i We terminate this section with a result corresponding to the general case of a uniquely bounded set XA with a generating function u( ) (Denition 4.13, section 4.3.2): 8 9 > > < 
8X 2 XA > > < = u(X) = 
8X 2 @ XA (6.14) > > : > 
8X 2 (
© 2004 by Chapman & Hall/CRC

Theorem 6.6 Let the system (6.1) have generalised motions X(t X  i) 2 Ct(<
 Xa  I ) where XA is a closed uniquely bounded set with the generating function u, (6.14). In order for the set S , (6.15), to be estimate Eps (
XA
I ) of the domain Dps (
XA
I ) of practical stability of the system (6.1) with respect to f
XA (6:14)
Ig 0

it is sucient that

tu_ MAS = (1 ; )
8t 2 <
:

rrr

Proof. Let all the conditions hold. Hence, S is connected with a non-empty interior because 2 ]0
1] and due to Lemma 4.11 (section 4.3.3). The denitions (6.14) and (6.15) of XA and S yield v( ) = u( ) (in Theorem 6.2): @ =  and v_ vM S = 
vmA MAS = u_ MAS : These equations show that the condition tu_ MAS < (1 ; ) becomes @ ; t_v vM S < vmA MAS :

Hence, all the conditions of Theorem 6.2 are satised, which proves Theorem 6.6. Notice that the condition of the theorem is satised for = 1, i.e. for S = XA, as soon as u_ MAS  0.

Example 6.2 In order to illustrate the meaning of \at most nite subset Z of S " in the Theorem 6.1 and Theorem 6.2 and its usefulness, we consider the problem of nding the greatest and the largest set S to be an estimate of Dps (
XA
I ) of simple linear second order system (X 2 <2):

2 ;3:1 dX = 4 dt

2

;2 ;3:1

3 5 X


with

XA = fX : kX k  eg: Evidently, I =  in this example. The function v is selected as v(X) = ln kX k2
so that v(X) 2 C(
© 2004 by Chapman & Hall/CRC

and, for X 6= 0,

" ;3:1 2 # 1 1 T T v_ (X) = kX k2 2X X_ = kX k2 2X X= ;2 ;3:1

Hence,

= kX2k2 ;3:1(x21 + x22)] = ;6:2:

v_ MAS = ;6:2: The function v obeys the condition 2a) of Theorem 6.2 for any neighbourhood S of X = 0 because then Z = f0g  S . The largest neighbourhood S and the greatest obeying the conditions 1) and 2b) of the same theorem are to obey @ ; t_v vM S  vmA MAS for all t 2 <
, which in this case means: vM S  2 ; t(;6:2) or vM S  2 + 6:2t for all t 2 <
. Since min 2 + 6:2t : t 2 <
] = (2 + 6:2t)jt=0 and (t = 0) 2 <
then v M S should obey v M S  2. This is satised by the set XA itself. Hence, = 1 and S = Dps (1
XA
f0g) = XA :

Example 6.3 61] Let function f( ) in (6.1) have the next specic form: 2 ;0:2 3

66 0 66 0 4

f(X
i) =

0

77 0 7 ;1:0 2:0 75 X + 0 ;15

2 N  (X)] 3 3 66 77 0 N  (X)] 7 6 77
0 75 66 _ 77 6 2 ;100 4N  (X)
_ (X
X)]5

2 ;0:2 0 0 6 + 64 0 ;2:0 0 0

where

0

3

1

1

2

2

3

N4 4(X
i)]

3 2  (X) 3 2 0:5 0:5 0 7 66  (X) 77 66 0 1 0 77 6 7 6 66 0 777 X
64  (X) 775 = 66 0 1 4 1 jx j 10; jx j 5  (X
i) i 7 8 6:9
 3:75 9 > > > > = < 13:8 N ( ) = > 
j j  3:75 >
7:5 > >  : 1 2 3

4

3

3

1

1

1

1

1

;6:9
  ;3:75 1

© 2004 by Chapman & Hall/CRC

3

1

N1(σ1)

6.9 -3.75

σ1

3.75 -6,9

Figure 6.4: Nonlinearity N1 () graph.

sign σ2

1 0

σ2

-1

Figure 6.5: Nonlinearity sgn() graph. .

N3(σ3,σ3)

2 -4

-2 2

4

-2

Figure 6.6: Nonlinearity N3 () graph.

© 2004 by Chapman & Hall/CRC

σ3

N4(σ4)

12 -6

0

-2

6

2

σ4

-12

Figure 6.7: Nonlinearity N4 () graph.

N2 (2) = 60 sign 2


8 ( j j > 4
or ) 9 > > > > 2 sgn 

> > < = j j > 2
 _ < 0 N (
_ ) = > ( j j < 4
 _ > 0 or ) >
> > > : 0
j j < 2  8 9 12 sgn 
j j  6
> > > > < = N ( ) = > 3(j j ; 2)sgn 
2  j j  6
>
> > :  0
j j  2 3

3

3

3

3

3 3

3

3 3

3

3

4

4

4

4

4

4

4

4

i(t) = (2:5 sin2 3t):

In this example

8 2 jx j 3 2 16 39 > > > 66 77 66 77> < = XA = >X : X 2 <
64 jx j 75  64 30 75> : > > : jx j 70  1

3

2 3

The set XA can be expressed in the form (6.9) for which we set  = 2
b1 = 8
b2 = 15
b3 = 35
so that     XA = X : X 2 <3
max jxb ij : i = 1
2
3   : i © 2004 by Chapman & Hall/CRC

x3

70 -30 16

x2

-16 30

-70

XA

x1

Figure 6.8: Set XA .

The function H( ) is found as 2 N1(1) 0 66 ;0:2 ; 0:1 1 66 66 ;0:1 N1(1 1) ;1 ; 20 N2(2 2) H(X
i) = 66

66 64

4

4

2

4

3

3

3

7
; 17  100 N ( ) jx j 777 3

4

4

75 N 4 (4 ) ;15 ; 0:1  jx1j 4 4

0

2

3T

;250 N ( ) sin 3t 7 77 N (
 _ ) 2  ; 777 3

so that the matrix A, (5.6), is determined by 0 ;0:5 0:2 0 1 B C A=B B@ 0 ;41:0 2 CCA : 0 30:0 ;15 In this case, 0 ;0:5 0:2 0 1 0 8 1 0 ;1 1 CC BB CC BB CC BB ; 1365 ; 41:0 2 15 c = Ab = B 0 = B CA < 0: C B C @ A@ A @ ;75 0 30:0 ;15 35 Hence, ci < 0 for any > 0. The condition ci  (1 ; i )bi, i = 1
2
3, of Theorem 6.5 is satised for any 2 ]0
1]. The result is that the set S = XA is (itself) the (maximal) estimate of Dps (
XA  i), i.e. Dps (
XA  i) = XA
2 ]0
1]: The preceding examples show that although necessity of the conditions of Theorem 6.1 through Theorem 6.6 is not proved, these theorems can in a certain case exactly determine the domain Dps (
XA  I ). © 2004 by Chapman & Hall/CRC

Notice also that Dps (1
XA I ) = XA means that XA is a positively invariant set.

6.4 Estimate of the domain of practical stability with settling time s

In the analysis of the domain Dp (
s
XA
XF
I ) of system practical stability with settling time s , we use both the set XA of permitted states at any time t 2 <
and the set XF of allowed states since the settling time s has passed. Theorem 6.7 Let the system (6.1) have generalised motions X(t X0 i) 2 Ct(<
 X a  I ). In order for a compact set S , S XA , to be estimate Ep (
s
XA
XF
I ) of the system practical stability with the settling time s with respect to f
XF
Ig: S = Ep (
s
XA
XF
I ), it is sucient that 1) S is connected with non-empty interior: S 6= , and S XA , 2) there are a connected compact subset EF of XF with non-empty connected interior, at most nite subset Z of S \ EF and a function v : <

t

0

Zt 0

@ ;v ()d  vmA M S for all t 2 <
,

()d  vm@ EF ; vM S for all t 2 T
s .

rrr

Proof. Let all the conditions of the theorem statement hold. Hence, all the conditions of Theorem 6.1 also hold so that X(t X  i) 2 XA, 8(t
X
i) 2 <
 S  I  i.e. S = Eps (
XA
I ). From the conditions under c) after integrating D v(t
X) we derive vt
X(t X  i)] < v(0
X ) + vm@ EF ; vM S
8(t
X
i) 2  s
S  I : This and v(0
X )  vM S , 8X 2 S , imply vt
X(t X  i)] < vm@ EF
8(t
X
i) 2  s
S  I
0

0

+

0

0

0

0

0

0

or

0

X(t X0  i) 2 EF
8(t
X0
i) 2  s
S  I
due to the condition b). This result, EF XF and the condition 1) verify all requirements of (b) of Denition 6.1: S = Epc (
s
XF
I ). This and S = Eps (
XA
I ) prove S = Eps(
s
XA
XF
I ) (ii-c) of Denition 6.1]. © 2004 by Chapman & Hall/CRC

Usually we are interested in a practical system stability with settling time s with respect to f
XA
XF
Ig so that Dp (
s
XA
XF
I )  XF and @ Dp (
s
XA
XF
I ) \ @ XF = . Therefore, we accept EF XF  S
@ XF \ @ S = 
(6.17) and v_ MAEF = sup D+ v(t
X
i) : (t
X
i) 2 <
 (XA ; EF )  I ] 2 <: (6.18) Theorem 6.8 Let the system (6.1) have generalised motions X(t X0 i) 2 Ct(<
 S  I ) and the sets EF , S and XF be compact and connected with non-empty interiors. Let (6.17) and (6.18) hold. In order for the set S to be the estimate Ep (
s
XA
XF
I ) of the domain of the system practical stability with settling time s with respect to f
XA
XF
Ig: S = Ep (
s
XA
XF
I ), it is sucient that 1) there exist a function v : <

2) vM S (0)  vm@ A (t) ; t_vM AS for all t 2 <
, and 3) vM S (0)  vm@ EF (t) ; t_vM AEF for all t 2  s
.

rrr

Proof. Let all the conditions hold. Hence, all conditions of Theorem 6.2 hold, which means that S = Eps (
XA
I ) and X(t X  i) 2 S , 8(t
X
i) 2 <
 S  I . 0

0

By replacing v_ M AS by v_M AE in the proof of Theorem 6.2 we show that vX(t X0  i)]  v(0
X0 ) + t_vM AE
8(t
X0
i) 2 <
 S  I
which together with the condition 2) yields vX(t X0 i)]  v(0
X0 ) + vm@ E (t) ; vM S (0)  vm@ E (t)
8(t
X0
i) 2  s
S  I : The last inequality is due to vM S (0) v(0
X0 ), 8X0 2 S . The preceding result and the condition 1b) imply X(t X0  i) 2 EF
8(t
X0
i) 2  s
S  I
and X(t X0  i) 2 XF
8(t
X0
i) 2  s
S  I
due to EF XF . This and the properties of S to be compact, connected with nonempty interior show that all conditions of b) of Denition 6.1 are satised so that S = Epc (
s
XF
I ). This and S = Eps (
s
XF
I ) prove S = Ep (
s
XA
XF
I ). © 2004 by Chapman & Hall/CRC

Let

Q = (Qij ) 2
(6.19a)

Q = ij sup hii(X
i) : (X
i) 2 (XA ; XF )  I ] + + (1 ; ij ) max f0
sup hij (X
i) sign xj : (x
i) 2 (XA ; XF )  I ]g


(6.19b)

XF = fX : bT jX j  g
b > 0
 2 ]0
1:

(6.20)

S = fX : bT jX j  g
2 ]0
1]


(6.21)

and Fig. 6.1.

Theorem 6.9 Let the system (6.5) have generalised motions X(t X  i) 2 Ct(<
 S I ). Let the matrix Q and sets XF and S be dened by (6.19), (6.20) and (6.21), respectively. In order for the set S to be the estimate Ep (
s
XA (6.2)
XF
I ) of the domain Dp (
s
XA (6.2)
XF
I ) : S = Ep (
s
XA(6.2)
XF
I ), it is sucient that 0

all the conditions of the Theorem 6.3 hold and

t  ( ; )
8t 2  s



(6.22)

 q q  max bi
b i
i i

(6.23a)

: : : qn)T = QT b:

(6.23b)

where

=

n X i=0

and q = (q1 q2

rrr

Proof. Let all the conditions hold. Therefore, S = Dps(
XA
I ) (Theorem 6.3) and X(t X0 i) 2 XA , 8(t
X0
i) 2 <
 S  I . We accept EF = XF and use S , (6.21), to nd @ = 
vM S (0) = vM S = 
vm@ E (t) = vm@ E = vmF (6.24a) and (6.24b) v_MAF =  due to (6.19) and (6.23). From (6.22) and (6.24) follows that the condition 2) of Theorem 6.8 is satised. All other conditions of the same theorem are also satised because the sets S and XF are O-uniquely bounded with the same generating function u = v. Altogether, S = Epc (
s
XF
I ) that together with S = Eps (
XA
I ) proves S = Ep (
s
XA(6.2)
XF
I ). © 2004 by Chapman & Hall/CRC

Let (Fig. 6.2)





S = X : X T PX  
2 ]0
1]


XF = X : X T PX  
 2 ]0
1
P = P T = (pij ) > 0


and

(6.25) (6.26)

M( ) = mij ( )] = H T ( )P + PH( )


(6.27a)

mij = ij sup wii(X
i) : (X
i) 2 (XA ; XF )  I ] + + (1 ; ij ) max f0
sup wij(X
i) sign (xi xj ) : (x
i) 2 (XA ; XF )  I ]g
(6.27b) M = (mij ) 2
8 (M) > <  (P) => :  (M) (P)

9 > = > (M)  0: 

if (M) 0
if

(6.28b)

Then, the set S is estimate Ep (
s
XA(6.7)
XF
I ) of Dp (
s
XF
I ):

S = Ep (
s
XA (6.7)
XF
I ):

rrr

Proof. Under the conditions of the theorem statement we calculate for v(X) = X T P X:

@ = 
v_ vM S (0) = 
vmF MAF = 
which together with (6.28) verify the condition 2) of Theorem 6.8. Since the sets S and XF are O-uniquely bounded then they satisfy all other conditions of the same theorem, too. Therefore, S = Ep (
s
XA(6.7)
XF
I ).

If, (Fig. 6.3),



XF = X : max

 jx j i bi





: i = 1
2
: : :
n  
 2 ]0
1


b > 0


    S = X : max jxb ij : i = 1
2
: : :
n  
2 ]0
1]
i

© 2004 by Chapman & Hall/CRC

(6.29) (6.30)

then we use the matrix Q (6.19) and the vector q (6.23b) in order to specify conditions under which the set S , (6.30), is the estimate of Dp (
s
XA
XF
I ). Let R = (rij ) 2
(6.31b)

p2 : : : pn )T = RT b:

(6.31c) Theorem 6.11 Let the system (6.5) have generalised motions X(t X0  i) 2 Ct(<
 S  I ), Theorem 6.5 hold, the sets XF , (6.29), and S , (6.30), and the vector p, (6.19), (6.31) obey:





t max pbi : i = 1
2
: : :
n   ;
8t 2  s
: (6.32) i Then, the set S is estimate Ep (
s
XA(6.9)
XF
I ) of Dp (
s
XA(6.9)
XF
I ): S = Ep (
s
XA (6.9)
XF
I ):

rrr Proof. Let all the condition be valid. Theorem 6.5 guarantees S = Eps (
XA
I ) and X(t X  i) 2 XA , 8(t
X
i) 2 <
 S  I . In this case: 0

vM S (0) = 


0

@ vMF

 pi



= 
v_ MAF =  max b : i = 1
2
: : :
n : i These results and (6.32) show that the condition 2) of Theorem 6.8 holds. Since the sets XA, XF and S areO-uniquely boundedwith the same generating function u = v, v(X) = max jxb ij : i = 1
2
: : :
n , and numbers ,  and are i positive, then the sets satisfy all the conditions of Theorem 6.8 that implies S = Ep (
s
XA(6.9)

XF
I ). The preceding results are based on properties of O-uniquely bounded sets and are generalised as follows: Theorem 6.12 Let the system (6.5) have generalised motions X(t X0  i) 2 Ct(<
 XA  I ) and all the conditions of Theorem 6.6 hold. Let the sets XA , S and XF be uniquely bounded with the same generating function u, (6.14), and be determined

by (6.14), (6.15) and (6.33), respectively,

XF = fX : u(X)  g
 2 ]0
1: (6.33) In order for the set S to be estimate Ep (
s
XA
XF
I ) of the domain Dp (
s
XA
XF
I ) of the system practical stability with the settling time s with respect to f
XA
XF
Ig it is sucient that t u_ MAF  ( ; )
8t 2 <
: (6.34) rrr © 2004 by Chapman & Hall/CRC

Proof. Let all the conditions hold. Then all the conditions of Theorem 6.8 are satised because:

1) unique boundedness of XA , XF and S , and
 2 <+ , and (6.14), (6.15) and (6.33) guarantee that these sets are compact, connected with non-empty interiors, and obey the condition 1.b) of Theorem 6.8, 2) their generating function u is well dened and continuous on XA , 3) the condition t u_ MAS  (1 ; ), 8t 2 <
, of Theorem 6.6 shows that the condition 2) of Theorem 6.8 is fullled for u = v, and 4) the condition (6.34) implies validity of condition 3 of Theorem 6.8.

Example 6.4 Let a second order system of the form (6.1) be dened by

    3 2 x + 2x ; x sin 6 x sin 6 x 7 dX = ; 66 4     75 = f(X): 1

dt

Given:

2

1

2

2

;2x + x + x sin 6 x sin 6 x 1

2

1

2





1





= 20
s = 4
XA = X : kX k2  e10
XF = X : kX k2  e
I = : What is the largest number 2 <+ such that the set S = fX : kX k2  e g is estimate Ep (20
4
fX : kX k2  e10g
fX : kX k2  eg
) of the domain Dp (20
4
fX : kX k2  e10 g
fX : kX k2  eg
)? Since f(x
i)  f(X) 2 C (1) (<2) then X(t X0  0) 2 Ct(<20  <2 ). The sets




X = X : kX k  e
F

S = X : kX k  e
2 <


XA = X : kX k  e 2

10

2

2

+

(6.35)

are compact, connected and with non-empty interiors. Let v(X) = ln kX k2


(6.36)

v(X) 2 C(<2 ; f0g)


(6.37)

so that

© 2004 by Chapman & Hall/CRC

which yields Z = f0g. This shows that the function v is not positive denite. But, positive deniteness of the function v is not required in the framework of the practical stability and we may continue with testing it. The results are:



(6.38a)

@ = v@ = 1 for EF = XF
(6.38b) @ XF = X : kX k  e ) vmF mEF

(6.38c) S = S = X : kX k  e ) v@ (0) = v@ =
@ = 10
@ XA = X : kX k2  e10 ) vmA 2

2

MS

so that

MS

v(X) = ln kX k2 < vm@ EF = 1 < v(Y ) = ln kY k2
8(X
Y ) 2 EF  (<2 ; EF ): (6.39) Further,

   3 2 ; x ; 2x + x sin 6 x sin 6 x 7 6 D v(X) = v(X) _ = kX1k 2X T 64     75 = ;2
1

+

2

2

1

2

+2x1 ; x2 ; x1 sin 6 x2 sin 6 x1

2

8(X 6= 0) 2 <
2

so that v_ MAEF = ;2:

(6.40)

We use Theorem 6.8 and we will determine the set S so that all the conditions are satised. It was shown above that the sets XA , XF and S have the required properties. From (6.37) and (6.39) it follows that the function v obeys a) and b) of the condition 1) of the theorem. The set S = fX : kX k2  e g should be determined, i.e. should be calculated so that the conditions 2) and 3) of the theorem also hold, which take the following form due to (6.38):











@ (t)  10 ; t v_ 2) vM S (0) =  vmA MAS = ;2
8t 2 <20


that is that

 10 + 2t
8t 2 <20: Hence, for (t = 0) 2 <20,

 10:







(6.41)

h

i

3) vM S (0) =  vm@ EF (t)  1 ; t v_ MAEF = ;2
8t 2 4
20


© 2004 by Chapman & Hall/CRC

or

 1 + 2t
8t 2 4
20: For (t = 4) 2 4
20] follows

 9:

(6.42)

Now, (6.41) and (6.42) determine the largest = 9, or equivalently the estimate

S = Ep (
s
XA
XF
) of the form fX : kX k  e g as S = Ep (20
4
fX : kX k  e g
fX : kX k  eg
) = fX : kX k  e g: 2

2

10

2

2

9

6.5 Conclusion The practical stability criteria permit an aggregation function v to be non-denite with non-semidenite Dini derivative. The function v need not be continuous everywhere. The criteria are expressed in a very simple form in terms of extremal values of v and D+ v on compact sets. They are suitable for direct calculations. The criteria show qualitative and quantitative trade-o among the sets XA, XF , E( ) , the settling and nal time and the form of aggregation function v used. The choice is straightforward in the case of the sets being uniquely bounded, where their generating functions should be taken for the function v.

© 2004 by Chapman & Hall/CRC

Chapter 7

Comparison systems and vector norm-based Lyapunov functions 7.1 Introductory comments and de nitions 7.1.1 Presentation

This last chapter is devoted to the stability study of large scale systems whose models are not completely specied, either because of an undetermined disturbing input, or because the identication of some varying parameters is not possible. This is represented by the equation: dX = g(X
d) g :
© 2004 by Chapman & Hall/CRC

Some examples a) Equation (1.18) of Section 1.2, or (3.4) of Section 3.1:

dX = f(X
i)
dt related to the practical stability concept, belongs to that type of equation (with k = 1). In this case, d = i(t) is an input disturbance, and Sd 
c) In some cases, equation (7.1) can be written as: dX = f(X) + d(X) (7.2) dt where f is a known function, and d 2 Sd 
© 2004 by Chapman & Hall/CRC

7.1.2 Comparison systems

The notion of comparison system originates in the theory of dierential (and integral) inequalities, which inequalities directly follow from the use of Lyapunov's methods. Since the beginning of the century, many works have been devoted to related topics (see for example description and references in 88], Chapter 2). The \comparison concept" has also been dened as a formal property of qualitative concepts 92] that we shall use in the sequel, under the following reduced expression 162]:

Denition 7.1 Let two systems (S) dX dt = g(X
d) (C) dZ dt = h(Z
X
d)

g :
(7.6)

h :
(7.7)

and sets X 
i) Notation P (C
Z
A) ) Q(S
X
B) signies \If Z has a property (or concept) P for system (C) with regard to the ordered list of arguments A (related to concept P) then X has the property (or concept) Q for system (S) with regard to the ordered list of arguments B (related to concept Q)". ii) (C
Z
A) is a comparison system of (S
X
B) with regard to property or concept P if

P (C
Z
A) ) P(S
X
B):

Example 7.1 The rst Lyapunov method can be stated in term of comparison systems, since the asymptotic stability of Z = 0 for dZ = AZ () dt implies the asymptotic stability of X = 0 for 8 dX > < dt = AX + b(X)
() > : lim bT (X)b(X) = 0: X !0 X T X We can say that (()
f0g) is a comparison system of (()
f0g) with regard to the asymptotic stability property and to the instability property.

Example 7.2 Let the system (S) be dened by: " ;1 # dX = 1 ; x ; x X
X 2 <
dt 1 1;x ;x 2 1

2 2

2

2 1

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2 2

(7.8)

x2 1.5 1

X

0.5

1

-1

-1.5

-0.5

0.5 -0.5

x1

1.5

-1

-1.5

Figure 7.1: Trajectories of system (7.8).

(see Fig. 7.1) and the system (C) be dened by: dz = 2z(1 ; z)
z 2 <: (7.9) dt Consider the positive denite function z(x) = x21 + x22
then it veries equation (C). z = 0 is an unstable equilibrium of system (C), and ze = 1 is an asymptotically stable one (see Fig. 7.2), since the solution of (C) is z(t t0
z0) = z + (1 ; z z)0exp (t ; t) : (7.10) 0

0

0

D(fze g) = < is the asymptotic stability domain of ze = 1. The convergence of z to Z = fze g (for z > 0) implies that state X tends to the unit circle X = fX 2 < : z(X) = 1g, for any initial condition X 2 D(X ) = < ; f0g. Then, asymptotic stability of Z for system (C) with the asymptotic stability domain < implies asymptotic stability of X for system (S) with the asymptotic stability domain < ; f0g, or asymptotic stability (C
f1g
< ) ) asymptotic stability (S
fX 2 < : x + x = 1g
< ; f0g) and 1) (C
fze = 1g) is a comparison system of (S
fX 2 < : x + x = 1g) with +

0

2

0

2

+

2

2

+

2 1

2 2

2

2

2 1

2 2

regard to the asymptotic stability property. 2) (C
fze = 1g
<+) is a comparison system of (S
fX 2 <2 : x21 + x22 = 1g
<2 ; f0g) with regard to the asymptotic stability domain concept. Example 7.3 We still consider systems (7.8) and (7.9), now related to exponential stability. Equation (7.10) implies that ze = 1 has the domain De = 0:5
+1 of exponential stability with regard to ( = 2
= 1) (see Denition 2.8, Chapter 2). Then (C
fze = 1g
De = 0:5
+1
 = 2
= 1) is a comparison system of (S
fX 2 <2 : x21 + x22 = 1g
fX 2 <2 : x21 + x22 0:5g
 = 2
= 1) with regard to the exponential stability domain concept.

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z

1 0.5 0.01 0

t

Figure 7.2: Trajectories of system (7.9).

7.1.3 Dierential inequalities, overvaluing systems

Basic results concerning dierential inequalities were provided by Wa,zewski 206], who gave the necessary and su cient hypothesis ensuring that the solution of a system described by: dX = F (t
X) dt with initial condition X0 at t0 with function F (t
X) verifying inequality F (t
X)  G(t
X) is overvalued by the solution of the so called \overvaluing system": dZ = G(t
Z) dt with initial condition Z0 X0 at t0 , or in other words, conditions on function G(t
Z) that guarantee: Z(t t0
Z0 ) X(t t0
X0 ): These conditions and other linked results are the topic of this section. Denition 7.2 a) A function f :
© 2004 by Chapman & Hall/CRC

Remark: 1) This denition also exists for time-dependent functions f(t
X
V ),

that are involved in time-varying systems. 2) \Quasi-increasing" is also called \increasing with respect to the diagonal elements" 92], or \uniformly nonsingular monotone" 88], or \satisfying the Wa,zewski conditions". Example 7.4 a) Any continuous function f :
b) Any continuous function f :
Denition 7.4 The system (7.13): dOS = h(OS d)
h :
(7.13)

is a (T
S
Sd )-local (direct) overvaluing system of system (7.1) i i) the solutions OS(t t0
OS0  d) of (7.13) and X(t t0
X0 d) of (7.1) exist for X0  S , OS0  S , t 2 T , d 2 Sd , and are continuous with respect to time

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ii) OS(t t0
OS0 d) X(t t0
X0 d) for any OS0 and X0 in S such that OS0 = X0 , any t 2 t0
!), and any d 2 Sd . iii) S is (T
Sd )-positively invariant for (7.1) and (7.13).

Remarks: 1) The mention \direct", referring to the existence of reverse overvaluing

systems that are used in the instability study 162], shall be omitted in the following. 2) In the previous denitions 7.3 { 7.4, arguments T
S or Sd can be omitted: { if T = < and S =
Theorem 7.1 Let two systems be with time continuous solutions: dX = g(X)g :
(7.14) (7.15)

Then the system (7.15) is a global overvaluing system of system (7.14) i h is globally quasi-increasing. rrr

Theorem 7.2 Consider the disturbed system (7.1), and let V (X) :
a continuous function with right-hand gradient, components of which are positive denite functions, such that along the solutions of (7.1) the following holds:

Dt V (X) = f(X
d)  h(X
V (X)) 8d 2 Sd
8X 2
(7.16)

Suppose that system (7.17):

8 dX > > < dt = g(X
d) > > : dV = h(X
V ) dt

(7.17a) (7.17b)

has a unique solution, continuous with regard to time, for d 2 Sd . Then (7.17) is an Sd -overvaluing system of system (7.18):

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8 dX > > < dt = g(X
d) > > : Dt V (X) = DX V ]T g(X
d)  h(X
V (X)) +

+

i h is globally quasi-increasing with regard to the variable V .

(7.18a) (7.18b)

rrr

Proof. This result follows from 13]. Denition 7.5 System (7.17) is a decoupled Sd -overvaluing system of system (7.18) if and only if it is a Sd -overvaluing system of (7.18) with special form (7.19):

8 dX > > < dt = g(X
d) > > : dV = h(V (X)) dt

(7.19a) (7.19b)

which means that (7.16) comes down to

Dt V (X) = f(X
d)  h(V (X)) 8d 2 Sd
8X 2
Corollary 7.1 If system (7.1) and function V satisfy the hypothesis of Theorem 7.1, then system (7.19) is Sd -decoupled overvaluing system of (7.18) i h(V ) is quasiincreasing. In this case, any motion Z(t t0
Z0) of

8 dZ > < dt = h(Z) > : Z = V (X )

(7.20)

Z(t t0
Z0 ) V X(t t0
X0 d)]
8d 2 Sd :

(7.21)

0

0

veries

This corollary obviously follows from Theorem 7.2 and Denition 7.4. It gives a way to have a suitable stability study of system (7.1) by comparing it with a simpler system (7.20): simpler, because (7.20) is k-order (k  n) and is not disturbed. However, this result does not concern general (S
T
Sd )-local overvaluing systems, that are to be worked out in the next sections, leading to stability domains estimations. Before this, we have to study some particular quasi-increasing functions.

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7.1.4

M matrices

;

Denition and properties

Example 7.4 shows that any function dened by f(X) = MX + q where M is a constant matrix with nonnegative o-diagonal elements, and a constant vector, is quasi-increasing. Such matrices M = (mij )ij =1:::n, mij 0, 8i 6= j, present strong and useful properties that are recalled here, and that shall be useful in the following sections.

Denition 7.6 48] A constant real square matrix A is called: i) a Z -matrix if and only if all its o-diagonal elements are non-positive, which is also denoted A 2 Z and a Z  -matrix if and only if o-diagonal elements are negative, this is A 2 Z . ii) a ;Z -matrix i ;A 2 Z, which is also denoted A 2 ;Z, and a ;Z  -matrix if and only if ;A 2 Z , A 2 ;Z . iii) a P -matrix if and only if all its leading principal minors are positive, which is also denoted A 2 P,

8 > > < P = >A = (aij )ij > :

 a : : : a  9 > n   >  =   . . . . : a > 0
: : :
> 0 :   :::n . .   > >  an : : : ann   11

=1

1

11

1

iv) an M -matrix, (Metzlerian), or A 2 M, if and only if both A 2 Z and A 2 P. v) the opposite of an M -matrix, or an ;M -matrix, or A 2 ;M, if and only if ;A 2 M.

A characteristic of ;M-matrices is to have nonnegative o-diagonal elements, and a negative diagonal.

Theorem 7.3 48] Let M be a constant n  n matrix with non-negative elements (M 2 ;Z). Then the proposition \M is a ;M matrix" is equivalent to any of the following propositions:

1) Any eigenvalue of M has a negative real part. 2) Any real eigenvalue of M is negative. 3) M veries the Koteliansky conditions (that is, ;M 2 P):

 m m < 0
 m

11

11

21

m   > 0
m  12 22

  m  m  m

11 21 31

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m12 m13   m22 m23  < 0
: : :  m32 m33 

   m : : : m k   ..  > 0 : : : (;1)n det M > 0: (7.24) (;1)k  ... .    mk : : : mkk  11

1

1

4) There exists a constant vector X 0 such that MX < 0. 5) There exists a constant vector X > 0 such that MX < 0. 6) There exists a diagonal matrix $ with positive diagonal such that N = M$ is a matrix with dominant negative principal diagonal, i.e.:

nii +

X j 6=i

jnij j < 0 8i = 1 : : :n (with here, jnij j = nij ):

7) M ;1 exists and all its coecients are non-positive, i.e.: ;M ;1 0 8) If N 2 Z and N  M , then N ;1 exists. 9) For any vector X 0, X 6= 0 there exists an index i such that if Y = MX , xiyi < 0. 10) If d(A) denotes the diagonal of A, then for each diagonal matrix R such that R ;d(A), the inverse R;1 exists and (R;1 (A ; d(A))) < 1, where ( ) is the spectral radius (i.e. the maximum of the moduli of all eigenvalues). 11) There exists a permutation matrix P such that P MP T = T1 T2 , T1 lowertriangular matrix, T2 upper-triangular, T1 and T2 2 Z and have strictly negative diagonal.

rrr

Further properties of ;M matrices 1) If M 2 ;M, then there exists a negative real eigenvalue (M) of M, called the importance value of M, such that the real part of any eigenvalue of M is at most (M). 2) If M 2 ;M, there exists a nonnegative eigenvector u(M) 0 associated to

(M) and called the importance vector of M. 3) If M 2 ;M and M is irreducible, then u(M) > 0, and in Theorem 7.3 one can choose X = u(M) (in proposition 5) and $ = diag(u(M)) (in proposition 6).

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Dynamic systems dened by ;M matrices

We consider here the linear system dened as in (7.22) by dX = MX + q
M(n  n) is a ; M matrix
q and X 2 0 and an n-vector c such that c q, the sets:



9 Ba = X 2 >



+ + n n = Ba = X 2 <+ : 0  X < a
B a = X 2 < + : 0  X  a
> (7.27)

> A = BXe = X 2 >  A+ = B +Xe = X 2
rhomboid: D1 () = X 2
parallelepiped: D1 () = X 2 0 and such domains are neighbourhoods of Xe . Finally, let K be the simplicial cone 195] dened as: K = a 2
8X 2 @ Dj ()
X(t X ) 2 exp ; (M) t @ Dj (): 0

© 2004 by Chapman & Hall/CRC

0

(7.30)

rrr

Proof. Let M = (mij ), q = (qi). First point (i) is obvious. Proof of (ii) :

Existence of K: Let u > 0 such that Mu < 0,  > 1 and a = ;M ; q + u. Then a > q and ;M a > q, which proves K = 6 . Invariance of B a : Let a = ( : : :n)T > q, b = ;M a = ( : : :n )T q, and X(t) = (x : : :xn )T 2 @ B a . Then there is an index i such that at time t, jxij = i and jxj j  j for j =6 i. Since M 2 ;Z, we have 1

1

1

1

Dt jxij  +

n X i=1

mij jxj j + qi 

n X i=1

mij j + qi = i + qi  0


and jxij cannot increase at time t. Then, any trajectory of (7.25) crossing @ Ba enters B a or lies on @ B a . This proves the positive invariance of B a .

Invariance of B +a : Suppose that at time t, X(t) 2 @B +a , and X(t) 62 @Ba . this means that there is an index i such that at time t, xi (t) = 0, and 0  xj (t)  j . Then

dxi(t) = X m x + q 0
ij j i dt j 6=i

and X(t) cannot go out of B +a . This, together with positive invariance of B a , proves the positive invariance of B+a . Proof of (iii): This can be found in 195], in the case q = 0, and is easy to generalise for q 0 by setting Y = X + M ;1 q, Y (t
Y0 ) = exp (Mt) Y0 , and K comes down to

fa 2 0g. Proof of (iv):

Domains D (): Let the positive (semi)denite function v dened by v (X) = u(M T )T jY j. Then Dt v (X)  u(M T )T M jY j = (M)v (X)  0, and v (X) exponentially tends to zero as t ! +1. Domains D1 (): Let I = fi 2 f1
2 : : :ng
ui(M) =6 0g and v1(X) = max u;i jyij i2I = u;i0 jyi0 j. 1

+

1

1

1

1

1

1

1

Dt v1 (X)  u;i0 +

Dt v1 (X)  +

1

X

X mi0 j jyj j uuj  u;i01 v1 (X) mi0 j uj j

j 2I j 2I ; 1 ui0 v1 (X) (M)ui0 = (M)v1 (X):

Then v1 (X) exponentially tends to zero as t ! +1. © 2004 by Chapman & Hall/CRC

Example 7.5 Let the linear system be dened by " ! " ! dX = ;2 1 X + 1 : dt 1 1 ;1

(7.32)

This yields (see Fig. 7.3): "2! Xe =
K = fa = (a1
a2 )T : 1  1 + a1  a2  2a1 ; 1g
3

A = fX = (x
x )T : 0  x  2
0  x  3g
+

For a =

"5! 7

1

2

1

2

2 K
B+a = fX = (x
x )T : 0  x  5
0  x  7g: p T T T 1

2

1

2

An importance vectorp of M = M is u = u(M) = u(M ) = (2
1+ 5) , associated with (M) = ;3 +2 5 .

p D () = fX = (2 + y
3 + y )T : 2jy j + (1 + 5)jy j  g
p D1 () = fX = (2 + y
3 + y )T : jy j  2
jy j  (1 + 5)g: 1

1

2

1

2

1

1

2

2

Several trajectories of (7.32) are given on Fig. 7.3, which shows invariant domains K
A
A+
Ba

B+a
Du ().

Relation between quasi-increasing functions and ;M matrices

The stability theory literatures suggest a tight connection between Wa,zewski's conditions and ;M matrices. However, until 163] (see also 162], 164]), this link always appeared in an implicit manner, and the following Theorem 7.5 aims at clarifying the existing implications. @f (X) Let us suppose that f(X) has a bounded Jacobian matrix fX (X) = @X for X 2 S (connected subset of
K

x2

a

B-a+

7

D∞(α) 3 2

D1(α) +

A

1

Xe q

0

1

2

3

5

x1

A B-a Figure 7.3: Domains for Example 7.5 (eqn. 7.32).

P 3: Xe is hyperbolic (that is, fX (Xe ) has no eigenvalue with zero real part) and asymptotically stable P 4: fX (Xe ) 2 ;M.

Theorem 7.5

P1 ) P2 P 20 ) P 1 P 200 ) P 1 with N = S P 1 and P 3] ) P 4:

rrr

Corollary 7.2 If f(X) = MX + q then f is quasi-increasing and X = ;M ; q is asymptotically stable, i M is a ;M matrix. 1

7.2 Vector norm-based comparison systems 7.2.1 De nition and aim of vector norms

The concept of vector norm (V.N.) was introduced by Robert (1964) in order to solve numerical analysis problems linked with linear recurrences. It was applied, together

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with the concept of vector Lyapunov's function (initiated by Bellman (1962) and Matrosov (1962)) to an aggregation of large-scale systems and to their stability analysis, by Borne, Gentina, Laurent, Gruji&c, Richard (see for example 57], 87], 173]) with further applications to constrained control 169], practical stability 164], and dierential-dierence equations 34], 35], 21]. Denition 7.7 Let E =
E = E + E + E + : : : + Ek
Ei =
2

3

by Xi :

Xi = PiX


(7.33)

where Pi is a projection operator from E into Ei . Let pi be a scalar norm (i = 1
2
: : :
k) dened on the subspace Ei , and p be a vector norm (V.N.) of dimension k and with the i-th component:

pi(X) = pi(Xi ) p(X) :
E = E  E  E  : : :  Ek : 1

2

3

In the following, we shall use R.V.N. for \regular vector norm", and k X i=1

ni = n
Xi = (xj )j 2Ii


'

i=1:::k

Ii = f1
: : :
ng:

The aim of this section is to give a general way of computing overvaluing systems that are based on an R.V.N. p(X), in other words, of computing a function h in such a way that system (7.35):

8 > > < dX dt = g(X
d)
> > : dp = h(X
p)
dt

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(7.35a) (7.35b)

is a local overvaluing system of

8 > > < dX dt = g(X
d)
> > : Dt p(X) = DX p]T h(X
d): +

+

(7.36a) (7.36b)

Then (7.35b) will be used as a comparison system of (7.35a), since its solution veries Z(t)  p(X(t)), provided that initial conditions are compatible: Z(t0 ) = p(X(t0 )). However, the stability study of (7.35b) has to be carried out without solving (7.35a), and this will motivate the use of decoupled overvaluing systems (see Denition 7.5) that are such that h depends only on Z:

8 > > < dX dt = g(X
d), > > : dZ dt = h(Z).

(7.37a) (7.37b)

This could have been obtained by means of the general vector Lyapunov's functions instead of V.N., however the main advantage of V.N. is to give direct and systematic computations.

7.2.2 A rst statement

Lemma 7.1 a) Consider the system (7.1), and let a R.V.N. p(X) and a constant positive vector c 2
Dt (p(X))  h(X
p(X)) 8X 2 Xc: +

(7.39)

Suppose that the system

8 > dX > < dt = g(X
d), > > : dZ = h(X
Z) dt

(7.40a) (7.40b)

has a unique continuous solution for d 2 Sd . Then a sucient condition for system (7.40) to be a (Xc  Zc
Sd)-local overvaluing system of

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8 > dX > < dt = g(X
d)
> > : Dt p(X) = Dx p]T g(X
d) +

(7.41a) (7.41b)

+

is that Zc is Xc -positively invariant for (7.40b), which means

Z0 = p(X0 ) 2 Zc ) Z(t) 2 Zc 8t t0
8X 2 Xc:

(7.42)

b) If system (7.40) is decoupled, that is, if (7.39) can be written as:

Dt p(X)  h(p(X))
8X 2 Xc


(7.43)

+

with h a locally quasi-increasing function on Zc , then the above condition is that Zc is positively invariant for (7.44):

dZ = h(Z): dt

(7.44)

rrr

Proof. a) The proof refers to Denition 7.4, with notations:

0 B X(t t
X  d) X(t t
X  d) = B B @

1 9 > > CC > CA 2 S = Xc  Zc  > > p(X(t t
X  d)) > = (7.45) > 0 1 > > > X(t t
X  d) C B > C > OS(t t
OS  d) = B 2 S
@ A >  Z(t t
X
Z  d) 0

0

0

+

0

0

0

0

0

0

0

0

0

0

Z0 = p(X0 ): (7.46) i) Solution X(t t0
X0  d) exists and is continuous by hypothesis. ii) Since (7.39) holds with h(X
Z) locally quasi increasing w.r.t. Z, as long as X remains in Xc and Z in Zc , similarly to Theorem 7.2 we have Z0 = p(X0 ) ) Z(t t0
X0
Z0 d) p(X(t t0
X0 d))
(7.47) © 2004 by Chapman & Hall/CRC

and for any t t0 OS0 = X0 ) OS(t t0
OS0  d) X(t t0
X0  d)

(7.48)

supposed that OS and X remain in S and d 2 Sd . iii) Let us suppose that (7.42) holds, and consider Z0 = p(X0 ) < c. Then 0  Z(t t0
X0
p(X0) d) < c
8t t0
8d 2 Sd :

(7.49)

At time t = t0 , X0 is in Xc, therefore (7.47) holds until time t1 when p(X(t1 )) = c. However, such a time t1 does not exist because of (7.49). So, (7.47) remains true for any t t0. This proves 0  p(X(t t0
X0 d))  Z(t t0
X0
p(X0) d)


(7.50)

and Sd -positive invariance of Xc for system (7.1). b) The decoupled case appears as a particular case of (a).

7.2.3 Computation of overvaluing systems General formula Let the system (7.1) be decomposed into dX = A(X
d)X + b(X
d)
dt A(X
d) and v(X
d) bounded for bounded X, and any d 2 Sd :

9 =  (7.51)

This decomposition is obviously non-unique: as we shall see afterwards, it is preferable to work it out in such a way that dX = A(X
d)X dt

(7.52)

has its equilibrium X = 0 stable, or attractive, or both. Algorithms will also be proposed in order to optimize this decomposition. Then dene for any i
j 2 f1
2
: : :
kg the following notation: © 2004 by Chapman & Hall/CRC

9 > > Y yi = Pi Y
> > > bi(X
d) = Pi b(X
d)
> > > > i (X) = sup Dxi pi (xi)]bi (X
d)
> > d2Sd > > > i = sup i (X)
> x2S > > > Aij (X
d) = Pi A(X
d)pj
> > T Dyi pi (yi )] Aij (X
d)yj > mij (X
Y
d) =
> = pj (yj ) > > ij (X) = sup mij (X
Y
d)
> Y > d d > > > ij = sup ij (X)
> X 2S > > > M(X) = ij (X)]ij 2f :::kg
> > > > q(X) = i(X)]i2f :::kg
> > > > M = ij ]ij 2f :::kg
> > > >  q = i]i2f :::kg: 2
+

+

(7.53)

2S 2S

1

1

1

1

If A and b are not continuous w.r.t. X, then M(X) and q(X) may be discontinuous then dene two functions M and q continuous on S such that: M M(X) M(X) and q q(X) q(X): (7.54) Properties. By construction, if p is a R.V.N., then for any X in S , the following holds: (7.54) 1) q q(X) 0
q(X) q(X) 0 2) M(X) 2 ;Z
M(X) 2 ;Z
M 2 ;Z
(7.55) + 3) Dt p(X(t))  M(X)p(X) + q(X) (7.56) is veried for any motion of (7.1) inside S , for d 2 Sd . 4) The function dened by h(X
Z) = M(X)Z + q(X) (7.57) is locally quasi-increasing with regard to variable Z on S 
© 2004 by Chapman & Hall/CRC

Denition 7.8 i) Any matrix M(X) dened by (7.53) (7.54) is called an (S Sd )local overvaluing matrix of system (7.1) associated with the R.V.N. p(X).

ii) Any vector q(X) dened by (7.53) (7.54) is called an (SSd )-local overvaluing input vector of system (7.1) associated with the R.V.N. p(X). iii) Any pair (M(X)
q(X)) dened by (7.53) (7.54) is called an (S  Sd )-local overvaluing pair of (7.1) associated with the R.V.N. p(X). iv) If S =
Simpler formula with the usual Holder's norms and associated matrix measures

Though formulas (7.53), based on the gradient of the vector norm, are general ones, an equivalent calculation of overvaluing pairs (M
q) can be proposed, which is related to the notions of norm and measure of matrices 39]. The norm of a rectangular matrix A 2 C n
m , associated with two norms k k of vectors in C n and C m , is classically dened as: k kAk = supm kkAX (7.58) X k 2 <+ : X 2< The associated measure of a square matrix is: k ; 1 2 <: (A) = lim+ kI + "A (7.59) " "!0 For any matrices A and B of same dimension n  n, we have: Re i (A)  (A)
;(A)  Im i (A)  (;jA)
(A + B)  (A) + (B)
(7.60) (A)  kAk
(A) = (A) 8 0: Considering a matrix A = (aij ) 2 C n
n, and a vector q = (ui ) 2 C n , the followingX sums up the formulas that concern Holder's norms 1, 2 and 1. The notation here obviously means the sum for i = 1 to i = n, max means the max i i for i = 1 to n, and (A) are the eigenvalues of A.

Holder's norm 1 (sum norm) X kuk = juij
1

© 2004 by Chapman & Hall/CRC

i

kAk = max 1

X

j

i

jaij j


(7.62)

0 1 X  (A) = max B Re (ajj ) + jaij jC @ A: j 1

i i6=j

Holder's norm 2 (Euclidean norm) kuk = 2

"X ! = ui

1 2

2



i




=

kAk = max i (AA ) 2

1 2

i


where i (A) is the ith eigenvalue of A

(7.63)

2 (A) = 12 max i (A + A ): i

Holder's norm 1 (max norm) kuk1 = max juij
i

kAk1 = max i

X j

jaij j


1 (A) = max Re (aij ) + i

(7.64)

X i i6=j

jaij j:

The following inequalities are well known: kX k1  kX k2  kX k1  pnkX k2  pnkX k1  pnkX k1 :

Assumption 7.1 When all the scalar components pi(X) of p(X) are of the same Holder's type 1, 2 or 1, the application of formula (7.53) leads to the following formulation of the overvaluing pair (M(X)
q(X)): 9 ii(X) = sup (Aii (X
d)) > >

> > = ij (X) = sup kAij (X
d)k for i 6= j d2Sd > > > ki(X) = sup kbi(X
d)k  d2Sd d2Sd

(7.65)

where the measure  and norms k k refer to the corresponding scalar Holder's norm. © 2004 by Chapman & Hall/CRC

Remark. In the particular case p(X) = jX j (then, k = n), the expression is very simple:

9 > > d > = ij (X) = sup jaij (X
d)j
for i 6= j d2Sd > > > ki (X) = sup jbi (X
d)j:  d2Sd

ii (X) = sup aii(X
d)
d2S

(7.65bis)

Proof. We give here only a proof for the Euclidean norm pi(Xi ) = kXi k2 = kXi k = (XiT Xi )1=2
i = 1
: : :
k: Xit

X

Aij ( )Xj T (AT ( ) + Aii( ))Xi X XiT bi( ) j = 6 i ii Dt+ pi(Xi ) = i 2(X + + T (XiT Xi )1=2 (XiT Xi )1=2 i Xi )1=2

X ij ( )kkXj k + kXi kkbi( )k
  (Aiik(X))kkXi k + kXi kkAkX k kX k 2

2

i

j 6=i

that is,

Dt kXi k   (Aii ( ))kXi k + +

2

X j 6=i

i

i

kAij ( )kkXj k + kbi( )k:

In other cases (norms 1 and 1), analogous demonstrations can be drawn, or deduced from the expressions given in 55] and generalised in 162]. This can again be expressed by developing (7.65): for pi(Xi ) = kXi k1, 8i = 1
: : :
k:

9 > > > > > > > = ij (X) = sup max jarsj
for i
j = 1
: : :
k
i 6= j > d2Sd s2Ii r2Ii > > > > X > > ki (X) = sup jbrsj
 d2Sd r2Ii 2 3 X ii (X) = sup max 64arr + jarsj75
for i = 1
: : :
k d2Sd s2Ii r Ii " X r=#s 2 6

© 2004 by Chapman & Hall/CRC

(7.66)

for pi(Xi ) = kXi k1 , 8i = 1
: : :
k:

2 3 X ii(X) = sup max 64arr + jarsj75
for i = 1
: : :
k d2Sd r2Ii s Ii s=r 2 3 X ij (X) = sup max 4 jarsj5
for i
j = 1
: : :
k
i 6= j 2 6

d2Sd r2Ii s2Ij

ki(X) = sup max jbrsj
d2Sd s2Ij

9 > > > > > > > = > > > > > > > 

(7.67)

for pi(Xi ) = kXi k2, 8i = 1
: : :
k. The expression (7.65) together with (7.63) is not easy to compute if A strongly depends on X or d, since eigenvalues are to be calculated. However, one can use the following equations, which are less precise but easier to compute:

2 3 X ii(X) = sup max 64ass + 12 jasr + ars j75 d2Sd s2Ii r Ii r=s  X X 1

9 > > > > > > > = ij (X) = 2 sup max jasr j] + max jars j] > d2Sd s2Ii r2Ii > > > "X ! = > > > ki(X) = sup bi :  d2Sd s2I 2 6

(7.68)

1 2

2

i

This expression of ij was given in 55] or 87].

7.2.4 Overvaluation lemma

The above formulas and properties simplify and make explicit the expression of Lemma 7.1, leading to the following Lemma 7.2.

Lemma 7.2 Let the system (7.1) (7.51) have a unique continuous solution X(t t
X  d). Let a R.V.N. p(X) and a constant positive vector c 2
0

a) A sucient condition for the system (7.69):

8 dX > > < dt = g(X
d) = A(X
d)X + b(X
d) > > : dZ = M(X)Z + q(X) = h(X
Z) dt

© 2004 by Chapman & Hall/CRC

(7.69a) (7.69b)

to be an (Xc  Zc
Sd )-local overvaluing system of (7.70):

8 dX > > < dt = g(X
d) > > : Dt p(X) = DX p]T g(X
d) +

+

(7.70a) (7.70b)

is that Zc is Xc -positively invariant for (7.69b) (then X is considered as a disturbance), which means:

Z0 = p(Xo ) 2 Zc ) Z(t) 2 Zc 8t t0
8X 2 Xc :

(7.71)

b) A sucient condition for the decoupled system (7.72):

8 dX > > < dt = g(X
d) > > : dZ = MZ + q dt

(7.72a) (7.72b)

to be (Xc Zc
Sd )-local overvaluing system of(7.70) is that M is asymptotically stable and c belongs to the simplicial cone K dened by (7.29) this, i.e.: 1) M veries the Koteliansky conditions (Theorem 7.3, proposition 3) 2) ;M c q (7.73) c) Dening Ze = ;M ;1 q, ((7:72b)
Zc
Ze) is a comparison system of ((7:1)
Xc
XZe ) with regard to the asymptotic stability property with the estimate of asymptotic stability domain, i.e.: \Ze is asymptotically stable for (7.72b), with Zc an estimate of its asymptotic stability domain
Proof.

a) M(X) and q(X) are continuous w.r.t. time, whenever X is the continuous solution of (7.1) (7.51). Then (7.69b) admits a unique continuous solution. Properties 3 (7.57) and 7 (7.58) ensure that h(X
Z) fulls hypothesis of Lemma 7.1 (quasi-increasingness and overvaluation (7.39)) Therefore, Lemma 7.1 (a) proves Lemma 7.2 (a). b) Solutions of (7.72b) are continuous, and Theorem 7.4 (ii) proves that Zc is positively invariant with respect to (7.72b). The function MZ + q is quasiincreasing, and thus Lemma 7.1 (b) proves Lemma 7.2 (b). c) If Ze is asymptotically stable, Ze is invariant, and any motion (7.1) initialised in Xc remains in Xc and tends to XZe as t tends to innity. © 2004 by Chapman & Hall/CRC

Example 7.6 Consider the system (7.74), with the R.V.N. p of size n: dX = A(X)X + b(X) dt p(X) = (jx1j
: : :
jxnj)T : This yields (see for example (7.65bis)):

0 a (X) BB BB ... BB M(X) = B BB jai (X) BB .. B@ .

: : : ja1i(X)j .. ... . : : : aii (X) 1 .. . jan1(X)j : : : jani(X)j 11

: : : ja1n(X)j .. . : : : jain(X)j .. ... . : : : ann(X)

(7.74) (7.75)

1 CC 0 jb (X)j 1 CC BB ... CC CC CC q(X) = BBB jbi(X)j CCC CC B@ .. CA . CC j b (X) j n A 1

(7.76) M = sup M(X)
q = sup q(X): X 2Xc X 2Xc If, for Xc =
(7.24), then all motions of (7.74) converge inside of X 2
Example 7.7 Let us consider

dX = A(X
d)X
dt 2 ;2 cos x2 66 6 ;0:2 ;1:5 A(X
d) = 66 64 0:1 2 0:3 ;0:1 0:4

0:2 0:7 1 ;0:4 0:6 sign x3 ;1:3 0:3 2 1:1 ;2

3 77 77 77 5

(7.77)

d = ( 1
2 )
Sd = fd
kdk1  1g ( i.e. j 1 j  1
j 2 j  1)
and dene the max-vector norm p(X) as "X ! "x ! "x ! " kX k ! 1 1 3 1 1 X=
X1 =
X3 =
p(X) = : X2 x2 x4 kX2 k1 (7:77bis) Then, for d 2 Sd 11(X) = max f;2 + jcos x2j
j ; 0:2j ; 1:5g  11 = ;1
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12(X
d) = max fj0:2j + j0:7 1j
j ; 0:4j + j0:6 sign x3jg  12 = 1
21(X
d) = max fj0:1 2j + j0:3j
j ; 0:1j + j0:4jg  21 = 0:5
22(X) = max f;1:3 + j0:3 2j
j1:1j ; 2g  22 = ;0:9
and the solutions of dZ = dt

" ;1

1 0:5 ;0:9

!

Z = MZ:

(7.78)

Z0 = p(X0 ) verify p(X(t))  Z(t) for any d varying in Sd . In this case, M veries the Koteliansky conditions, which leads to the conclusion that X = 0 is globally asymptotically stable for (7.77).

7.3 Vector norms and Lyapunov stability criteria The following is a direct elaboration of Lemma 7.2, which constitutes a basic result for vector-norms-based Lyapunov stability criteria.

7.3.1 Stability of equilibrium points

In the rst part, it is assumed that for any disturbance d in Sd , X = 0 is a stationary point of (7.1). Then: g(0
d) = 0 8d 2 Sd :

(7.79)

Then, system (7.1) can be described under the form: dX = A(X
d)X (7.80) dt with A(X
d) bounded for any d 2 Sd and any X in some neighbourhood S of X = 0. Notice that X = 0 is also an equilibrium point of (7.1) and (7.80).

Theorem 7.6 Let S 
of system (7.80) associated with p, calculated by means of equations (7.53) or (7.65), or in other words such that:

Dt p(X)  Mp(X) 8t 2 <
8X 2 S
8d 2 Sd
+

and suppose that M is irreducible.

© 2004 by Chapman & Hall/CRC

Let c 2 0 and ;M c > 0. Let u(M) (resp. u(M T )) > 0 a positive importance vector of M (resp. M T ), associated with the importance value

(M), and dene the three types of open sets:

Xc = fX 2 0
D () = fX 2 0
D1 () = fX 2 0: Then, considering that disturbances d remain in Sd : 1

(7.81) (7.82) (7.83)

i) if M is stable, then X = 0 is stable for (7.80) and any set D1 () or D1 () included in S is an estimate Es of its stability domain ii) if M is asymptotically stable, then X = 0 is asymptotically stable for (7.80) and any set Xc strictly included in S is an estimate Eas of its asymptotic stability domain iii) if M is asymptotically stable, then X = 0 is exponentially stable for (7.80) and any set D1() or D1 () strictly included in S is an estimate Ees of its exponential stability domain with regard to (
= ; (M)) iv) if S =
Remark: Since the denition of exponential stability refers to the Euclidean norm k k , coe cient  in (iii) depends on the relation between the norms pi and the Euclidean norm. For example, if all pi (i = 1 : : :k) are of the same type k k! (! = 1
2 or 1), we have:  = 1 for pi = k k
p  = n for pi = k k
p  = n for pi = k k1 : 2

2

1

In this case, (iii) means ; kX(t X0 )k!  exp (M)t kX0 k! :

Proof. ii). Consider the set Xc, with ;M c > 0. The pair (M
q = 0) is an (S Sd )overvaluing pair, then it also is a (Xc  Sd )-overvaluing pair since Xc  S . Thus,

Lemma 7.2 (b and c) ensures that if one denes system dZ = MZ (7.84) dt then ((7:84)
Zc
f0g) is a comparison system of ((7:80)
Xc
f0g) with regard to the asymptotic stability property with the estimate of domains. Since Z = 0 is asymptotically stable for (7.84) with Zc an estimate of its asymptotic stability domain, then point (ii) holds.

© 2004 by Chapman & Hall/CRC

iii). Let us consider the set D1 ()6= S and the positive function v1(X) = u(M T )T p(X)


(7.85)

that veries, for any X in D1 ():

D v (X)  u(M T )T Mp(X) = (M)v (X): (7.86) Now, D () = fX 2
1

1

1

1

1

=

=

v1 (X) = max pi (X)u;i 1 = u;i01 pi0 i=1:::k

(7.87)

with notation i0 = i0 (X), u(M) = u = (ui )i=1:::k . Then,

D v1 (X)  u;i0 +

1

k X j =1

k X mi0 j pj (Xj ) uuj  u;i01 v1 (X) mi0 j uj j j =1

and nally

D v1 (X)  (M)v1 (X): (7.88) Now, D1 () = fX 2
=

i). We suppose M is stable, then v1 (X) (7.85) and v1 (X) (7.87) still are Lyapunov functions, since for any X in S , and from (7.86), (7.88) D+ v1(X)  0 and D+ v1 (X)  0 then X = 0 is stable, and positive invariance of D1 ()  S and D1 ()  S ensures that both are estimates of its stability domain.

Theorem 7.7 Let S 
where M(X) is calculated with (7.53) applied to (7.80). Let c > 0 such that Xc  S (see eqn. (7.38a)) and the corresponding Zc 
dZ = h(Z) dt

(7.90)

is stable (respectively asymptotically stable), then Xe = 0 is a stable equilibrium for (7.80) (respectively asymptotically stable).

© 2004 by Chapman & Hall/CRC

z2 p (x)=⎮x ⎮ 2 2

ε(ze)

z1

ze=0

ε(xe)

p1(x)=⎮x1⎮

Figure 7.4: Estimates relative to Theorem 7.7.

ii) if E (Ze ) is an estimate of the (respectively asymptotic) stability domain of Ze = 0 for (7.90), then the set

E (Xe ) = fX 2
(7.91) is an estimate of the (respectively asymptotic) stability domain of Xe = 0

rrr

for (7.80).

Theorem 7.8 (Borne and Gentina criterion) 87], 173] Let S 
such that:

D p(X)  M(X)p(X)
+

with the following properties: 1) M(X) is irreducible for all X in S  2) there is " > 0 such that M(X) + "Ik is a ;M -matrix 3) the non constant elements of M(X) are grouped into one unique column. Let

(M(X1 )) = sup f (M(X))g  ;" < 0 the upper (with regard to X 2 S ) X 2S importance value of M(X), obtained for X = X1 (possibly not unique). Then, as soon as disturbances d remain in Sd : i) X = 0 is exponentially stable for system (7.1) (with rate = ") and any set D1() included in S : D1() = fX 2
© 2004 by Chapman & Hall/CRC

ii) if S =
rrr

7.3.2 Stability of sets

In this part, X = 0 may not be an equilibrium point, and if it is, it is not necessarily a stable one. However, the following theorem provides the conditions for the existence and estimation of stable sets and associated domains. We consider the system described by (7.1), which can be decomposed into (7.51) for computation of overvaluing pairs (7.53), (7.54) or (7.65). For example, decomposition (7.51) can correspond to the natural decomposition (7.2) in which vector b( ) represents the eect of the disturbance d 2 Sd : dX = f(X) + d(X) = A(X)X + d(X): (7.92) dt In this case, if there is no disturbance, X = 0 is an equilibrium, but the inuence of disturbance creates a dierent behaviour (displacement of the equilibrium or limit cycle, or others) that is to be estimated.

Theorem 7.9 (Borne and Richard criterion) Let S 
Dt p(X)  Mp(X) + q 8t 2 <
8X 2 S
8d 2 Sd
and suppose that M is irreducible and is a ;M -matrix. +

Dene the set





(7.93)

A = X 2 q is an estimate of the asymptotic stability domain of A iii) any set D () = fX 2
© 2004 by Chapman & Hall/CRC

1

z2 p2(x)

K

D1(α) Ze

D1(α')

S 0

X'

Xc

z1 p1(x)

Figure 7.5: Variant of Theorem 7.9.

Remark: a variant of this statement can be provided by using the property that D () (in eqn. (7.28)) is exponentially contractive for linear systems (variable Z).

D () = Z 2
1

1

basis of rhomboids (7.96). In order for domain D1 () to lead to an invariant domain X dened by X = fX 2
0 u; BB ; Ze = ;M q   B B@ ... 1

1

1

u;k 1

1 CC CC : A

(7.97)

Theorem 7.10 Let S 
where M(X), q(X) are calculated by means of (7.53), (7.54) or (7.65) applied to a decomposition (7.51) of system (7.1). Then, considering that the disturbances d remain in Sd :

© 2004 by Chapman & Hall/CRC

z2 p2(x)=⎮x2⎮

ε(A)

ε(ze) S

A ze=0

z1 p1(x)=⎮x1⎮

Figure 7.6: Illustration of Theorem 7.10.

i) if the point Ze 0 is a stable equilibrium (respectively asymptotically stable) of the system:

dZ = h(Z) dt

(7.99)

and if the set A 
A = fX 2
(7.1). ii) if E (Ze ) is an estimate of the stability (respectively asymptotic stability) domain of Ze for (7.99), such that:

A  E (A) = fX 2
(7.101)

then E (A) is an estimate of the stability (respectively asymptotic stability) domain of set A for system (7.1). rrr

Remark: in hypothesis (7.101), A  E (A) is equivalent to fZ 2
overvaluing pair of system (7.1), or in other words such that:

D p(X)  M(X)p(X) + q(X)
+

with a matrix M(X) verifying properties 1), 2), 3) of Theorem 7.8, i.e. such that M(X) + "Ik (" > 0) is an irreducible ;M -matrix, with non constant elements grouped into one unique column. We dene = (M(X1 )) as in Theorem 7.8, u = u(M T (X1 )) a positive associated importance vector of M T (X1 ), and the sets:   n T ; 1 T A = X 2 < : u p(X)  ; sup (u q(X))
(7.102) X 2S

D () = fX 2
© 2004 by Chapman & Hall/CRC

(7.103)

We moreover assume that A  S . Then, considering that the disturbances d remain in Sd : i) A is exponentially stable for system (7.1) (with rate = ") ii) D1() is an estimate of the asymptotic stability domain of A iii) if S =
Proof. The proof is based on the Lyapunov function: v1 (X) = uT p(X): We have:

D v (X)  uT (M(X)p(X) + q(X))
+

1

and due to the particular form of M, this yields:

D v (X)  v (X) + uT q(X): +

1

1

The comparison scalar system

_ = + sup (uT q(X)) X 2S

then converges towards e = ;1 sup (uT q(X)), which denes A. X 2S

7.3.3 Examples

Example 7.8 (Theorem 7.9) Let the disturbed nonlinear system 0 1 ; x ; x (X) + 0:1 1 dX 2 1

2 2

@ dt = (X) + 0:5 2 (

with Sd = d(X) =

1

1 ; x21 ; x22

" (X) !

A = g(X
d) " 0:9 !)

(7.104)

: jd(X)j  : (7.105) 2 (X) 0:5 (notice that the same study is still available if 1 and 2 are functions of both state and time t 162]). Consider p(X) = jX j, and dene two overvaluing pairs on the basis of a separation of <2 in two sets X c , c = ( 1
2 )T : © 2004 by Chapman & Hall/CRC

1

x2 1

-1

3

2

1

x1

-1

Figure 7.7: Simulations of Example 7.8.

1) if X 2 X c , we use the decomposition:

0 1; ; g(X
d) = @

2 2

1 (X) + 0:1

2 (X) + 0:5

1 ; 12 ; 22

2 1

1 0 x ( ; x + ; x ) 1 AX +@ A 1

2 1

2 1

2 2

2 2

x2 ( 12 ; x21 + 22 ; x22)

(7.106) which admits the following (X c  Sd )-local overvaluing pair (see formula (7.65bis)) M1 =

" 1; ; 2 1

1

!

1

2 2

1 ; 12 ; 22


q1 = p2 ( 12 + 22 )3=2 3 3

"1! 1

:

(7.107) 2) if X 62 X c , we directly use the writing (7.104) of g(X
d), that leads to the following (<2 ; Xc  Sd )-local overvaluing pair: M2 =

" 1; ; 2 1

1

!

1

2 2

1 ; 12 ; 22


q2 =

"0! 0

:

(7.108)

Dening M = max fM1
M2g and q = max fq1
q2g leads to the following Sd -local overvaluing pair: " 1 ; 2 ; 2 ! 1 1 2 M = M1 = M2 =
q = q1: (7.109) 1 1 ; 12 ; 22 M is a ;M-matrix i c is such that: cT c = 12 + 22

© 2004 by Chapman & Hall/CRC

> 2:

(7.110)

Under this hypothesis, Theorem 7.9 (iii) ensures that the set A: " 1 !) ( 2 2 3=2 2( + ) (7.111) A = X 2 <2 : jX j  ;M ;1q = p 1 2 2 2 3 3( 1 + 2 ; 2) 1 is globally asymptotically stable, that is it contains all possible asymptotic behaviours of (7.104) for any d 2 Sd (7.105). Now, we can carry on the study by minimizing the vector ;M ;1 q with regard to c = ( 1
2 )T , under the constraint (7.110). This yields for 12 + 22 = 6, ( " p2 !) Aopt = X 2 <2 : jX j  p : (7.112) 2 Simulation  particular cases of behaviour corresponding to  of Fig. 7.72 shows t d(X) = 0:9 cos t
0:5 1 + t2 , and t0 = 0 they are taken from 162].

Example 7.9 (Theorem 7.9) Consider the fourth-order nonlinear system dened

on Example 7.7 (7.77), with additional input disturbance d0: dX = A(X
d)X + d0(X)
X 2 <4
dt (d
d0) 2 Sd  Sd
Sd = fd0 2 <4 : jdj < 0:4g: The R.V.N. p(X) dened as in (7.77bis) as: " kX k ! 1 1 p(X) = kX2k1 leads to the Sd -local overvaluing pair: " ;1 1 ! " 0:4 ! M=
q=
0:4 0:5 ;0:9 hence: " 1:9 ! ; 1 ;M q =
1:5 and Theorem 7.9 (iii) ensures that the set A: A = fX 2 <4 : jX jT  (1:9
1:9
1:5
1:5)g is globally asymptotically stable for (d
d0) 2 Sd  Sd . 0

(7.113)

0

(7.114)

(7.115)

0

Example 7.10 (Theorem 7.10) Let us study the 2nd order system: " # dX = ;2 + jx j + 8 10; jx j X
dt 1;x 1

3

© 2004 by Chapman & Hall/CRC

2

2

7

2 1

2

3

(7.116)

z2 (C)

200

3 2

ze+

1

0 1

2

3

1.6 107

ze-

z1

Figure 7.8: Equilibria and phase portrait of system (7.116). x2

ε(A)

(C)

199

1.1 0.5

A

1.6 107

x1

Figure 7.9: Set A and its asymptotic stability domain estimate, for system (7.116) with any d 2 Sd .

d = ( 1 (X)
2 (X)
3 (X))T is the disturbance vector continuous w.r.t. X, Sd = fd : <2 ! <3 : j 1(X)j < 10;2
j 2 (X)j  0:9
j 3(X)j  0:5g: Considering p(X) = jX j leads to: " ;2 + 10;2z2 0:9 + 8 10;7z3 # 2 h(Z) = Z
(7.117) 0:5 1 ; z12

D p(X)  h(p(X)):

(7.118) Moreover, h is quasi-increasing, and cancels at the origin. Then the preliminary hypotheses of Theorems 7.7 and 7.10 are fullled. Studying (7.117) shows that Z = 0 is an unstable equilibrium, and Theorem 7.7 cannot be applied. However, there exists a positive asymptotically stable equilibrium at " 0:501 ! Ze + =
(7.119) 1:107 +

© 2004 by Chapman & Hall/CRC

0

+

σ

sinσ

-

(s+1)(s+3) s(s+2)(s+4)

y

Figure 7.10: Output feedback of Example 7.11.

with asymptotic stability domain described on Fig. 7.8. System (7.117) actually presents two asymptotically stable equilibria Ze+ , Ze (one positive, one negative) and two unstable ones (the origin and 1:6 107
200]T ), that can be obtained and studied by numerical means since all equilibria are hyperbolic. The manifold (C) denes a boundary of the asymptotic stability domain of Ze+ . Then, Theorem 7.10 leads to the (open) estimate E (A) (see Fig. 7.9) of the asymptotic stability domain of

A = fX 2 < : jx j < 0:501
jx j < 1:107g: 2

1

2

(7.120)

Example 7.11 (Theorem 7.6) We consider the system described on Fig. 7.10, with the aim that y tends to zero. It can be represented, using arrow-matrix modelling 12], 172]: dX = A(X)X dt 2 ;1 0 3 2 3 1:5 x 1 6 77 6 7 1:5 A(X) = 664 0 ;3 75 X = 64 x2 75 : (7.121) x3 =  1 1 ;2 ; sin  We study the equilibrium X = 0. The R.V.N. p(X) = jX j leads to the following S -local overvaluing matrix (where  is a positive parameter):

2 ;1 0 6 M = 64 0 ;3 1

1:5 1:5 1 ;2 ; m

3 77
m= inf sin  = sin  for   3 (7.122) 5 j j<   2

S = fX 2 < : jx j < g: 3

3

(7.123)

In order to guarantee that M is asymptotically stable, it is necessary and su cient that: det M = ;m = ; sin  < 0 © 2004 by Chapman & Hall/CRC

(7.124)

x2 3π

D1(α) D∞(α) −π

π

0

x3

π 3π/2

x1

−3π

Figure 7.11: Estimates of the asymptotic stability domain for Example 7.11.

thus  < , and we dene

S = S () = fX 2 < : jx j <   g: 3

(7.125)

3

Using Theorem 7.6 (i), we obtain the following domain estimates: D1 () = fX 2 <3 : jx1j + 13 jx2j + jx3j < g6= S ()

(7.126)

8 0 3=2 19 > > < =   B C C =2 D1 () = >X 2 < : jX j < B  S () = 2 @ A> 6 :  

(7.127)

3

=

that are depicted on Fig. 7.11. Obviously, D1 ()  D1 () is also an estimate.

Example 7.12 (Theorem 7.8) Consider the equation (with f and f piecewise 1

continuous functions)

dX = dt

" ;1:5 f (X) ! 1

2

f2 (X)

X:

2

(7.128)

Using p(X) = jX j and Theorem 7.8 shows that X = 0 is globally exponentially stable (with exp (;t) convergence) if det

" ;0:5 jf (X)j ! 1

2

1 + f2 (X)

that is, f2 (X) < ;1 ; 4jf1(X)j. © 2004 by Chapman & Hall/CRC

> 0
8X 2
(7.129)

It is globally asymptotically stable if, for a su ciently small " det

" ;1:5 jf (X)j ! 1

2

f2 (X)

= ;1:5 f2(X) ; 2jf1(X)j " > 0:

Example 7.13 (Theorem 7.8)

0 ;3 + (X) dX = B B @ dt

1 CC X + " (X) !
A

x21 + x22 64 ;1

1

x21 x21 + x22

2

3 (X)

Sd = f(

) : j i j < 1 8i = 1
2
3g: Choosing p(X) = jX j, S = fX 2 <
X T X = x + x  64g, leads to 1

2

3

2

D p(X) 

2 1

" ;2 1 !

(7.130)

(7.131) (7.132)

2 2

"1!

p(X) + for all X in S
(7.133) 1 ;1 1 which has been studied in Example 7.5, eqn. (7.32). Theorem 7.8 with variant of eqn. (7.96) proves that the set: +

A = fX 2 < : jx j  2
jx j  3g

(7.134) is asymptotically stable for any disturbance i in the admissible set, and that the sets Xc , c 2 K, hence the set E (see Fig. 7.12) union of sets Xa such that aT a = 64, are estimates of the asymptotic stability domain of A. This estimate can be enlarged by using the variant (eqn. (7.96)) of Theorem 7.8, which is represented by domain E 0. 2

1

2

7.4 Vector norms and practical stability criteria with domains estimation The denition of a decoupled overvaluing system, related to an R.V.N. p(X) and to a (supposed here) constant overvaluing pair (M
q) allows the comparison of the behaviours of p(X) with the solution: ; 

(t t0
p(X0 )) = exp M(t ; t0 ) p(X0 ) + M ;1q ; M ;1 q
(7.135) of the linear system: 8 _ < = M + q
(7.136) : (t0) = p(X0): This yields the following statement:

© 2004 by Chapman & Hall/CRC

x2

8

7

K

B-a+ D1(α)

D∞(α)

S

3 2

A+

1 -8

q 0

ε

ε'

Xe

1

2

8

5

3

x1

A

-8

Figure 7.12: Estimates of the asymptotic stability domain for Example 7.13.

Theorem 7.12 164] Let S 
sets:





X = X 2
XA = X 2 q  S
XF = X 2 q  S 0

0

0

0

+

(7.137)

+

(7.138)

+

(7.139)

Then system (7.1) is practically stable with settling time s with respect to

f+1
X
XA
XF
Sd g provided that p obeys the following relations: p  pA
 pA pF exp (M s) p + M ; q ; M ; q: 0

0

0

0

1

1

(7.140) (7.141)

rrr

Proof. Lemma 7.2 (b) ensures that XA and XF are positively invariant, since M veries the Koteliansky conditions and c = pA or pF veries: ;M c > q. This proves stability aspect. We have for all d in Sd , for all X0 in S , and all t t0 = 0  exp (Mt) p(X0 ) + M ;1q ; M ;1 q p(X(t X0  d)): Let X0 2 X0, then p(X0 ) < p0, and for any t > s we have:  p(X(t X0  d)) < exp (M s) p0 + M ;1q ; M ;1 q  pF : © 2004 by Chapman & Hall/CRC

This proves the contractiveness aspect.

Remark: this way of estimating practical stability domains always provides positively invariant domains X , which means that setting XA = X (i.e. pA = p ) makes no dierence in the calculations, compared to XA  X . Example 7.14 Let us consider again the system of Example 7.8: 0 1;x ;x 1 (X) + 0:1 dX = @ AX (7.142) dt (X) + 0:5 1;x ;x 0

0

0

0

2 1

2 2

1

2 1

2

with

2 2

Sd = d = (
)T : j (X)j  0:9
j (X)j < 0:5
8X 2 < : 1

2

1

(7.143)

2

2

We consider two kinds of studies: rst study: s, XA, XF are given, then nd X0. second study: s, XA, are given, then nd the smallest XF and then the largest corresponding X0. We shall consider the R.V.N. p(X) = jX j.

a) rst study: let

= 0:5s, XA = <2 or X0, pF = (2 2)T . The pair (M
q) is dened by (see (7.107)): " 1 ; 2 1 ! "1! 2 3 M() =
q() = p 
(7.144) 3 3 1 1 1 ; 2 with  > 0 any scalar parameter such that M() is asymptotically stable, i.e.: s

2 > 2 Condition ;MpF > q, from XF denition (7.139), leads to: 3 2 ; 2 > p 3 3 which includes condition (7.145). Condition (7.141) gives the condition relative to p0 :

0 1 2 e+1 e;1 BB 2 CC 6    2 2 B@ C p  642 exp 2 + e;1 e+1 A 2

0

2

2

© 2004 by Chapman & Hall/CRC

3



(7.145) (7.146)

  3 77 " 1 ! : 5

2 e ; exp 2 p 3 3(2 ; 2)

1

(7.147)

p2, ⎮x2⎮ 1053

720

XA =X0 -1053

XF

2

-2

1053

0 2

720

-2

p1,⎮x1⎮

-2

Figure 7.13: Practical stability of (7.142) with the settling time 0.5s, with regard to f+1, X0 , XA , XF , Sd g, rst study.

In order to maximize the right-hand side of inequality (7.147) within the constraint (7.146), we have to choose:  = 4:512, then p0 has to verify: " 1 0:462 ! "1! p0  1052:9 : (7.148) 1 0:462 1 Any domain Xp0 is convenient as an initial set, and then domain X0 dened by the union of the Xp0 such than p0 veries (7.147) is a possible set for practical stability. We also set XA = X0 (see Fig. 7.13).

b) second study: let

s = 0:5s, XA = X0, and we want to minimize XF (then pF ). We have now to verify (7.146) and (7.149):

0 1 e+1 e;1   2 B C 2 2 B C p + p  exp B 2 F @ e;1 e+1 C A 2

0

2

2

3



 

2 e ; exp 2 p 3 3(2 ; 2)

"1! 1

:

(7.149) Minimization p p of pF leads to the same result as in Example 7.8, that is, 2 = 6, pF = ( 2
2)T . The admissible initial set is then characterized by: " 1 0:462 ! " 2 ! p0  : (7.150) 0:462 1 2 This leads to the set X0 (Fig. 7.14), obtained by union of Xp0 such that p0 obeys (7.150).

© 2004 by Chapman & Hall/CRC

p2, ⎮x2⎮ 2

X0

(2) -2

-(2)

1/2

XF

1/2

0 (2)

-(2)

1/2

2

1/2

p1,⎮x1⎮

-2

Figure 7.14: Practical stability of (7.142) with the settling time 0.5s, with regard to f+1, X0 , XA , XF , Sd g, second study.

7.5 Conclusions This chapter was devoted to the vector-norms approach. It appears that such a tool addresses the dierent aspects of stability questions: the stability properties themselves (including practical stability with settling time), but also the related question of the domains estimation and, what was not presented here, the question of constrained control, that appears as a direct application of the preceding invariance properties. Vector norms constitute a simple case of vector Lyapunov functions, and the main presented results can be enlarged to this more general class, provided the considered vector Lyapunov functions verify the following connectivity property 161]:

 the considered V.L.F. p is regular, radially unbounded, and such that the related following set:

X c = fx 2
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{ On the other hand, the main advantage is to allow the unknown disturbing parameters to be taken into account. Lastly, it is important to remark that this chapter was presented in the framework of nonlinear time-invariant systems: but, the method of vector norms addresses more general classes of models, for example nonlinear time-varying systems , or nonlinear time-delay systems with possible time-varying delays 21], 35], 58].

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