Practical stability in dynamical systems

Chaos, Solitons and Fractals 11 (2000) 1087±1092

www.elsevier.nl/locate/chaos

Practical stability in dynamical systems Xiao-song Yang Institute for Nonlinear Science, Chongqing University of Posts and Telecommunication, Chongqing 400065, People's Republic of China Accepted 11 January 1999 Dedicated to Prof. Xiaoxin Liao on the occasion of his 60th birthday

Abstract Practical stability is of signi®cant practical importance in scienti®c and engineering problems but less investigated. In this paper we revisit the practical stability theory and present new results. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction The notion of practical stability in dynamical stability was discussed by Lasalle and Lefshetz [2] in the 1960s and then was treated by Liao [3]. Recently, this concept has been renewed by Kapitaniak and Brindley [1] in dealing with stability of chaotic attractors and synchronization. For readersÕ convenience, we ®rst recall the classical practical stability of equilibrium points of dynamical systems. Consider a dynamical system described by dx ˆ f …x; t†; dt

…1†

where f …0; t†  0; f 2 cr ‰B  R‡ ; Rn Š; R‡ ˆ ‰0; 1†, and B is a region in Rn containing x ˆ 0. Let the system (1) be under in¯uence of a permanently acting perturbation p…x; t† with kp…x; t†k 6 d so that the perturbed system is dx ˆ f …x; t† ‡ p…x; t†: dt

…2†

De®nition 1. Given a positive number d and two sets Q and Q0 such that 0 2 Q; Q0  Q  B and Q is bounded. If every solution x…t0 ; x0 ; t† to Eq. (2) satis®es x…t0 ; x0 ; t†  Q for 8x0 2 Q0 and 8p…x; t† with kp…x; t†k 6 d, then the equilibrium solution x ˆ 0 to Eq. (1) is called practically stable with respect to d, Q and Q0 . The notion of practical stability is relevant to number d and sets Q and Q0 . For practical systems, Q is the permissible state set and Q0 is the initial state set. Therefore, in order to study practical stability in a dynamical system, we should ®rst know: (i) the scope of the permissible state set; (ii) the amplitude of p…x; t† (i.e., what the number d is); (iii) to what extent we can control the initial conditions (i.e., how large the initial state set is). For this kind of stability, a known result is the following theorem. 0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 0 1 2 - 0

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Theorem A [2,3]. Suppose that Q0  Rn is a compact set containing 0 2 Rn . If there exists a function V …x; t† 2 cr ‰Rn  R‡ ; RŠ such that D‡ V …x; t† j…2† 6 0 8x 2 Qc0 and V …x1 ; t1 † < V …x2 ; t2 † 8x1 2 Q0 ;

8x2 2 Qc ;

t2 P t1 P 0;

where D‡ denotes the Dini derivative, and Sc means the complementary set of S. Then every solution x…t0 ; x0 ; t† to Eq. (2) is contained in Q for t P t0 thus the equilibrium solution x ˆ 0 to Eq. (1) is practically stable with respect to d, Q and Q0 . Another stability is the so-called practical stability in ®nite time period. De®nition 2. The solution x ˆ 0 to Eq. (1) is said to be practically stable with respect to d, Q and Q0 in ®nite time period ‰t0 ; T Š, if for every perturbation p…x; t† with kp…x; t†k 6 d the solution x…t0 ; x0 ; t† to Eq. (2) remains in Q for t 2 ‰t0 ; T Š and x0 2 Q0 . Theorem B [2,3]. If there exists a function function V …x; t† 2 cr ‰Rn  R‡ ; RŠ such that V 6 l0 for x 2 Q0 and V P l for x 2 Qc , furthermore if dV j…2† 6 …l ÿ l0 †=…T ÿ t0 †; dt then the solution x ˆ 0 to Eq. (1) is practically stable with respect to d, Q and Q0 in finite time period ‰t0 ; T Š. There is little work on practical stability of equilibrium points, to say nothing about chaotic attractors, in dynamical systems. Even in the above theorems there still is a common problem: the stability criteria are established in terms of perturbed systems and this is usually dicult or impossible because we do not know the exact form of perturbations. In addition, the practical stability should be an intrinsic property of a dynamical system. Therefore it is reasonable to establish the stability criteria in terms of the original unperturbed systems. Motivated by [1], we consider this problem again, and the purpose of this paper is to investigate general properties of dynamical systems which are practically stable and to establish some practical stability criteria for equilibrium points. 2. Practical stability of equilibrium points First let us see what conditions are necessary for the zero solution to Eq. (1) to be practically stable. 2.1. Necessary conditions The following fact is obvious. Proposition 2.1. S If the zero solution to Eq. (1) is practically stable with respect to d, Q and Q0 , then 1. x…Q0 † ˆ x2Q0 x…x†  Q 2. kf …x†k P d for some x 2 Q: Let kf kQ ˆ maxx2Q kf …x†k=kxk, we have the following result. Theorem 2.2. Suppose that the zero solution to Eq. (1) is globally asymptotically stable in Lyapunov' sense. If x ˆ 0 is practically stable with respect to d, Q and Q0 , then d=kf kQ 6 maxkxk: x2Q

…3†

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Proof. If this is not the case, then d=kf kQ > maxx2Q kxk and we can take a constant perturbation p with kpk ˆ d. Now there are two cases to be considered here. (i) There exists an equilibrium point x0 to Eq. (2). Since x ˆ 0 to Eq. (1) is globally asymptotically stable, the equilibrium point x0 to Eq. (2) is also globally asymptotically stable. Nonetheless, from f …x0 † ‡ p ˆ 0, it follows that kf kQ
kx0 k P kf …x0 †k ˆ kpk ˆ d: Therefore kx0 k P d=kf kQ > maxkxk; x2Q

which implies that x0 62 Q, leading to a contradiction by de®nition. (ii) There exists no point satisfying f …x† ‡ p ˆ 0. In this case, one constructs a function V …x† ˆ p
x and its derivative along the trajectories to Eq. (2) is dV j…2† ˆ p
f …x† ‡ kpk2 6ˆ 0: dt If there exists a point x such that p
f …x † ‡ kpk2 ˆ p
f …x † ‡ d2 ˆ 0 then d2 ˆ p
f …x † 6 kpk
kf …x †k 6 dkf kQ kx k; it follows that kx k P d=kf kQ > max kxk: x2Q

Therefore one sees that dV =dt j…2† > 0 or dV =dt j…2† < 0 for trajectories in Q. It follows that no solution to Eq. (2) is practically stable, contrary to the hypothesis. The proof is thus completed.  Under a slightly di€erent condition, inequality Eq. (3) still holds. Theorem 2.3. Suppose that Q0 ˆ Q and Q is diffeomorphic to the n-ball. If the zero solution to Eq. (1) is practically stable with respect to d, Q and Q0 , then d=kf kQ 6 maxkxk: x2Q

Proof. As similarly as before, suppose that d=kf kQ maxx2Q kxk. Take a constant perturbation p with kpk ˆ d. If there exists an x0 such that f …x0 † ‡ p ˆ 0, then d ˆ kf …x0 †k 6 kf kQ kx0 k: Therefore kx0 k P d=kf kQ > maxkxk x2Q

This implies that the equilibrium point x0 62 Q, consequently there exists no equilibrium point in Q. On the other hand, the practical stability guarantees that every solution to Eq. (2) with initial point in Q remains in Q for t P t0 . By Poincare±Hopf index Theorem [4], the vecto®eld f …x† ‡ p has at least one equilibrium point in Q. Thus we get a contradiction. 

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2.2. Sucient conditions Lemma 2.4. Suppose the permissible state set Q has nonempty C1 -differentiable boundary oQ. Denote by N …x† the normal unit vector to oQ at x 2 oQ, which points outwards. If for every perturbation p…x; t† with kp…x; t†k 6 d we have hf …x† ‡ p…x; t†; N …x†i < 0;

8x 2 oQ;

t P t0 ;

…4†

then the solution x ˆ 0 to Eq. (1) is practically stable with respect to d, Q and Q0 . Proof. Let y…t0 ; y0 ; t† be a solution to Eq. (2) with initial condition t ˆ t0 , y0 2 oQ0 . Suppose that there exists t1 > t0 such that y…t0 ; y0 ; t1 † 62 Q, then there exist t3 ; t2 ; t0 6 t3 < t2 6 t1 , such that y…t0 ; y0 ; t† 2 Q; y…t0 ; y0 t† 2 Q

t 2 ‰t0 ; t3 Š; t 2 …t3 ; t2 †:

Clearly, y…t0 ; y0 ; t3 † 2 oQ, and in a neighborhood of y…t0 ; y0 ; t3 †, the function H …t† ˆ hy…t0 ; y0 ; t†; N …y…t0 ; y0 ; t3 †i Is a increasing function for t 2 …t3 ÿ g; t3 ‡ g†, where g is a suciently small positive number. It follows that for t 2 …t3 ÿ g; t3 ‡ g†; 06

dH …t† ˆ dt



dy…t0 ; y0 ; t† ; N …y…t0 ; y0 ; t3 † dt



ˆ hf …y…t0 y0 t†† ‡ p…y…t0 ; y0 t†; t†; N …y…t0 ; y0 ; t3 †i In particular, this is true at t ˆ t3 , i.e.,hf …y…t0 y0 t†† ‡ p…y…t0 ; y0 t†; t†; N …y…t0 ; y0 ; t3 †i P 0, which is in contradiction to Eq. (4). Theorem 2.5. Suppose that (1) satisfies hf …x†; N …x†i < ÿd 8x 2 oQ; then the zero solution x ˆ 0 to Eq. (1) is practically stable with respect to d, Q and Q0 . Proof. For x 2 oQ, we have hf …x† ‡ p…x; t†; N …x†i ˆ hf …x†; N …x†i ‡ hp…x; t†; N …x†i hÿd ‡ kp…x; t†k
kN …x†k ˆ ÿd ‡ d
1 ˆ 0: It follows from the above lemma that the statement is true. Corollary 2.6. Suppose that B…x† : Q ˆ f x 2 Rn : B…x† 6 0. If

the

permissible

state

hf …x†; gradB…x†=kgradB…x†ki < ÿo; x 2 oQ ˆ Bÿ1 …0†: Then the zero solution x ˆ 0 to Eq. (1) is practically stable. Let us consider a special but natural case.

 set

Q

is

described

by

a

C1 -function

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3. k-practical stability In this section we consider a natural situation where the permissible state set is a ball, that is, Q ˆ B…k† ˆ fx 2 Rn : kxk P kg. For convenience, we call the practical stability in this situation k-practical stability. In this case it is easy to establish a Lyapunov theory for k-practical stability of equilibrium points. Theorem 3.1. Suppose that there exists a positive definite Lyapunov function V …x† 2 c1 ‰Rn ; RŠ with the following properties: (i) V …x† > V …x‡ †; for kxk > kx‡ k ˆ g; (ii) j V …x† ÿ V …y† j 6 kkx ÿ yk; j 6 ÿ hV …x†, (iii) dV dt …1† (iv) h
minkxkˆg V …x† > dk, where h and k are positive numbers. Then x ˆ 0 to Eq. (1) is k-practically stable. Proof. On the one hand, the derivative along the trajectories to Eq. (2) is dV j…2† 6 ÿ hV …x† ‡ kkp…x; t†k: dt On the other hand, for a trajectory y…t0 ; y0 ; t† to Eq. (2), if there exists t1 P t0 such that y…t0 ; y0 ; t1 † 62 Q; then there exists a maximal s > 0, such that for t 2 ‰t1 ; t1 ‡ s†; y…t0 ; y0 ; t† 62 Q and for t 2 ‰t1 ; t1 ‡ s†, dV j…2† 6 ÿ hV …y…t0 ; y0 ; t†† ‡ kkp…y…t0 ; y0 ; t†; t†k; dt < ÿh
min V …x† ‡ dk; kxkˆg

…5†

< 0: This implies that V …y…t0 ; y0 ; t††decreases for t 2 ‰t1 ; t1 ‡ s† and reaches the region Q. It is easy to see from Eq. (5) that when y…t0 ; y0 ; t1 † goes into Q, it can never get out. The proof can be completed by readers. Let us consider a special case for practical stability in ®nite time period. Suppose that Q0 ˆ D…r0 †; Q ˆ D…k†; where D…r0 †…D…k†† denotes the ball centered at the origin with radius r0 …k†. Now suppose r0 < k and let Qc ˆ Q ÿ Q0 . We have the following assertion. Theorem 3.2. If Eq. (1) satisfies hx; f …x†i 6 …k ÿ r0 †=2…T ÿ t0 † ÿ dkxk

8x 2 Qc ;

then the zero solution to Eq. (1) is k-practically stable with respect to Q0 and d in finite time T ÿ t0 . Proof. Consider the Lyapunov function V …x† ˆ hx; xi, then we have dV j…2† ˆ hx; f …x†i ‡ hx; p…x; t†i 6 …k ÿ r0 †=2…T ÿ t0 † ÿ dkxk ‡ hx; p…x; t†i: dt Since hx; p…x; t†i 6 kxk
kp…x; t†k 6 dkxk, we have dV j…2† 6 …k ÿ r0 †=…T ÿ t0 †; dt therefore along the trajectory y…t0 ; y0 ; t† to Eq. (2) with y0 2 Q0 , we have V ……y…t0 ; y0 ; t†† 6 k ÿ r0 ‡ V ……y…t0 ; y0 ; t0 ††;

t0 6 t 6 T

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Notice the form of this Lyapunov function we can conclude that a solution with initial point in Q0 remains in Q in ®nite time period T ÿ t0 thus completing the proof.  It is clear that similar results can be established for discrete dynamical systems. References [1] [2] [3] [4]

T. Kapitaniak, J. Brindley, Practical stability of chaotic attractors, Chaos, Solitons and Fractals 9 (1998) 43±50. J. Lasalle, S. Lefschetz, Stability by Liapunov, Direct Method and Application, Academic Press, New York, 1961. X. Liao, Stability Theory and its Applicatiions, Huazhong Normal University Press, 1988 (in Chinese). J.W. Milnor, Topology from the Di€erentiable View Point, The University Press of Virginia, Charlottesville, 1965.