Stability for dynamical systems with first integrals: A topological criterion

Systems & Control Letters 19 (1992) 461-465 North-Holland

461

Stability for dynamical systems with first integrals: A topological criterion * D. Aeyels Department of Systems Dynamics, Universiteit Gent, Grote steenweg Noord 6, B-9052 Zwijnaarde, Belgium

R. Sepulchre Center for Systems Engineering and Applied Mechanics (CESAME), Universitd Catholique de Louvain, Bat. Maxwell, Place du Levant 3, B-1348 Louvain.la-Neuve, Belgium

Received 16 March 1992 Revised 19 June 1992 Abstract: This paper provides a new criterion for stability of equilibrium points of topological flows in the presence of first integrals. The result generalizes a well known criterion for stability of such systems obtained by means of Liapunov theory. Keywords: Topological flows; autonomous differential equations; asymptotic stability; stability; first integrals.

1. Introduction T h e m a i n goal o f this p a p e r is to p r e s e n t a c r i t e r i o n for c h e c k i n g t h e stability of an equilibr i u m p o i n t of a t o p o l o g i c a l flow in t h e p r e s e n c e of first integrals. It can b e f o r m u l a t e d as follows: the asymptotic stability o f the e q u i l i b r i u m with r e s p e c t to p e r t u r b a t i o n s b e l o n g i n g to t h e level sets of t h e first i n t e g r a l s c o n t a i n i n g t h e equilibCorrespondence to: R. Sepulchre, Center for Systems Engineering and Applied Mechanics (CESAME), Universit6 Catholique de Louvain, B~t. Maxwell, Place du Levant 3, B-1348 Louvain-la-Neuve, Belgium. Fax: +39-10-472272, Email: [email protected]. * The results presented in this paper have been obtained within the framework of the Belgian Program on Concerted Research Actions and on Minister's Office, Science Policy Programming. The scientific responsability rests with its authors. The first author also acknowledges support by the National Fund for Scientific Research in Belgium (F.K.F.O Grant) and by the E.C. Science Project nr SC1-0433-C(A). This work was done while the first author was visiting the CESAME.

rium p o i n t ( c o n d i t i o n a l a s y m p t o t i c stability) ensures the stability o f the e q u i l i b r i u m with r e s p e c t to a r b i t r a r y p e r t u r b a t i o n s ( u n c o n d i t i o n a l stability). T h e p r o o f will b e given in S e c t i o n 2. A n a t u r a l a p p l i c a t i o n a r e a of t o p o l o g i c a l flows d e a l s with the solutions o f ( a u t o n o m o u s ) differential e q u a t i o n s . In this f r a m e w o r k , t h e p r o b l e m of c h e c k i n g t h e stability of an e q u i l i b r i u m p o i n t in the p r e s e n c e of first integrals has a long history (see [5] for a survey a n d d e t a i l e d b i b l i o g r a p h i c a l notes). Existing stability results a r e essentially b a s e d on t h e c o n s t r u c t i o n of suitable L i a p u n o v functions f r o m t h e k n o w l e d g e o f the first integrals. W e show in Section 3 how t h e p r e s e n t c r i t e r i o n g e n e r a l i z e s s o m e results o f this classical approach. In S e c t i o n 4, we give an a l t e r n a t i v e p r o o f of the c r i t e r i o n u n d e r t h e a d d i t i o n a l a s s u m p t i o n t h a t the J a c o b i a n m a t r i x o f t h e first i n t e g r a l s has full r a n k at the e q u i l i b r i u m point. This implies t h a t in t h e n e i g h b o u r h o o d of t h e e q u i l i b r i u m t h e r e exists a r e g u l a r f i b r a t i o n for the level surfaces. T h e p r e s e n t a p p r o a c h relaxes this a s s u m p t i o n a n d allows for m o r e c o m p l e x situations. Finally, we discuss s o m e n o n - a u t o n o m o u s extensions of the result.

2. A stability result for topological flows with first integrals L e t F b e a t o p o l o g i c a l space. A flow on F is a c o n t i n u o u s function d e f i n e d f r o m F × ~ ~ F a n d d e n o t e d by (y, t ) - ~ y . t which satisfies the following p r o p e r t i e s : for all y ~ F a n d s, t ~ ~, y " 0 = y, Y" ( s + t ) = ( y - s ) .

(la) t.

W e also use t h e n o t a t i o n U ' t i m a g e at t i m e t by t h e flow of a a n d y ' [ t ~ , t2] to d e n o t e t h e arc image of a point y ~ F during the

0167-6911/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

(ib) to d e n o t e the subset U of F d e f i n e d by t h e interval of t i m e

462

D. Aeyels, R. Sepulchre / Stabili~ for dynamical systems with first integrals

[tl, t2]. A local flow can be defined on a subset S or F if for 7 ~ S there exists a neighbourhood U of 7 and an s > 0 such that ( U n S ) . [ 0 , e) is contained in S. For 7 E F , the set 7"[0, ~) is called the positive trajectory of YIn the following, we consider a local flow @ defined on an open set of R" containing zero, and we assume that zero is an equilibrium point for the flow, i.e. 0. t = 0 Vt ~ E. For x ~ R ~ and u ~ R, we use the notation B(x, u) to denote the u-open ball centered at x. We also denote B± the open ball centered at zero and of radius e, B~ its closure and S~ the boundary of B~. As usual, the equilibrium point zero is said to be stable if for each e > 0, there exists a fi > 0 such that B~ • t a B E for all positive t. If in addition all trajectories converge asymptotically to zero in an neighbourhood of zero, the equilibrium is said to be asymptotically stable. Furthermore, we assume that the flow possesses k continuous first integrals (or constants of motion) that are denoted by the continuous function

G :~" -~ ~k : x ~ G( x)

=

(Gl( x) . . . G k ( X) ).

For any h ~ R k, we define the h-level set as the invariant set

Lh={x~R'lG(x

) =h}.

We assume without loss of generality that the origin belongs to L0, i.e. the function G ( x ) vanishes at the origin. By definition, each level set L h is invariant with respect to the flow. This allows us to define a flow restricted to the h-level set L h which will be denoted by @h. As a consequence, the origin is also an equilibrium point of the flow @0 and it makes sense to speak about the stability of the origin with respect to @0, that is with respect to perturbations restricted to the same level set. Theorem 1. If the origin is an asymptotically stable equilibrium of the flow qb°, then it is a stable equilibrium of the flow c19. Proof (ex absurdo). Suppose 0 is unstable. Then, there exists an e > 0 for which we can construct in B~ a sequence (Pi)i~>~ ~ 0 such that for each i, the positive trajectory of p, does not stay within B~ for all positive times. Let q~ denote the (first) exit point from B~ of the positive trajectory of Pi.

All the qi are in the compact set S t and thus there exists a subsequence (qj)j>~ converging to some q ~ S~. Noticing that G(qj) = G(pj) for each j, this implies that q belongs to L 0 by continuity of the first integrals. Now the asymptotic stability of the origin for the system restricted to L 0 implies that there exists some fi > 0 (and smaller than ½s) such that every trajectory starting in B~ n L o never leaves Be/2 and moreover converges to zero. For j large enough, the positive trajectory of pj starts in B~ and hence intersects $6 before reaching S~; denote by rj the last intersection point of the positive trajectory starting at P1 with S~, before reaching the point qj. In this way we obtain a third sequence (rj)y>_~ a subsequence of which converges to r ~ S~. Remark that r belongs to S~ ca L0; this means that the positive trajectory of r never leaves B~/2 and eventually enters Bs. By continuity of the flow, there exists a finite T > 0 such that r . T belongs to B6. Pick a u ~> 0 such that B ( r . T, u) a B6. By L e m m a 1 (see Appendix), there exists a neighbourhood V(r) of r such that if y belongs to V(r), the positive trajectory of y does not leave a u-tubular neighbourhood of the positive trajectory of r before the time T. Notice that the tubular neighbourhood is taken such that there is no intersection with S,:. For j large enough, rj belongs to V(r). Hence the positive trajectory of rj does not leave the u-tube before the time T and in particular rj. T B s. However, by construction, the positive trajectory of rj ~ S~ has no other intersection with the compact set S s before reaching the point qj on S~. This means that the positive trajectory of rj leaves the open ball B~ before entering the open ball B~. But this is not possible without leaving the above constructed u-tube, which leads to a contradiction. [] R e m a r k that the assumption of asymptotic stability in L 0 is trivially satisfied when the origin is isolated in L 0. This leads to the following corollary: Corollary 1. If the origin is the unique solution of the equation G( x ) = 0 in a neighbourhood of zero, then the origin is stable. For a direct proof of Corollary 1, we refer to [1].

D. Aeyels, R. Sepulchre / Stability for dynamical systems with first integrals

Remark. We want to stress that no differentiable manifold structure has been assumed on the level sets. This point is discussed in Section 4.

3. Connections with the Liapunov approach In the context of differential equations, the problem of checking stability in the presence of first integrals has a long history motivated by the study of stationary motions of mechanical (Hamiltonian) systems. The classical approach is to construct suitable Liapunov functions from the knowledge of the first integrals. In particular, it is sufficient for proving stability to combine the first integrals into a single positive definite function because in this case the derivative along the solutions vanishes. The condition that G ( x ) = 0 has (locally) no zeros outside the origin (Corollary 1) is clearly equivalent to the fact that [IG(x)ll is (locally) positive definite. As a consequence, II G(x)II is in this case a simple Liapunov function for checking the stability of the equilibrium. In fact, the condition of Corollary 1 is sharp because when it is not satisfied, it is never possible to construct a positive definite function by combining the first integrals. This is contained in the following result:

Theorem 2 (Pozharitskii [4]). There exists a continuous function V: E k ~ ~ such that V(G(x)) is positive definite if and only if II G(x)II is positive definite, where I1" II is any norm in ~ . T h e E n e r g y - C a s i m i r m e t h o d [1,2] (or Chetaev's method, see [5]) proves to be helpful in many situations to examine the stability of an equilibrium point. In [1] it is shown that it actually gives a sufficient condition for the assumption of Corollary 1 to be satisfied. Hence Theorem 1 is a generalization of these classical results to the case where there exists a non-trivial intersection of the level sets at the origin.

4. Regularity at the equilibrium point In this section, we assume that the flow q~ is defined by the solutions of a differential equation A:=f(x),

f ( 0 ) = 0,

x ~ ~",

(2)

463

and we assume that the flow exhibits differentiable first integrals, denoted by G : ~n ~ ~k. The main interest of Theorem 1 is in the absence of regularity assumptions on the first integrals; for completeness, we give an alternative approach under the following assumption: Regularity Assumption. The Jacobian matrix of G(x) evaluated at zero is of full rank. This will be called the 'regular case'. In such a situation, by the implicit function theorem, the level sets L h are (locally) homeomorphic to each other. This means that equation (2) can be (locally) parametrized on each level set L h by the value h or equivalently can be (locally) rewritten as

= g ( z , h), h=0,

z~

(3a)

n-~,

h~

~.

(3b)

From Theorem 1, we immediately get the following: Corollary 2. If 2 = g(z, O) is asymptotically stable at the origin z = 0, then the origin (z, h) = (0, 0) of system (3) is stable. This can also be deduced from the following Liapunov argument: the asymptotic stability of z = 0 for the restricted system 2 = g(z, O) defined on i n-k implies the existence of a Liapunov function V(z) which strictly decreases along the flow q~0 in a neighbourhood of the origin; for e > O , let V~ the set of points z where V is smaller than e; for small e > O, there exists a r E > 0 such that the derivative of V ( z ) along the (positive) trajectories of the flow q~ is non-positive in the set {(z, h ) I V ( z ) = e

and Ih[ 0, the image of V~ x [-r~., r E] by the flow q~ is contained in V~ x [ - r ~ , r E] for all positive times. This yields the stability of (z, h ) = (0, 0) for q~. On the contrary, when the Jacobian matrix of G(x) is not of full rank at the origin, equation (2) does not necessarily admit the parametrized form

D. Aeyels, R. Sepulchre / Stabilityfor dynamicalsystems withfirst integrals'

464

(3) (the level sets may be no longer homeomorphic to each other). However, Theorem 1 remains valid. For a simple illustration, consider the following example:

Example 1. Suppose 2 = -x(x 2-ye), 3) = y ( x 2 _ y 2 ) ,

(4a) X ~ ~, y E [~.

(4b)

The system admits a first integral G(x, y ) = x y the Jacobian of which is not regular at the origin. The level set L 0 corresponds to the x-axis. Since the system is clearly asymptotically stable on L 0, Theorem 1 shows stability at the origin.

5. Concluding remarks (i) The standard stability results in the context of differential equations with first integrals provide sufficient conditions for an isolated intersection of the first integrals at the equilibrium point; we have seen that this implies stability. When the intersection is non trivial, one wonders what kind of additional property to the conditional stability of the equilibrium point leads to the unconditional stability of this equilibrium. The additional property treated in this paper is the conditional attractivity of the equilibrium which may be seen as a generalization of the case where the intersection of the first integrals is isolated. It emphazises the strong implications of attractivity, at the topological level, i.e. independent of regularity assumptions; this holds in spite of the fact that attractivity is in no way a structural property: for instance consider the system

2=x(h2-x2),

h=0;

Suppose (i) g(z, h, t) is locally Lipschitzian in (z, h), uniformly with respect to t; (ii) the equilibrium z = 0 of 2 = g( z, O, t) is uniformly asymptotically stable, (rid the equilibrium h = 0 of it = s(h) is stable. Then the equilibrium (z, h ) = (0, O) is uniformly stable. This means that in the regular case, and under a Lipschitz condition, the result of Theorem 1 can be directly extended to non-autonomous differential equations which exhibit autonomous first integrals. Other non-autonomous extensions (including the non-regular case and non-autonomous first integrals) will be considered by K. Peiffer in a forthcoming paper. Finally, we remark that even in the regular case, an extension of Theorem 1 for non-autonomous differential equations requires the introduction of a uniform Lipschitz condition. This is illustrated by the following example:

Example 2. Suppose 2 = - x + sin(tv),

(6a)

1~=0,

(6b)

x~R,

y~.

Assumptions (i) and (ii) of Theorem 3 are satisfied. Nevertheless, the origin of system (6) is unstable.

Acknowledgments We thank Karl Peiffer for fruitful discussions about the Liapunov approach and an anonymous referee for his contribution in Section 4.

Appendix

the point x = 0 is an unstable equilibrium on each level set Lh, h ~ 0, and a (globally) attractive equilibrium on the x-axis, i.e. on L 0. (ii) In the regular case, the main result (Theorem 1) must also be compared to the following result of 'total stability':

Lemma. Consider the flow cl) defined in Section 1. If U is an 'e-tubular neighbourhood' of the arc x ' [0, T], then there exists a neighbourhood V of x such that

y~V

~

Ix't-y'tl N (sufficiently large), we have that B cB(X'tn, e). Therefore we have that

B'(-t,)cB(x't,,e)'(-t,),

n>U.

(9)

If t n ---,~, then by assumption, the right hand side

465

tends to the single point x. On the other hand, by continuity of the flow as a function of t, the left hand side tends to B . ( - D which has positive diameter. This is a contradiction. []

References [1] D. Aeyels, On stabilization by means of the EnergyCasimir method, Systems Control Lett. 18 (1992) 325-328. [2] A.M. Bloch and J.E. Marsden, Stabilization of rigid body dynamics by the Energy-Casimir method, Systems Control Lett. 14 (1990) 341-346. [3] A. Isidori, Nonlinear Control Systems (Springer-Verlag, Berlin, 1988). [4] G.K. Pozharitskii, On the construction of Liapuriov functions from the integrals of the equations of the perturbed motion (Russian), Prikl. Mat, Meh. 22 (1958) 145-154. [5] N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov's Direct Method (Springer-Verlag, New-York, 1977).