On Equivalence of Exponential and Asymptotic Stability Under Changes of Variables

We show that uniformly global asymptotic stability for a family of ordinary diierential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables.


Introduction
Lyapunov's notion of (global) asymptotic stability of an equilibrium is a key concept in the qualitative theory of di erential equations and nonlinear control.In general, a far stronger property is that of exponential stability, which requires decay estimates of the type \kx(t)k ce t kx(0)k."In this paper, we show that, for di erential equations evolving in nite-dimensional Euclidean spaces R n (at least in spaces of dimensions 6 = 4; 5) the two notions are one and the same under coordinate changes.Of course, one must de ne \coordinate change" with care, since under di eomorphisms the character of the linearization at the equilibrium is invariant.However, if we relax the requirement that the change of variables be smooth at the origin, then all obstructions disappear.The basic ingredient of the construction we are about to present relies on the existence of smooth Lyapunov functions V .The coordinate transformations are constructed via \projecting" along the gradient ow of V onto a level set V 1 (c).The result now relies on the fact that this level set is di eomorphic to the standard sphere, which is true except for those cases where the Poincar e conjecture is still open.This explains why we have to exclude n = 4; 5 from our statements.Closely related to our work is the fact that all asymptotically stable linear systems are equivalent (in the sense just discussed) to _ x = x; see e.g.1].

Problem Statement
Throughout the paper, k k denotes the usual Euclidean norm, and \smooth" means C1 .For a di erentiable function V : R n !R the expression L f d V (x) denotes the directional derivative LARS GR UNE, EDUARDO D. SONTAG, FABIAN WIRTH DV (x)f(x; d).We consider the family of di erential equations _ where f : R n D !R n is continuous and for x 6 = 0 locally Lipschitz continuous in x, where the local Lipschitz constants can be chosen uniformly in d 2 D R m .We assume that D is compact and that f(0; d) = 0 for all d 2 D and let D denote the set of measurable functions from R to D. Then we say that the zero state is uniformly globally asymptotically stable (UGAS) if there exists a class KL function 1 such that, for each d( ) 2 D, every maximal solution is de ned for all t 0 and k (t; x; d( ))k (kxk; t) ; 8t 0 : (2.2) Note that while our general assumptions on f do not guarantee uniqueness of solutions through zero, assumption (2.2) implies that (t; 0; d) 0 is the unique solution with initial condition x = 0, for all d 2 D and thus the same is true for every initial condition.
An apparently stronger formulation of (2.2) is the following.We call the zero position of ( 2 Extending the concepts in 1, p. 207] to our nonlinear setting, we will call a homeomorphism T : R n !R n a change of variables if T (0) = 0, T is C 1 on R n , and T is di eomorphism on R n n f0g (i.e., the restrictions of T and of T 1 to R n n f0g are both smooth).Given a change of variables T and a system (2.1), we may consider the transformed system _ y(t) = f(y(t); d(t)) ; (2.4) where, by de nition, f(y; d) = DT (T 1 (y))f(T 1 (y); d) : In other words, system (2.4) is obtained from the original system by means of the change of variables y = T (x).Observe that the new system again satis es the general requirements.
It is our aim to show that for dimensions n 6 = 4; 5 the following assertions are true.Given a system of the form (2.1) satisfying (2.2) there exists a transformed system that satis es (2.3).In this sense, global asymptotic stability is equivalent to global exponential stability under nonlinear changes of coordinates.Furthermore, one may obtain transformed systems where the constants de ning the exponential stability property can be chosen to be the special values c = = 1.

Statement of Results
The main tool for our construction of T is the use of an appropriate Lyapunov function V .In fact, we can obtain T for a whole class of functions as stated in the following proposition.
Recall that a function V : R n !R is called positive de nite if V (0) = 0 and V (x) > 0 for all x 6 = 0, and proper if the set fx j V (x) g is bounded for each 0. Proposition 3.1 Let n 6 = 4; 5 and let V : R n !R be a proper, positive de nite C 1 function.Assume furthermore that V is smooth on R n n f0g with nonvanishing gradient.Then there exists a class K 1 function which is smooth on (0; 1) and satis es (s)= 0 (s) s and a change of variables T : R n !R n with T (0) = 0 such that Ṽ (y) := V (T 1 (y)) = (kyk) : (3.5) Outline of proof: Let denote the smooth ow determined by _ x = rV (x) 0 krV (x)k 2 ; Fix c > 0 and de ne the smooth map : R n n f0g !V 1 (c) by (x) = (c V (x); x).Now observe that the properties of V imply that V 1 (c) is a homotopy sphere (cf.also 6, Discussion after Theorem 1.1]), so that V 1 (c) is di eomorphic to S n 1 for n 6 = 4; 5 ( see 2] for n = 1; 2; 3, 5, S9, Proposition A] for n 6).Now T is given by T (0) = 0 ; and T (x) = 1 (V (x)) S( (x)) ; x 6 = 0 : It is straightforward to see that T satis es (3.5).For the remaining statements see 3].Theorem 3.2 Let n 6 = 4; 5 and consider any system (2.1) on R n which is UGAS (2.2).Then, (2.1) can be transformed into a system (2.4) that is UGES (2.3).In particular, the constants in (2.3) can be chosen to be c = 1; = 1.
satisfes L f d W (x) W (x):Applying Proposition 3.1 to W , we obtain for each d 2 D and y 6 = 0 h f(y; d); yi = kyk 0 (kyk) L fd W(y) kyk 0 (kyk) W(y) = kyk 0 (kyk) (kyk) kyk 2 : .1) uniformly globally exponentially stable (UGES), if there exist constants c 1; > 0 such that If the origin is no common xed point for all values d 2 D then (2.2) is impossible.In this case, however, still a useful notion of stability is possible which is known as input-to-state stability.For this stability concept similar results to those discussed in this paper can be obtained, 3].