Asymptotic stability equals exponential stability, and ISS equals finite energy gain - if you twist your eyes

In this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables. The same is shown for input-to-state stability and input-to-state exponential stability, and for input-to-state exponential stability and a nonlinear $H_\infty$ estimate.


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." (See for instance 16] for detailed discussions of the comparative roles of asymptotic and exponential stability in control theory.) 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 (which we take to be the origin) is invariant. However, if, in the spirit of both structural stability and the classical Hartman-Grobman Theorem (which, cf. 23], gives in essence a local version of our result in the special hyperbolic case), we relax the requirement that the change of variables be smooth at the origin, then all obstructions disappear. Thus, we ask that transformations be in nitely di erentiable except possibly at the origin, where they are just continuously di erentiable. Their respective inverses are continuous globally, and in nitely di erentiable away from the origin.
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]. The basic idea of the proof in 1] is based upon projections on the level sets of Lyapunov functions, which in the linear case of course be taken to be quadratic (and hence have ellipsoids as level sets). It is natural to use these ideas also in the general nonlinear case, and Wilson's paper 36], often cited in control theory, remarked that level sets of Lyapunov functions are always homotopically equivalent to spheres. Indeed, it is possible to obtain, in great generality, a change of coordinates rendering the system in normal form _ x = x (and hence exponentially stable), and several partial versions of this fact have appeared in the literature, especially in the context of generalized notions of homogeneity for nonlinear systems; see for instance 6,25,15,27,24].
It is perhaps surprising that, at least for unperturbed systems, the full result seems not to have been observed before, as the proof is a fairly easy application of results from di erential topology. (Those results are nontrivial, and are related to the generalized Poincar e conjecture and cobordism theory; in fact, the reason that we only make an assertion for 6 = 4; 5 is closely related to the fact that the original Poincar e conjecture is still open.) Note, however, that it has been common practice in the papers treating the nonlinear case to use the ow generated by the original system to de ne an equivalence transformation, thereby reducing the regularity of the transformation to that of the system. Here we use the ow generated by the (normalized) Lyapunov function itself, which yields more regular transformations. In addition, and most importantly, our poof also allows for the treatment of perturbed systems (for which the reduction to _ x = x makes no sense).
Lyapunov's notion is the appropriate generalization of exponential stability to nonlinear di erential equations. For systems with inputs, the notion of input to state stability (ISS) introduced in 29] and developed further in 5,9,13,14,17,18,26,28,32,33] and other references, has been proposed as a nonlinear generalization of the requirement of nite L 2 gain or, as often also termed because of the spectral characterizations valid for linear systems, \ nite nonlinear H 1 gain" (for which see e.g. 2,11,12,34]). We also show in this paper that under coordinate changes (now in both state and input variables), the two properties (ISS and nite H 1 gain) coincide (again, assuming dimension 6 = 4; 5).
We do not wish to speculate about the implications of the material presented here. Obviously, there are no \practical" consequences, since nding a transformation into an exponentially stable system is no easier than establishing stability (via a Lyapunov function). Perhaps these remarks will be of some use in the further theoretical development of ISS and other stability questions. In any case, they serve to further justify the naturality of Lyapunov's ideas and of concepts derived from his work.

Setup
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 . Let D denote the set of measurable, locally essentially bounded functions from R to D. For any x 0 2 R n and any d( ) 2 D, there exists at least one maximal solution of (1) for t 0, with x(0) = x 0 . By abuse of notation, we denote any such solution, even if not unique, as (t; x 0 ; d( )), t 2 I(x; d( )), where I(x; d( )) is its existence interval. Throughout the paper, k k denotes the usual Euclidean norm, and \smooth" means C 1 . For a di erentiable function V : R n ! R the The general framework a orded by the model (1) allows us to treat simultaneously classical di erential equations (the case when D = f0g) and more generally robust stability of differential equations subject to perturbations (when functions in D are seen as disturbances which do not change the equilibrium, as in parameter uncertainty), as well as systems with inputs in which elements of D are seen as exogenous tracking or regulation signals, or as actuator errors (in which case, the continuity properties of (x; d) 7 ! ( ; x; d) are of interest).
In light of these applications, we now describe the appropriate stability concepts.
For the rst, assume that D is compact and that f(0; d) = 0 for all d 2 D. Then we say that the zero state is uniformly globally asymptotically stable (UGAS) if there exists a class KL function 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) (2) for all t 0. As usual, we call a function : 0; 1) ! 0; 1) of class K, if it satis es (0) = 0 and is continuous and strictly increasing (and class K 1 if it is unbounded), and we call a continuous function : 0; 1) 2 ! 0; 1) of class KL, if it is decreasing to zero in the second and of class K in the rst argument. (It is an easy exercise, cf. e.g. 20], to verify that this de nition is equivalent to the requirements of uniform stability and uniform attraction stated in \" " terms.) Note that while our general assumptions on the right hand side f do not guarantee uniqueness of solutions through zero, the added assumption of asymptotic stability implies that (t; 0; d) 0 is the unique solution with initial condition x = 0, for all d 2 D. As a consequence, since away from zero we have a local Lipschitz condition, solutions are unique for each given initial state and d 2 D. If the origin is no common xed point for all values d 2 D then (2) is impossible. In this case, however, still a useful notion of stability is possible. We call the system (1) (globally) input-to-state stable (ISS), if there exists a class KL function and a class K 1 function such that all solutions of (1) satisfy for all d( ) 2 D and all t 0. Formulation (3) is the most frequently used characterization of the ISS property. Note that with~ = 2 and~ = 2 inequality (3) immediately implies ) ; hence this \max" formulation can be used as an equivalent characterization.
Two apparently stronger formulations of these properties are obtained if we replace (kxk; t) by ce t kxk, more precisely we call the zero position of (1) uniformly globally exponentially stable (UGES), if there exist constants c 1; > 0 such that k (t; x; d( ))k ce t kxk (4) holds for all d( ) 2 D and all t 0, and we call the system input-to-state exponentially stable (ISES), if there exist a class K 1 function and constants c 1; > 0 such that for all d( ) 2 D and all t 0. (As usual, these de nitions use appropriate constants c; > 0. In this paper, however, we will see that we can always work with \normalized" versions choosing c = 1; = 1. For the (ISES) property we use the \max" formulation because it allows a further implication as stated in Theorem 5, below. Observe that (5) implies (3) with (kxk; t) = ce t kxk.) 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 (1), we may consider the transformed system _ y(t) =f(y(t); d(t)) ; (6) where, by de nition,f (y; d) = DT(T 1 (y))f(T 1 (y); d) : In other words, system (6) 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: f(y; d) is continuous, and it is locally Lipschitz on x for x 6 = 0, uniformly on d.
It is our aim to show that for dimensions n 6 = 4; 5 the following assertions are true. Given a system of the form (1) satisfying (2) or (3), respectively, there exists a transformed system that satis es (4) or (5), respectively. 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.
Furthermore we show that if system (1) is ISES (5) with c = = 1 then there exists a homeomorphism R : R m ! R m on the input space with R(0) = 0 that is a di eomorphism on R m n f0g such that the transformed system with v = R(d) satis es the following \L 2 to L 2 " nonlinear H 1 estimate: t Z 0 k (s; x; v( ))k 2 ds kxk 2 + t Z 0 kv(s)k 2 ds: (8) Since (8)

Construction of the coordinate transformation
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. The next result says in particular that any such function may look like kxk 2 under a coordinate change. This implies in particular that the level sets under coordinate change are spheres. It may therefore not come as a surprise that a basic ingredient of the proof is related to the question of whether level sets of Lyapunov functions in R n are di eomorphic to the sphere S n 1 . This question is solved except for the two special cases of dimensions n = 4 and n = 5, though in the case n = 5 it is at least known that the statement is true if only homeomorphisms are required. (For the case n = 4 this question is equivalent to the Poincar e conjecture; see 36].) Proposition 1 Let n 6 = 4 and let V : R n ! R be a proper, positive de nite C 1 function. Assume furthermore that V is smooth on R n nf0g with nonvanishing gradient. Then for each class K 1 function which is smooth on (0; 1) there exists a homeomorphism T : R n ! R n with T(0) = 0 such thatṼ (y) := V (T 1 (y)) = (kyk) : In particular this holds for (kyk) = kyk 2 . If n 6 = 4; 5 then T can be chosen to be a di eomorphism on R n nf0g. Furthermore, in this case there exists a class K 1 function which is smooth on (0; 1) and satis es (s)= 0 (s) s such that T is C 1 with DT(0) = 0.
PROOF. For the function V the right hand side of the normed gradient ow _ x = rV (x) 0 krV (x)k 2 is well de ned and smooth for x 6 = 0. Denote the solutions by (t; x). Then V ( (t; x)) = V (x)+t, and thus since V is proper and rV (x) 6 = 0 for x 6 = 0 for a given initial value x 2 R n is well de ned for all t 2 ( V (x); 1), thus also smooth (see e.g. 10, Corollary 4.1]). Fix c > 0. We de ne a map : R n n f0g ! V 1 (c) by (x) = (c V (x); x) : Obviously is smooth, and since the gradient ow crosses each level set V 1 (a); a > 0 exactly once it induces a di eomorphism between each two level sets of V , which are C 1 manifolds due to the fact that V is smooth away from the origin with nonvanishing gradient. Now observe that the properties of V imply that V 1 (c) is a homotopy sphere (cf. also 36, Discussion after Theorem 1.1]), which implies that V 1 (c) is di eomorphic to S n 1 for n = 1; 2; 3 (see e.g. 22, Appendix] for n = 2, 7, Theorem 3.20] for n = 3; n = 1 is trivial). For n 6 we can use the fact that the sublevel set fx 2 R n j V (x) cg is a compact, connected smooth manifold with a simply connected boundary, which by 21, x9, Proposition A] implies that the sublevel set is di eomorphic to the unit disc D n , hence V 1 (c) is di eomorphic to S n 1 . Thus for all dimensions n 6 = 4; 5 we may choose a di eomorphism S : V 1 (c) ! S n 1 .

Main Results
Using the coordinate transformation T we can now prove our main results.
Theorem 2 Let n 6 = 4; 5 and consider any system (1) on R n which is UGAS (2). We suppose that the set D R m is compact. Then, (1) can be transformed into a system (6) that is UGES (4).
In particular, the constants in (4) can be chosen to be c = 1; = 1.
PROOF. Under our assumptions, by 20, Theorem 2.9, Remark 4.1] there exists a smooth function V : R n ! R for (1) such that for some class K 1 function 1 . Furthermore, there exist class K 1 functions 2 ; 3 such that 2 (kxk) V (x) 3 (kxk) : (10) Now let 4 be a C 1 function of class K 1 which is smooth on (0; 1) and satis es 0 4 (0) = 0, such that 4 (a) minfa; 1 To be precise, the results in that reference make as a blanket assumption the hypothesis that f is locally Lipschitz, not merely continuous, at x = 0. However, as noted in e.g. 35], the Lipschitz condition at the origin is not used in the proofs.
Obviously is smooth on (0; 1); furthermore is of class K 1 and by 26, Lemma 12] is a C 1 function on 0; 1) with 0 (0) = 0. Thus de ning W(x) := (V (x)) we obtain a C 1 Lyapunov function, which is smooth on R n nf0g, for which an easy calculation shows that Applying Proposition 1 to W, using the class K 1 function with (s)= 0 (s) s we obtain for each d 2 D and y 6 = 0 hf(y; d); yi = kyk 0 (kyk) Lf dW (y) kyk 0 (kyk)W (y) = kyk 0 (kyk) (kyk) kyk 2 : Clearly the overall inequality also holds for y = 0 so that we obtain d dt ky(t)k 2 = 2hf(y(t); d(t)); y(t)i 2ky(t)k 2 and hence ky(t)k 2 e 2t ky(0)k 2 , i.e. the desired exponential estimate. 2 Theorem 3 Let n 6 = 4; 5 and suppose that the system (1) on R n is ISS (3) with some class K 1 function and some class KL function . Then (1) is can be transformed into a system (6) that is ISES (5) with constants c = = 1.
PROOF. By 32, Theorem 1] y there exists a C 1 function V which is smooth on R n n f0g and a class K 1 function such that kxk > (kdk) ) L f d V (x) Now Proposition 1 yields a parameter transformation T such thatW (y) = W(T 1 (y)) = (kyk) and (s)= 0 (s) s. Now choose a class K 1 function such that kT 1 (y)k (kyk) and de ne~ = 1 .
Then a straightforward calculation yields kyk >~ (kdk) ) Lf dW (y) W (y): (12) Similar to the proof of Theorem 2 this implies k~ (t; y; d( ))k e t kyk as long as k~ (t; y; d( ))k >~ (sup 0 t kd( )k) which yields the desired estimate. 2 Theorem 4 Consider the system (1) on R n being ISES (5) with some class K 1 function and c = = 1. Then there exists a homeomorphism R : R m ! R m on the input space with R(0) = 0, that is a di eomorphism on R m n f0g, such that the the transformed system (7) satis es the nonlinear H 1 estimate (8).
PROOF. From (5) it is immediate that for any d( ) 2 D, any x 2 R n , and any T > 0 we have kxk e T ( sup 0 T kd( )k) ) k (t; x; d( ))k e t kxk for all t 2 0; T] : (13) Now consider the function W(x) = kxk 2 . Then (13)

Remarks
Note that, in general, for our results to be true we cannot expect T to be di eomorphic on the whole R n . Consider the simplest case where f does not depend on d and is di erentiable at the origin. If T were a di eomorphism globally, then DT 1 (0) would be well-de ned, which implies that Df(0) = @ @y y=0 DT(T 1 (y))f(T 1 (y)) = DT(0)Df(0)DT 1 (0) and so the linearizations in 0 are similar; in particular, the dimension of center manifolds remains unchanged.
Actually, if one wants the exponential decay to be e t , even for linear systems one cannot obtain a di eomorphism T. As an example, consider the one-dimensional system _ x = x=2. Here one uses the change of variables y = T(x) given by T(x) = x 2 ; x > 0; T(0) = 0 and T(x) = x 2 ; x < 0 to obtain _ y = y. Note that T is C 1 with DT(0) = 0. The inverse of this T is given by T 1 (y) = p y; y > 0; T 1 (0) = 0 and T 1 (y) = p y; y < 0 which is smooth only away from the origin, though continuous globally.
An example for the case of nontrivial center manifolds is given by the system _ x = x 3 . Let us rst note that for this system there is no transformation in the class we consider such that the transformed system is of the form _ y = y. The reason for this is that we would have _ T(x) = _ y = y = T(x), so at least for x > 0 V = T is a Lyapunov function with the property that _ V (x) = V 0 (x)x 3 = V (x). It is readily seen that the solutions of this di erential equation (in x and V ) are V c (x) = c exp 1 2x 2 , for c 2 R. However, the image of 0; 1) under such V c yields a bounded set, so that these functions are no candidate for coordinate transforms on R. Nonetheless a coordinate transform according to our requirements can now be easily built: Take any K 1 function with 0 > 0 on (0; 1) so that with via the symmetrization ( x) := (x) we get a smooth function on R. Now de ne T(x) := (x)V 1 (x); x 0; T(x) := (x)V 1 (x); x < 0 : Then for y 6 = 0 we have _ y = _ T(x) = (1+ 0 (x)x 3 (x) )T (x) < y, so that the transformed system decays at least exponentially with constants c = 1; = 1. Again note that the requirement DT(0) = 0 is vital, in fact all orders of derivatives vanish in 0.
A basic ingredient of the proof of Theorem 2 is the construction of a Lyapunov function with the property _ V V . Actually, one may even, under restricted conditions, obtain the equality _ V = V . It should be noted that already in 3] it is shown that for dynamical systems with globally asymptotically stable xed point a continuous Lyapunov function with the property V ( (t; x)) = e t V (x) exists, see also Chapter V.2 in 4]. Note, however that in these references only systems with trajectories de ned on R are considered, which does not include the previous example. Indeed, if f(x; d) = f(x) is independent of d 2 D and the system _ x = f(x) is backward complete we can can also de ne a coordinate transformation based on a di erent W than the one used in the proof of Theorem 2: In this case the function W(x) = exp t(x) with t(x) de ned by V ( (t(x); x)) = 1 is positive de nite, proper, and satis es L f W(x) = W(x), thus W( (t; x)) = W(x) t. Since V 1 (1) = W 1 (1) we still nd a di eomorphism S as in the proof of Proposition 1. Deviating from this proof, instead of the gradient ow we now use the trajectories of the system, i.e. we de ne (x) = (W (x) 1; x) yielding W( (x)) = W(x) (W (x) 1) = 1. Thus from we can construct T as in the proof of Proposition 1, and obtain W(T 1 (y)) = kyk 2 . Furthermore the de nition of implies that each trajectory f (t; x) j t 2 Rg is mapped onto the line f S( (x)) j > 0g and consequentlyf(y) = y, i.e. we obtain a transformation into the linear system _ y = y. Note, however, that with this construction the coordinate transformation will in general only have the regularity of f (e.g. a homeomorphism if f is only C 0 ), which is inevitable since it transforms f into a smooth map. Moreover, this construction cannot be generalized to systems with disturbances.
Since we are not requiring that the inverse of a change of variables be itself a change of variables (because one may, and in fact does in our constructions, have DT(0) = 0, in which case T 1 is not di erentiable at the origin), the way to de ne a notion of \equivalence" is by taking the transitive and symmetric closure of the relation given by such changes of variables.
That is, we could say that system (1) is equivalent to a system (6) if there exist k 2 N and maps f 0 = f; f 1 ; : : : ; f k =f : R n D ! R n , all satisfying the assumptions on f, with the following properties: For each i = 0; : : : ; k 1 there exists a change of variables T as above such that f l (y; d) = DT(T 1 (y))f m (T 1 (y); d), where l = i; m = i + 1 or l = i + 1; m = i.
Finally, regarding our notion of system transformation, note that even if f(0; d) 6 = 0 for some d 2 D for the original system (1), then under the assumption DT(0) = 0 we havef(0; d) = 0 for all d 2 D for the transformed system. This implies that even if the original system had unique trajectories through zero, the transformed system cannot have this property.