A Smooth Converse Lyapunov Theorem for Robust Stability

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1 A Smooth Converse Lyapunov Theorem for Robust Stability Yuandan Lin Department of Mathematics Florida Atlantic University Boca Raton, FL 3343 Eduardo D. Sontag Department of Mathematics Rutgers University New Brunswick, NJ Yuan Wang Department of Mathematics Florida Atlantic University Boca Raton, FL 3343 Abstract. This paper presents a Converse Lyapunov Function Theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unified and natural manner, it () allows arbitrary bounded time-varying parameters in the system description, (2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets.. Introduction. This work is motivated by problems of robust nonlinear stabilization. One of our main contributions is to provide a statement and proof of a Converse Lyapunov Function Theorem which is in a form particularly useful for the study of such feedback control analysis and design problems. We provide a single (and natural) unified result that:. applies to stability with respect to not necessarily compact invariant sets; 2. deals with global (as opposed to merely local) asymptotic stability; 3. results in smooth (infinitely differentiable) Lyapunov functions; 4. most importantly, applies to stability in the presence of bounded time varying parameters in the system. (This latter property is sometimes called total stability and it is equivalent to the stability of an associated differential inclusion.) The interest in stability with respect to possibly non-compact sets is motivated by applications to areas such as output-control (one needs to stabilize with respect to the zero set of the output variables) and Luenberger-type observer design ( detectability corresponds to stability with respect to the diagonal set {(x, x)}, as a subset of the composite state/observer system). Such applications and others are explored in [6], Chapter 5. Smooth Lyapunov functions, as opposed to merely continuous or once-differentiable, are required in order to apply backstepping techniques in which a feedback law is built by successively taking directional derivatives of feedback laws obtained for a simplified system. (See for instance [9] for more on backstepping design.) Finally, the effect of parameter uncertainty, and the study of associated Lyapunov functions, are topics of interest in robust control theory. An application of the result proved in this paper to the study of input to state stability is provided in [27]. Supported in part by US Air Force Grant AFOSR Supported in part by NSF Grant DMS Keywords: Nonlinear stability, Stability with respect to sets, Lyapunov function techniques, Robust stability. Running head: Converse Lyapunov Theorem for Robust Stability AMS(MOS) subject classifications: 93D05, 93D09, 93D20, 34D20

2 .. Organization of Paper. The paper is organized as follows. The next section provides the basic definitions and the statement of the main result. Actually, two versions are given, one that applies to global asymptotic stability with respect to arbitrary invariant sets, but assuming completeness of the system that is, global existence of solutions for all inputs and another version which does not assume completeness but only applies to the special case of compact invariant sets (in particular, to the usual case of global asymptotic stability with respect to equilibria). Equivalent characterizations of stability by means of decay estimates have proved very useful in control theory see e.g. [25] and this is the subject of Section 3. Some technical facts about Lyapunov functions, including a result on the smoothing of such functions around an attracting set, are given in Section 4. After this, Section 5 establishes some basic facts about complete systems needed for the main result. Section 6 contains the proof of the main result for the general case. Our proof is based upon, and follows to a great extent the outline of, the one given by Wilson in [3], who provided in the late 960s a converse Lyapunov function theorem for local asymptotic stability with respect to closed sets. There are however some major differences with that work: we want a global rather than a local result, and several technical issues appear in that case; moreover, and most importantly, we have to deal with parameters, which makes the careful analysis of uniform bounds of paramount importance. (In addition, even for the case of no parameters and local stability, several critical steps in the proof are only sketched in [3], especially those concerning Lipschitz properties and smoothness around the attracting set. Later the author of [2] rederived the results, but only for the case when the invariant set is compact. Thus it seems useful to have an expository detailed and self-contained proof in the literature for the more general cases.) A needed technical result on smoothing functions, based closely also on [3], is placed in an Appendix for convenience. Section 7 deals with the compact case, essentially by reparameterization of trajectories. An example, motivated by related work of Tsinias and Kalouptsidis, is given in Section 8 to show that the analogous theorems are false for unbounded parameters. Obviously in a topic such as this one, there are many connections to previous work. While it is likely that we have missed many relevant references, we discuss in Section 9 some relationships between our work and other results in the literature. Relations to work using prolongations are particularly important, and are the subject of some more detail in Section Definitions and Statements of Main Results. Consider the following system: () ẋ(t) =f(x(t), d(t)), where for each t IR, x(t) IR n and d(t) D, and where D is a compact subset of IR m, for some positive integers n and m. The map f :IR n D IR n is assumed to satisfy the following two properties: f is continuous. f is locally Lipschitz on x uniformly on d, that is, for each compact subset K of IR n there is some constant c so that f(x, d) f(z, d) c x z for all x, z K and all d D, where denotes the usual Euclidian norm. Note that these properties are satisfied, for instance, if f extends to a continuously differentiable function on a neighborhood of IR n D. Let M D be the set of all measurable functions from IR to D. We will call functions d M D time varying parameters. For each d M D, we denote by x(t, x 0,d) (and sometimes simply by x(t) if there is no ambiguity from the context) the solution at time t of () with x(0) = x 0. This is defined on some maximal interval (T x 0,d,T+ x 0,d ) with T x 0,d < 0 <T+ x 0,d +. Sometimes we will need to consider time varying parameters d that are defined only on some interval I IR with 0 I. In those cases, by abuse of notation, x(t, x 0,d) will still be used, but only times t I will be considered. 2

3 T x 0,d The system is said to be forward complete if T + x 0,d =+ for all x0 all d MD. Itisbackward complete if = for all x0 all d MD, and it is complete if it is both forward and backward complete. We say that a closed set A is an invariant set for () if x 0 A, d M D, T + x 0,d =+ and x(t, x0, d) A, t 0. Remark 2.. An equivalent formulation of invariance is in terms of the associated differential inclusion (2) ẋ F (x), where F (x) ={f(x, d), d D}. The set A is invariant for () if and only if it is invariant with respect to (2) (see e.g. []). The notions of stability to be considered later can be rephrased in terms of (2) as well. We will use the following notation: for each nonempty subset A of IR n, and each ξ IR n, we denote ξ A def = d(ξ, A) = inf d(ξ, η), η A the common point-to-set distance, and ξ {0} = ξ is the usual norm. Let A IR n be a closed, invariant set for (). We emphasize that we do not require A to be compact. We will assume throughout this work that the following mild property holds: (3) sup ξ IR n { ξ A } =. This is a minor technical assumption, satisfied in all examples of interest, which will greatly simplify our statements and proofs. (Of course, this property holds automatically whenever A is compact, and in particular in the important special case in which A reduces to an equilibrium point.) Definition 2.2. System () is (absolutely) uniformly globally asymptotically stable (UGAS) with respect to the closed invariant set A if it is forward complete and the following two properties hold:. Uniform Stability. There exists a K -function δ( ) such that for any ε 0, (4) x(t, x 0,d) A ε for all d M D, whenever x 0 A δ(ε) and t Uniform Attraction. For any r, ε > 0, there is a T>0, such that for every d M D, (5) x(t, x 0,d) A <ε whenever x 0 A <rand t T. For the definitions of the standard comparison classes of K and KL functions, we refer the reader to the appendix. Observe that when A is compact the forward completeness assumption is redundant, since in that case property (4) already implies that all solutions are bounded. In the particular case in which the set D consists of just one point, the above definition reduces to the standard notion of set asymptotic stability of differential equations. (Note, however, that this definition differs from those in [3], and [3], which are not global.) If, in addition, A consists of just an equilibrium point x 0, this is the usual notion of global asymptotic stability for the solution x(t) x 0. Remark 2.3. It is an easy exercise to verify that an equivalent definition results if one replaces M D by the subset of piecewise constant time varying parameters. Remark 2.4. Note that the uniform stability condition is equivalent to: there is a K -function ϕ so that x(t, x 0,d) A ϕ( x 0 A ), x 0, t 0, and d M D. (Just let ϕ = δ.) 3

4 The following characterization of the UGAS property will be extremely useful. Proposition 2.5. The system () is UGAS with respect to a closed, invariant set A IR n if and only if it is forward complete and there exists a KL-function β such that, given any initial state x 0, the solution x(t, x 0,d) satisfies (6) x(t, x 0,d) A β( x 0 A,t), any t 0, for any d M D. Observe that when A is compact the forward completeness assumption is again redundant, since in that case property (6) implies that solutions are bounded. Next we introduce Lyapunov functions with respect to sets. For any differentiable function V :IR n IR, we use the standard Lie derivative notation L fd V (ξ) def = V (ξ) x f d(ξ), where for each d D, f d ( ) is the vector field defined by f(, d). By smooth we always mean infinitely differentiable. Definition 2.6. A Lyapunov function for the system () with respect to a nonempty, closed, invariant set A IR n is a function V :IR n IR such that V is smooth on IR n \A and satisfies. there exist two K -functions α and α 2 such that for any ξ IR n, (7) α ( ξ A ) V (ξ) α 2( ξ A ); 2. there exists a continuous, positive definite function α 3 such that for any ξ IR n \A, and any d D, (8) L fd V (ξ) α 3( ξ A ). A smooth Lyapunov function is one which is smooth on all of IR n. Remark 2.7. Continuity of V on IR n \A and property. in the definition imply: V is continuous on all of IR n ; V (x) =0 x A; and V :IR n onto IR 0 (recall the assumption in equation (3)). Our main results will be two converse Lyapunov theorems. The first one is for general closed invariant sets and assumes completeness of the system. Theorem 2.8. Assume that the system () is complete. Let A IR n be a nonempty, closed invariant subset for this system. Then, () is UGAS with respect to A if and only if there exists a smooth Lyapunov function V with respect to A. The following result does not assume completeness but instead applies only to compact A: Theorem 2.9. Let A IR n be a nonempty, compact invariant subset for the system (). Then, () is UGAS with respect to A if and only if there exists a smooth Lyapunov function V with respect to A. 3. Some Preliminaries about UGAS. It will be useful to have a restatement of the second condition in the definition of UGAS stated in terms of uniform attraction times: Lemma 3.. The uniform attraction property defined in Definition 2.2 is equivalent to the following: There exists a family of mappings {T r} r>0 with onto for each fixed r>0, T r :IR >0 IR >0 is continuous and is strictly decreasing; for each fixed ε>0, T r(ε) is (strictly) increasing as r increases and lim r T r(ε) = ; such that, for each d M D, (9) x(t, x 0,d) A <ε whenever x 0 A <rand t T r(ε). 4

5 Proof. Sufficiency is clear. Now we show the necessity part. For any r, ε > 0, let (0) A r, ε def = { T 0: x 0 A <r, t T, d M D, x(t, x 0,d) A <ε } IR 0. Then from the assumptions, A r, ε for any r, ε > 0. Moreover, A r, ε A r, ε2, if ε ε 2, and A r2,ε A r,ε, if r r 2. Now define T r(ε) def = inf A r, ε. Then T r(ε) <, for any r, ε > 0, and it satisfies T r(ε ) T r(ε 2), if ε ε 2, and T r (ε) T r2 (ε), if r r 2. So we can define for any r, ε > 0, () T r(ε) def = 2 ε T r(s) ds. ε ε/2 Since T r( ) is decreasing, T r( ) is well defined and is locally absolutely continuous. Also (2) Furthermore, (3) d T r(ε) dε T r(ε) 2 ε T r(ε) ε ε/2 ds = T r(ε). = 2 ε T r(s) ds + 2 ( Tr(ε) ε 2 ε 2 T ε ) r( 2 ) ε/2 [ = T r(ε) 2 ε ] T r(s) ds + [ ( Tr(ε) ε ε ε T ε r 2)] ε/2 = [ Tr(ε) ε T ] r(ε) + [ ( Tr(ε) ε T ε r 0, a.e., 2)] hence T r( ) decreases (not necessarily strictly). Since T ( ) (ε) increases, from the definition, T ( ) (ε) also increases. Finally, define (4) T r(ε) def = T r(ε)+ r ε Then it follows that onto for any fixed r, T r( ) is continuous, maps IR >0 IR >0, and is strictly decreasing; for any fixed ε, T r(ε) is increasing as r increases, and lim r T r(ε) =. So the only thing left to be shown is that T r defined by (4) satisfies (9). To do this, pick any x 0 and t with x 0 A <rand t T r(ε). Then t T r(ε) > T r(ε) T r(ε). Hence, by the definition of T r(ε), x(t, x 0,d) A <ε,as claimed. 3.. Proof of Characterization via Decay Estimate. We now provide a proof of Proposition 2.5. [ =] Assume that there exists a KL-function β such that (6) holds. Let c def = sup β(, 0), and choose δ( ) tobeanyk -function with δ(ε) β (ε), any 0 ε<c, 5

6 where β def denotes the inverse function of β( ) = β(, 0). (If c =, we can simply choose δ(ε) def = β (ε).) Clearly δ(ε) is the desired K -function for the uniform stability property. The uniform attraction property follows from the fact that for every fixed r, lim β(r, t) =0. t [= ] Assume that () is UGAS with respect to the closed set A, and let δ be as in the definition. Let ϕ( ) bethek-function δ ( ). As mentioned in Remark 2.4, it follows that x(t, x 0,d)) A ϕ( x 0 A ) for any x 0 IR n,anyt 0, and any d M D. def Let {T r} r (0, ) be as in Lemma 3., and for each r (0, ) denote ψ r = Tr. Then, for each r (0, ), ψ r :IR >0 IR >0 is again continuous, onto, and strictly decreasing. We also write ψ r(0) = +, which is consistent with that fact that lim ψr(t) =+. t 0 + (Note: The property that T ( ) (t) increases to is not needed here.) Claim: For any x 0 A <r,anyt 0 and any d M D, x(t, x 0,d) A ψ r(t). Proof: It follows from the definition of the maps T r that, for any r, ε > 0, and for any d M D, x 0 A <r, t T r(ε) = x(t, x 0,d) A <ε. As t = T r(ψ r(t)) if t>0, we have, for any such x 0 and d, (5) x(t, x 0,d) A < ψ r(t), t >0. The claim follows by combining (5) and the fact that ψ r(0) = +. Now for any s 0 and t 0, let (6) ψ(s, t) def = min { ϕ(s), inf r (s, ) } ψ r(t) Because of the definition of ϕ and the above claim, we have, for each x 0, d M D, and t 0:. (7) x(t, x 0,d) A ψ( x 0 A,t). If ψ would be of class KL, we would be done. This may not be the case, so we next majorize ψ by such a function. By its definition, for any fixed t, ψ(, t) is an increasing function (not necessarily strictly). Also because for onto any fixed r (0, ), ψ r(t) decreases to 0 (this follows from the fact that ψ r :IR >0 IR >0 is continuous and strictly decreasing), it follows that for any fixed s, ψ(s, t) decreases to 0 as t. Next we construct a function ψ :IR [0, ) IR 0 IR 0 with the following properties: for any fixed t 0, ψ(, t) is continuous and strictly increasing; for any fixed s 0, ψ(s, t) decreases to 0 as t ; ψ(s, t) ψ(s, t). Such a function ψ always exists; for instance, it can be obtained as follows. Define first (8) ˆψ(s, t) def = s+ s ψ(ε, t) dε. Then ˆψ(, t) is an absolutely continuous function on every compact subset of IR 0, and it satisfies ˆψ(s, t) ψ(s, t) s+ s 6 dε = ψ(s, t).

7 It follows that ˆψ(s, t) s = ψ(s +,t) ψ(s, t) 0, a.e., and hence ˆψ(, t) is increasing. Also since for any fixed s, ψ(s, ) decreases, so does ˆψ(s, ). Note that { } ψ(s, t) ψ(s, 0) = min inf r (s, ) ψ r(0), ϕ(s) = ϕ(s), (recall that ψ r(0) = + ), so by the Lebesgue dominated convergence theorem, for any fixed s 0, s+ lim ˆψ(s, t) = t s lim t ψ(ε, t) dε =0. Now we see that the function ˆψ(s, t) satisfies all of the requirements for ψ(s, t) except possibly for the strictly increasing property. We define ψ as follows: Clearly it satisfies all the desired properties. Finally, define ψ(s, t) def = ˆψ(s, s t)+ (s + )(t +). β(s, t) def = ϕ(s) ψ(s, t). Then it follows that β(s, t) isakl-function, and, for all x 0, t, d: x(t, x 0,d) A ϕ( x 0 A ) ψ( x 0 A,t) β( x 0 A,t), which concludes the proof of the Proposition. 4. Some Preliminaries about Lyapunov Functions. In this section we provide some technical results about set Lyapunov functions. A lemma on differential inequalities is also given, for later reference. Remark 4.. One may assume in Definition 2.6 that all of α,α 2,α 3 are smooth in (0, + ) and of class K.Forα and α 2, this is proved simply by finding two functions α, α 2 in K, smooth in (0, + ) so that α (s) α (s) α 2(s) α 2(s), for all s. For α 3, a new Lyapunov function W and a function α 3 which satisfies (8) with respect to W, but is smooth in (0, + ) and of class K, can be constructed as follows. First, pick α 3 to be any K -function, smooth in (0, + ), such that This is possible since α 3 is positive definite. Then let α 3(s) sα 3(s), s [0, α ()]. γ :IR 0 IR 0 be a K -function, smooth in (0, + ), such that γ(r) α (r) for all r [0, ]; γ(r) > α3(α (r)) for all r>. α 3(α (r)) 7

8 Now define β(s) def = s 0 γ(r) dr. Note that β is a K -function, smooth in (0, + ). Let W (ξ) This is smooth on IR n \A, and β α,β α 2 bound W as in equation (7). Moreover, def = β(v (ξ)). β (V (ξ)) = γ(v (ξ)) γ(α ( ξ A )), so (9) L fd W (ξ) =β (V (ξ))l fd V (ξ) γ(α ( ξ A ))α 3( ξ A ). We claim that this is bounded by α 3( ξ A ). Indeed, if s def = ξ A α (), then from the first item above and the definition of α 3, γ(α (s)) s α3(s) α ; 3(s) if instead s>α (), then from the second item, also γ(α (s)) α3(s) α 3(s). In either case, γ(α (s))α 3(s) α 3(s), as desired. From now on, whenever necessary, we assume that α,α 2,α 3 are K -functions, smooth in (0, + ). 4.. Smoothing of Lyapunov Functions. When dealing with control system design, one often needs to know that V can be taken to be globally smooth, rather than just smooth outside of A. Proposition 4.2. If there is a Lyapunov function for () with respect to A, then there is also a smooth such Lyapunov function. The proof relies on constructing a smooth function of the form W = β V, where β :IR 0 IR 0 is built using a partition of unity. Again let A IR n be nonempty and closed. For a multi-index ϱ =(ϱ,ϱ 2,...,ϱ n), we use ϱ to denote n ϱi. The following regularization result will be needed; it generalizes to arbitrary A the analogous (but i= simpler, due to compactness) result for equilibria given in [3, Theorem 6]. Lemma 4.3. Assume that V : IR n IR 0 is C 0, the restriction V IR n \A is C, and also V A = 0, V IR n \A > 0. Then there exists a K -function β, smooth on (0, ) and so that β (i) (t) 0 as t 0 + for each i =0,,... and having β (t) > 0, t >0, such that W def = β V is a C function on all of IR n. Proof. Let K,K 2,..., be compact subsets of IR n such that A int (Ki). For any k, let i= I k def = ( k +2, ) IR k def and I 0 = I. Pick for any k, a smooth (C ) function γ k :IR >0 [0, ] satisfying γ k (t) =0ift I k ; and γ k (t) > 0ift I k. Define for any k, { G k def = x IR n : x } k K i, V(x) clos I k. i= Then G k is compact (because of compactness of the sets K i and continuity of V ). Observe that each derivative γ (i) k has a compact support included in clos I k, so it is bounded. For each k =, 2,..., let c k IR satisfy 8

9 . c k ; 2. c k (D ϱ V )(x) for any multi-index ϱ k and any x G k ; and 3. c k γ (i) k (t), for any i k and any t IR >0. Choose the sequence d k to satisfy (20) 0 <d k <, k =, 2,... 2 k (k + )!c k k [ Let α :IR 0 IR 0 be a C function such that α 0on 0, ] [ ) and α on 3 2,. Define γ(0) def = 0 and (2) γ(t) def = d k γ k (t)+α(t), t >0. k= Notice that for any t (0, ), if k def = t denotes the largest integer t, then t I k, and t I j if j k, k. Hence the sum in (2) at most consists of three terms (for t the sum is just γ = α), and so γ is C at each t (0, ). Claim: For any i 0, lim γ(i) (t) =0. t 0 + Proof: Fix any i 0. Given any ε>0, let k 0 Z be such that ε> > 0. Let k 0 T def = min { k 0, i +, }. 3 We will show that t (0, T) = { γ (i) (t) < ε. Indeed, as 0 <t<min, k 0 k def = max{i +,k 0, 3}. So t i +, }, it follows that 3 γ (i) (t) d k γ (i) k (t)+d kγ (i) k (t), and noticing that i k <k = c k γ (i) k (t), ck γ (i) k (t), we have γ (i) (t) dk c k + d k c k 2k! + 2(k + )! < k! < k k 0 < ε, as wanted. Note also that if t ( 2, then γ(t) α(t) > 0; and if t 0, ), then γ(t) d k γ k (t) > 0 with 2 k def = 2, so the function t (22) β(t) def = t 0 γ(s) ds is also a K -function, smooth on (0, ). Furthermore, β satisfies β (i) (t) 0ast 0 + for each i =0,,... Finally, we show that W = β V is C. For this, it is enough to show that D ϱ 0 W (x n) 0asx n x A, for each multi-index ϱ 0 and each sequence {x n} IR n \A converging to a point x in the boundary of A. (In general, see e.g. [4] (p. 52), if A IR n is closed and ϕ :IR n IR satisfies that ϕ A =0,ϕ IR n \A is C, and for 9

10 each boundary point a of A and all multi-indices ϱ =(ϱ,ϱ 2,...,ϱ n), it holds that x a lim D ϱ ϕ(x) =0, then ϕ is x A C on IR n.) Pick one such ϱ 0 and any sequence {x n} with x n x A. If ϱ 0 = 0, one only needs to show that W (x n) 0, which follows easily from the facts that β K and V (x n) 0. So from now on, we can assume def that ϱ 0 = i. As A j=0int K j, x int K l for some l, and without loss of generality we may assume that there is some fixed l so that x n K l, for all n. Pick any ε>0. We will show that there exists some N such that n>n = D ϱ 0 W (x n) <ε. Let k Z be so that { ( ) } k>max i, log 2,l ε ( and let T 0, ) be such that T< 3 k +2. Observe that if t<t, then t I I k. As V is C 0 everywhere, V =0atA, V (x n) V ( x) = 0. So there exists N such that V (x n) <T whenever n>n. Fix an N like this. Then for any n>n, γ s (j) (V (x n)) = 0, j, s =, 2,...,k, (since γ s vanishes outside I s). Pick any j IN with j i, anyh IN with h i, and ϱ,...,ϱ h multi-indices such that ϱ µ i, µ =,...,h. Then for any q IN with q>k, by the way we chose c k, γ (j) q (V (x n)) cq, since q>k>i j. Also, if V (x n) I q, then again by the properties of the sequence c k, D ϱµ V (x n) c q, (since q>k>land x n K l imply x n K K q, and ϱ µ i<k<q). Therefore, for such q, if V (x n) I q, (23) γ (j) q (V (x n)) D ϱ V (x n) D ϱ h V (x n) c h+ q c i+ q < c q q. If instead it would be the case that V (x n) I q, then γ q (j) (V (x n)) = 0, and hence the inequality (23) still holds. Since we also have γ (j) (V (x n)) = q=k+ d qγ (j) q (V (x n)), (24) < γ (j) (V (x n)) D ϱ V (x n) D ϱ h V (x n) ( q=k+ ) 2 q (k + )! = 2 k (k + )! < ε (k + )!. q=k+ d qc q q < q=k+ 2 q (q + )! 0

11 Now observe that (D ϱ 0 W )(x) =(D ϱ 0 (β V )) (x) is a sum of i! terms (recall 0 <i= ϱ 0 ), each of which is of the form β (p) (V (x)) (D ϱ V )(x) (D ϱ h V )(x), where 0 <p i, h i, and each ϱ µ i. Each β (p) (V (x)) = γ (j) (V (x)), j = p i, so (24) applies, and we conclude (D ϱ 0 ε W )(x n) i! (k + )! <ε, (since k>i.) Now let us return to the proof of the Proposition 4.2. Proof of Proposition 4.2. Assume A, V and α,α 2,α 3 are as defined in Definition 2.6. Let β,w be as in Lemma 4.3. We show that W is a smooth Lyapunov function as required. def Let ˆα i = β α i,i=, 2. These are again K -functions, and they satisfy We define, for s>0: ˆα ( ξ A ) W (ξ) ˆα 2( ξ A ). ˇβ(s) def = min β (t) > 0. t [α (s),α 2 (s)] def Let also ˇβ(0) = 0. Define ˆα 3(s) def = ˇβ(s)α 3(s). Then ˆα 3 is a continuous positive definite function. Also, for any ξ IR n \A, which concludes the proof of the Proposition. L fd W (ξ) = β (V (ξ))l fd V (ξ) β (V (ξ))α 3( ξ A ) ˇβ( ξ A )α 3( ξ A )= ˆα 3( ξ A ), 4.2. A Useful Estimate. The following lemma establishes a useful comparison principle. Lemma 4.4. For each continuous and positive definite function α, there exists a KL-function β α(s, t) with the following property: if y( ) is any (locally) absolutely continuous function defined for t 0 and with y(t) 0 for all t, and y( ) satisfies the differential inequality (25) ẏ(t) α(y(t)), for almost all t with y(0) = y 0 0, then it holds that y(t) β α(y 0,t) for all t 0. s Proof. Define for any s>0, η(s) def dr =. This is a strictly decreasing differentiable function on α(r) (0, ). Without loss of generality, we will assume that lim s 0 + η(s) =+. If this were not the case, we could consider instead the following function: { ᾱ(s) def min{s, α(s)}, if 0 s<, = α(s), if s.

12 This function is again continuous, positive definite, satisfies ᾱ(s) α(s) for any s 0, and lim s 0 + s dr ᾱ(r) s dr r = +. Moreover, if ẏ(t) α(y(t)) then also ẏ(t) ᾱ(y(t)), so βᾱ could be used to bound solutions. Let 0 <a def = lim s + η(s). Then the range of η, and hence also the domain of η, is the open interval ( a, ). (We allow the possibility that a =.) For (s, t) IR 0 IR 0, define { β α(s, t) def 0, if s =0, = η (η(s)+t), if s>0. We claim that for any y( ) satisfying the conditions in the Lemma, (26) y(t) β α(y 0,t), for all t 0. As ẏ(t) α(y(t)), it follows that y(t) is nonincreasing, and if y(t 0) = 0 for some t 0 0, then y(t) 0, t t 0. Without loss of generality, assume that y 0 > 0. Let t 0 def = inf{t : y(t) =0} +. It is enough to show (26) holds for t [0, t 0). As η is strictly decreasing, we only need to show that η(y(t)) η(y 0)+t, that is, which is equivalent to (27) y(t) y0 dr α(r) dr α(r) + t, y0 y(t) dr α(r) t. From (25), one sees that t 0 t ẏ(τ) α(y(τ)) dτ dτ = t. Changing variables in the integral, this gives (27). It only remains to show that β α is of class KL. The function β α is continuous since both η and η are continuous in their domains, and lim r η (r) =0. It is strictly increasing in s for each fixed t since since both η and η are strictly decreasing. Finally, β α(s, t) 0ast by construction. So β α is a KL-function. 5. Some Properties of Complete Systems. We need to first establish some technical properties that hold for complete systems, and in particular a Lipschitz continuity fact. For each ξ IR n and T>0, let R T (ξ) def = { } η : η = x(t, ξ,d), d M D. This is the reachable set of () from ξ at time T. We use R T (ξ) to denote we write R T (S) def = R T (ξ), ξ S 0 R T (S) def = 2 0 t T R T (ξ). ξ S R t (ξ). If S is a subset of IR n,

13 In what follows we use S to denote the closure of S for any subset S of IR n. Proposition 5.. Assume that () is forward complete. Then for any compact subset K of IR n and any T>0, the set R T (K) is compact. To prove Proposition 5., we first need to make a couple of technical observations. Lemma 5.2. Let K be a compact subset of IR n and let T>0. Then the set R T (K) is compact if and only if R T (ξ) is compact for each ξ K. Proof. It is clear that the compactness of R T (K) implies the compactness of R T (ξ) for any ξ K. Now assume, for T>0and a compact set K, that R T (ξ) is compact for each ξ K. Pick any ξ K, and let U = {η : d(η, R T (ξ)) < }. Then U is compact. Let C be a Lipschitz constant for f with respect to x on U, and let r = e CT. For each d M D and each η with η ξ <r, let t = inf{t 0: x(t, η, d) x(t, ξ, d) }. Then, using Gronwall s Lemma, one can show that t T, from which it follows that R T (η) U, η ξ <r. Thus, for each ξ K, there is a neighborhood V ξ of ξ such that R T (V ξ ) is compact. By compactness of K, it follows that R T (K) is compact. Lemma 5.3. For any subset S of IR n and any T>0, R T ( S ) R T (S), R T ( S ) R T (S). In particular, R ( T S ) = R T (S). Proof. The first conclusion follows from the continuity of solutions on initial states; see [26], Theorem. The second is immediate from there. We now return to the proof of Proposition 5.. By Lemma 5.2, it is enough to show that R T (ξ) is compact for each ξ IR n and each T>0. Pick any ξ 0 IR n, and let τ = sup{t 0: R T (ξ 0) is compact }. Note that τ>0. This is because x(t, ξ 0,d) ξ 0 for any 0 t</m and any d M D, where M = max { f(ξ, d) : ξ ξ 0, d D}. We must show that τ =. Assume that τ<. Using the same argument used above, one can show that if R t (ξ 0) is compact for some t>0 then there is some δ>0such that R (t+δ) (ξ 0) is compact. From here it follows that R τ (ξ 0) is not compact. By definition, R t (ξ 0) is compact for any t<τ. Let τ = τ/2. Then there is some η R τ (ξ0) such that R (τ τ ) (η ) is not compact; otherwise, by Lemma 5.2, R ( (τ τ ) R ) τ (ξ0) would be compact. This, in turn, would imply that R τ (ξ 0) is compact, since R τ (ξ 0) R τ (ξ 0) R (τ τ ) (R τ (ξ 0)) R τ (ξ 0) R ( (τ τ ) R ) τ (ξ0). On the other hand, combining Lemma 5.3 with the fact that R t (R τ (ξ0)) is compact for any 0 t<τ τ, one sees that R t (η ) is compact for any 0 t<τ τ. Since η R τ (ξ0), there exists a sequence {z n} η with z n R τ (ξ 0). Assume, for each n, that z n = x(τ,ξ 0,d n) for some d n M D. For each d M D, and each s IR, we use d s to denote the function defined by d s(t) = d(s + t). Then by uniqueness, one has that for each n, x(s, z n, (d n) τ ) K for any τ s 0, where K = R τ (ξ 0). We want to claim next that, by compactness of K and Gronwall s Lemma, x( τ,η, (d n) τ ) ξ 0 = x( τ,η, (d n) τ ) x( τ,z n, (d n) τ ) 0, as n. 3

14 The only potential problem is that the solution x( τ,η, (d n) τ ) may fail to exist a priori. However, it is possible to modify f(x, d) outside a neighborhood of K D so that it now has compact support and is hence globally bounded. The modified dynamics is complete. Now the above limit holds for the modified system, and a fortriori it also holds for the original system. Choose n 0 such that (28) x( τ,η, (d n0 ) τ ) ξ 0 < /2. Let v = d n0, and let η 0 = x( τ,η, (d n0 ) τ ). Then, by continuity on initial conditions, there is a neighborhood U of η contained in B(η, ) such that (29) x ( τ,ξ,(v ) τ ) η 0 < /2, ξ U, where B(η, r) denotes the open ball centered at η with radius r. Combining (28) and (29), one has x( τ,ξ,(v ) τ ) U 0, ξ U, where U 0 = B(ξ 0, ). Let τ 2 = τ /2 =(τ τ )/2. Applying the above argument with ξ 0 replaced by η, τ replaced by (τ τ ), and τ replaced by τ 2, one shows that there exists some η 2 R τ 2 (η) such that R t (η 2) is compact for any 0 t<τ σ 2, and R (τ σ 2 ) (η 2) is not compact, where σ 2 = τ + τ 2, and there exist some v 2 defined on [0, τ 2) and some neighborhood U 2 of η 2 contained in B(η 2, ), such that x( τ 2,ξ,(v 2) τ2 ) U, ξ U 2. By induction, one can get, for each k a point η k, a neighborhood U k of η k contained in B(η k, ), and a function v k defined on [0, τ k ) (where τ k =2 k τ) such that R (τ σ k ) (η k ) is not compact, where σ k = τ + τ τ k = τ( 2 k ) τ; x( τ k,ξ,(v k ) τk ) U k, for any ξ U k. def Now define v on [0, τ) by concatenating all the v k s. That is, v(t) =v k (t) for t [σ k,σ k ) (with σ 0 = 0). Then v M D. For each k, let ζ k = x( σ k,η k, (v k ) σk ), where v k is the restriction of v to [0, σ k ). By induction, ) x ( (σ k σ i), η k, (v k ) σk U k i, for each 0 i k, from which it follows that ζ k U 0 for each k. By compactness of U 0, there exists some subsequence of {ζ k } converging to some point ζ 0 IR n. For ease of notation, we still use {ζ k } to denote this convergent subsequence. Our aim is next to prove that the solution starting at ζ 0 and applying the measurable function v does not exist for time τ, contradicting forward completeness. First notice that for any compact set S, there exists some k such that η k S. Otherwise, assume that there exists some compact set S such that η k S for all k. Let S = {η : d(η, S) }. The compactness of S implies that there exists some δ>0 such that R t (η) S for any η S, and any t [0, δ]. In particular, it implies that R (τ σ k ) (η k ) S for k large enough so that τ σ k <δ. This contradicts the fact that R (τ σ k ) (η k ) is not compact for each k. 4

15 Assume that x(τ,ζ 0,v) is defined. By continuity on initial conditions, this would imply that x(t, ζ k,v)is defined for all t τ and for all k large enough, and it converges uniformly to x(t, ζ 0,v). Thus, x(t, ζ k,v) remains in a compact for all t [0,τ] and all k. But x(σ k,ζ k,v)=x(σ k,ζ k,v k )=η k, contradicting what was just proved. So x(τ,ζ 0,v) is not defined, which contradicts the forward completeness of the system. Remark 5.4. For T>0and ξ IR n, let R T (ξ) ={η : η = x( T, ξ, d), d M D}, and R T (ξ) = R t (ξ). These are the reachable sets from ξ for the time reversed system t [ T,0] (30) ẋ(t) = f(x(t), d(t)). Similarly, one defines R T (S) and R T (S) for subsets S of IR n. If () is backward complete, that is, if (30) is forward complete, and applying Proposition 5. to (30), one concludes, for system (), that R T (K) is compact for any T > 0 and any compact subset K of IR n. In particular, for systems that are (forward and backward) complete, R T (K) R T (K) is compact for any compact set K and any T>0. Combining the above conclusion and Gronwall s Lemma, one has the following fact: Proposition 5.5. Assume that () is complete. For any fixed T > 0 and any compact K IR n, there is a constant C>0 (which only depends on the set K and T ), such that for the trajectories x(t, x 0,d) of the system (), x(t, ξ, d) x(t, η, d) C ξ η for any ξ, η K, any t T, and any d M D. 6. Proof of the First Converse Lyapunov Theorem. Proof. [ =] Pick any x 0 IR n and any d M D, and let x( ) be the corresponding trajectory. Then we have dv (x(t)) α 3( x(t) dt A ) α(v (x(t))), a.e. t 0, where α is the K -function defined by α( ) def = α 3(α 2 ( )). Now let β α be the KL-function as in Lemma 4.4 with respect to α, and define (3) ( β(s, t) def = α βα(α 2(s), t) ). Then β is a KL-function, since both α and α 2 are K -functions. By Lemma 4.4, ( V (x(t)) β α V (x0), t ), any t 0. Hence x(t) A β( x 0 A,t), any t 0. 5

16 Therefore the system () is UGAS with respect to A, by Proposition 2.5. [= ] We will show the existence of a not necessarily smooth Lyapunov function; then the existence of a smooth function will follow from Proposition 4.2. Assume that the system is UGAS with respect to the set A. Let δ and T r be as in Definition 2.2 and Lemma 3.. Define g :IR n IR by (32) g(ξ) def = inf t 0,d M D { x(t, ξ, d) A }. Note that, by uniqueness of solutions, for each t 0 > 0 and each d, it holds that x(t t 0,x(t 0,ξ,d), d t0 )=x(t, ξ, d), where d t0 is defined by d t0 (t) =d(t + t 0). Pick any d M D, ξ IR n, and t > 0. Let ξ = x(t,ξ,d). Then for any t<0, and v M D, x(t, ξ, v) =x(t t,ξ,v t #d t ), where v t #d t (s) = { d(s + t ), if t s 0, v(s + t ), if s< t. Thus, g(ξ) = inf x(t, ξ, v) A = inf x(t t, ξ,v t #d t ) t 0,v M D t 0,d M A D = inf τ t,v M D x(τ, ξ,v t #d t ) A inf τ 0,v M D x(τ, ξ,v) A = g(ξ ). This implies that (33) g ( x(t, ξ, d) ) g(ξ), t >0, d M D. Also one has (34) δ( ξ A ) g(ξ) ξ A. The second half of (34) is obvious from x(0, ξ,d)=ξ. On the other hand, if the first half were not true, then there would be some d M D and some t 0 0 such that δ( ξ A ) > x(t 0,ξ,d) A. Pick any 0 <ε< ξ A so that x(t 0,ξ,d) A <δ(ε). By the uniform stability property, applied with t = t 0 and x 0 = x(t 0,ξ,d), ξ A = x( t 0,x(t 0,ξ,d), d t0 ) A < ξ A, which is a contradiction. def For any 0 <ε<r, define K ε, r = { ξ IR n : ε ξ A <r }. Fact : For all ε and r with 0 <ε<r, there exists q ε, r 0, such that: ξ K ε, r, d M D, and t<q ε, r = x(t, ξ, d) A r. 6

17 Proof: If the statement were not true, then there would exist ε, r with 0 <ε<rand three sequences {ξ k } K ε, r, {t k } IR and d k M D with lim k t k = such that for all k: x(t k,ξ k,d k ) A <r. Pick k large enough so that t k >T r(ε), then by the uniform attraction property, ξ k A = x( t k,x(t k,ξ k,d k ), (d k ) tk ) A <ε, which is a contradiction. This proves the fact. Therefore, for any ξ K ε, r, g(ξ) = inf{ x(t, ξ, d) A : t [q ε, r, 0], d M D}. Lemma 6.. The function g(ξ) is locally Lipschitz on IR n \A, and continuous everywhere. Proof. Fix any ξ 0 IR n \A, and let s = ξ0 A 2. Let B (ξ 0,s) denote the closed ball centered at ξ 0 and with radius s. Then B (ξ 0,s) K σ, r for some 0 <σ<r. Pick a constant C as in Proposition 5.5 with respect to this closed ball and T = q σ, r. Pick any ζ,η B (ξ 0,s). For any ε>0, there exist some d η,ε and t η,ε [q σ,r, 0] such that g(η) x(t η,ε, η,d η,ε) A ε. Thus (35) g(ζ) g(η) x(t η,ε, ζ,d η,ε) A x(t η,ε, η,d η,ε) A + ε C ζ η + ε. Note that (35) holds for all ε>0, so it follows that g(ζ) g(η) C ζ η. Similarly, g(η) g(ζ) C ζ η. This proves that g is locally Lipschitz on IR n \A. Note that g is 0 on A, and for ξ A, η IR n : g(η) g(ξ) = g(η) η A η ξ, thus g is globally continuous. (We are not claiming that g is locally Lipschitz on IR n, though.) Now define U : IR n IR 0 by (36) U(ξ) def = { sup g ( x(t, ξ, d) ) } k(t), t 0,d M D where k : R 0 IR >0 is any strictly increasing, smooth function that satisfies: there are two constants 0 <c <c 2 <, such that k(t) [c,c 2] for all t 0; there is a bounded positive decreasing continuous function τ( ), such that k (t) τ(t) for all t 0. (For instance, (37) and (38) c + c2t +t is one example of such a function.) Observe that U(ξ) ( ) U(ξ) sup t 0 g(ξ) k(t) c 2g(ξ) c 2 ξ A, sup g(x(t, ξ, d)) k(t) cg(ξ) cδ( ξ t=0 A ). d M D 7

18 For any ξ IR n, since x(t, ξ, d) A β( ξ A,t), d, t 0, for some KL-function β, and 0 g(x(t, ξ, d)) x(t, ξ, d) A for all t 0, it follows that Thus there exists some τ ξ [0, ) such that U(ξ) = lim sup t + d In fact, we can get the following explicit bound. Fact 2: For any 0 < ξ A <r, U(ξ) = g(x(t, ξ, d))=0. sup 0 t τ ξ,d M D g(x(t, ξ, d)) k(t). sup 0 t t ξ,d M D g ( x(t, ξ, d) ) k(t), ( ) c where t ξ = T r δ( ξ 2c A ). 2 ( Proof: If the statement is not true, then for any ε>0, there exists some t ε >T c r 2c 2 δ( ξ A ) ) and some d ε such that So we have U(ξ) g(x(t ε,ξ,d ε))k(t ε)+ε. δ( ξ A ) c U(ξ) c g(x(t ε,ξ,d ε)) k(t ε)+ ε c c2 c g(x(t ε,ξ,d ε)) + ε c c2 x(t ε,ξ,d ε) c A + ε < δ( ξ A ) + ε. c 2 c Taking the limit as ε tends to 0 results in a contradiction. For any compact set K IR n \A, let t K def = max ξ K t ξ <. (Finiteness follows from Fact 2, as K {ξ : 0 < ξ A <r} for some r>0.) Lemma 6.2. The function U( ) defined by (36) is locally Lipschitz on IR n \A, and continuous everywhere. Proof. Forξ 0 A, pick up a compact neighborhood K 0 of ξ 0 so that K 0 A=. By (38), one knows that U(ξ) >r 0, ξ K 0, for some constant r 0 > 0. Let r = r0 and let 2c 2 { } K = K 0 η : η ξ 0 r 4C where C is such a constant that, (39) x(t, ξ, d) x(t, η, d) C ξ η, ξ, η K 0, 0 t t K0, d M D. In the following we will show that there exists some L>0 such that for any ξ, η K, it holds that (40) U(ξ) U(η) L ξ η. 8

19 First of all, for any ξ K and any ε (0, r 0/2), there exists t ξ,ε [0, t K0 ] and d ξ,ε M D, such that from which it follows that U(ξ) g(x(t ξ,ε,ξ,d ξ,ε ))k(t ξ,ε )+ε c 2 x(t ξ,ε,ξ,d ξ,ε ) A + ε, x(t ξ,ε,ξ,d ξ,ε ) A r. It follows from (39) that for any η K, x(t ξ,ε,η,d ξ,ε ) A x(t ξ,ε,ξ,d ξ,ε ) A x(t ξ,ε,ξ,d ξ,ε ) x(t ξ,ε,η,d ξ,ε ) r 2. By Proposition 5. one knows that there exists some compact set K 2 such that x(t, ξ, d) K 2, ξ K, t [0, t K ], and d M D. Again, applying Lemma 6. to the compact set K 2 {ζ : ζ A r /2}, one sees that g(x(t ξ,ε,ξ,d ξ,ε )) g(x(t ξ,ε,η,d ξ,ε )) C x(t ξ,ε,ξ,d ξ,ε ) x(t ξ,ε,η,d ξ,ε ), for some C > 0. Therefore, we have the following: U(ξ) U(η) g(x(t ξ,ε,ξ,d ξ,ε ))k(t ξ,ε )+ε g(x(t ξ,ε,η,d ξ,ε ))k(t ξ,ε ) c 2 g(x(t ξ,ε,ξ,d ξ,ε )) g(x(t ξ,ε,η,d ξ,ε )) + ε C c 2 x(t ξ,ε,ξ,d ξ,ε ) x(t ξ,ε,η,d ξ,ε ) + ε L ξ η + ε, for some constant L that only depends on the compact set K. Note that the above holds for any ε (0, r 0/2), thus, U(ξ) U(η) L ξ η, ξ, η K. By symmetry, one proves (40). To prove the continuity of U on IR n, note that for any ξ A, it holds that U(ξ) = 0, and so for all η IR n : U(ξ) U(η) = U(η) c 2 η A c 2 ξ η. The proof of Lemma 6.2 is thus concluded. We next start proving that U decreases along trajectories. Now pick any ξ A. Let h 0 > 0 be such that x(t, ξ, d) A ξ A, d D, t [0, h0], 2 where d denotes the constant function d(t) d. Such an h 0 exists by continuity. Pick any h [0, h 0]. For each d D, let η d = x(h, ξ, d). For any ε>0, there exist some t d,ε and d d,ε M D such that (4) U(η d ) g(x(t, η d,d d,ε ))k(t d,ε )+ε ( = g(x(t d,ε + h, ξ, d d,ε ))k(t d,ε + h) k(t d,ε + h) k(t d,ε ) k(t d,ε + h) ( U(ξ) k(t ) d,ε + h) k(t d,ε ) + ε, c 2 ) + ε 9

20 where d d,ε is the concatenation of d and d d,ε. Still for these ξ and h, and for any r> ξ A, define (42) Tξ,h r def = ( ) c max T r δ( x( t, ξ, d) 0 t h,d D 2c A ). 2 ( ( )) Claim: t d,ε + h Tξ,h, r for all d Dand for all ε 0, c 2 δ ξ A. 2 ( ( )) Proof: If this were not true, then there would exist some d and some ε 0, c 2 δ ξ A 2 t d, ε + h>tξ,h, r and hence in particular, for t = h and d = d it holds that ( ) c t d, ε + h>t r 2c δ( η d A ), 2 which implies that x(t d, ε,η d, d d, ε ) A = x(t d, ε + h, ξ, v) A < c 2c δ( η d A ), 2 where v is the concatenated function defined by such that { d, if 0 t h, v(t) = d d, ε (t h), if t>h. Using (38), one has δ ( ) η d A c U(η d) g(x(t d, ε c,η d, d d, ε ))k(t d, ε )+ ε c c2 c x(t d, ε,η d, d d, ε ) + ε < A c 2 δ ( ) ε η d A +, c ( ) which is a contradiction, since ε < c 2 δ ξ A cδ ( ) η d A. This proves the claim. 2 2 From (4), we have for any d Dand for any ε>0 small enough, U(x(h, ξ, d)) U(ξ) U(ξ) (k(t d,ε + h) k(t d,ε )) c 2 + ε = U(ξ) c 2 k (t d,ε + θh)h + ε, where θ is some number in (0, ). Hence, by the assumptions made on the function k, we have U(x(h, ξ, d)) U(ξ) U(ξ) τ(t d,ε + θh)h + ε U(ξ) τ(tξ,h) r h + ε. c 2 c 2 Again, since ε can be chosen arbitrarily small, we have U(x(h, ξ, d)) U(ξ) U(ξ) τ(tξ,h)h, r d D. c 2 Thus we showed that for any d and any h>0 small enough, U(x(h, ξ, d)) U(ξ) U(ξ) τ(tξ,h) r. h c 2 Since U is locally Lipschitz on IR n \A, it is differentiable almost everywhere in IR n \A, and hence for any d D and for any r> ξ A, (43) (44) U(x(h, ξ, d)) U(ξ) U(ξ) L fd U(ξ) = lim lim τ(t h 0 + h h 0 + ξ, r h) c 2 = U(ξ) ( ) τ lim c T r 2 h 0 + ξ, h = U(ξ) ( ( )) c τ T r δ( ξ c 2 2c A ) 2 cδ( ξ A ) ( ( )) c τ T r δ( ξ c 2 2c A ) 2 = ᾱ r( ξ A ), a.e., 20

21 where ᾱ r(s) = cδ(s) c 2 τ ( ( )) c T r δ(s). 2c 2 Now define the function ᾱ by ᾱ(s) = sup ᾱ r(s). r>s Note that ᾱ r(0) = 0 for any r>0, so ᾱ(0) = 0. Also, applying to r =2s, wehave ᾱ(s) c δ(s) c 2 τ ( ( )) c T 2s δ(s) > 0 2c 2 for all s>0. Notice that (44) holds for any r> ξ A, so it follows that for every d D, L fd U(ξ) ᾱ( ξ A ) for almost all ξ IR n \A. Now let ˇα(s) = cδ(s) 2s+ τ c 2 2s ( ( )) c T r δ(s) dr, 2c 2 for s>0, and let ˇα(0) = 0. Then ˇα is continuous on [0, ) (the continuity at s = 0 is because τ is bounded and δ(0) = 0), and for s>0, it holds that 0 < ˇα(s) c ( ( )) δ(s) c τ T 2s δ(s) c 2 2c 2 because of the monotonicity properties of T and τ. Furthermore, L fd U(ξ) ᾱ( ξ A ) ˇα( ξ A ), for almost all ξ IR n \A. By Theorem B. provided in the appendix, there exists a C function V :IR n \A IR 0 such that for almost all ξ IR n \A, V (ξ) U(ξ) < U(ξ) 2 and L fd V (ξ) 2 ˇα( ξ A ), d D. Extend V to IR n by letting V A = 0 and again denote the extension by V. Note that V is continuous on IR n. So V is a Lyapunov function, as desired, with α (s) = c 3c2 δ(s), α2(s) = 2 2 s and α3(s) = 2 ˇα(s). 7. Proof of the Second Converse Lyapunov Theorem. We need a couple of Lemmas. The first one is trivial, so we omit its proof. Lemma 7.. Let f :IR n D IR n be continuous, where D is a compact subset of IR l. Then there exists a smooth function a f :IR n IR, with a f (x) everywhere, such that f(x, d) a f (x) for all x and all d. Now for any given system not necessarily complete, consider the following system: Σ: ẋ = f(x, d), Σ b : ẋ = f(x, d). a f (x) f(x, d) Note that the system Σ b is complete since for all x, d. We let x b (, x 0,d) denote the trajectory a f (x) of Σ b corresponding to the initial state x 0 and the time-varying parameter d. The following result is a simple 2

22 consequence of the fact that the trajectories of Σ are the same as those of Σ b up to a rescaling of time. We provide the details to show clearly that the uniformity conditions are not violated. Lemma 7.2. Assume that A is a compact set. Suppose that system Σ is UGAS with respect to A. Then, system Σ b is UGAS with respect to A. Proof. Pick a time-varying parameter d M D and an initial state x 0 IR n. Let γ b (t) denote x b (t, x 0,d). Let τ γb (t) denote the solution for t 0 of the following initial value problem: (45) τ = a f (γ b (τ)), τ(0)=0. Since a f is smooth, and γ b is Lipschitz, a f γ b is locally Lipschitz as well. It follows that a unique τ γb (t) is at least defined in some interval [0, t ). Note that τ γb is strictly increasing, so t <+ would imply lim t t τγ b (t) =+. Claim: For every trajectory γ b of Σ b, τ γb (t) is defined for all t 0. Proof: If the claim is not true, then there exist some trajectory γ b of Σ b and some t > 0 such that lim t t τ γb (t) =. Now for t [0, t ), one has: (46) ( d dt γ b(τ γb (t)) = a f (γ b (τ γb (t))) f ( γ b τγb (t) ),d ( τ γb (t) )) d dt τγ (t) b ( ( = f γ b τγb (t) ),d ( τ γb (t) )). Thus γ b (τ γb (t)) is a solution of Σ on [0, t ). By the stability of Σ, it follows that γ b (τ γb (t)) A <δ ( x 0 A ), t [0, t ), where x 0 = γ b (0), and δ is the function for Σ as defined in Definition 2.2. (c.f. Remark 2.4.) Let c = δ ( x 0 A ), and let M = sup ξ A c a f (ξ). (M is finite because the set {ξ : ξ A c} is a compact set.) From here one sees that τ γb (t) Mt for any t [0, t ). This is a contradiction. Thus τ γb (t) is defined for all t 0. This proves the claim. Since a f (s) and, for every trajectory γ b of Σ b, τ γb (0) = 0, it follows that τ γb ( ) K for each trajectory γ b of Σ b. From (46), one also sees that if γ b (t) is a trajectory of Σ b, then γ b (τ γb (t)) is a trajectory of Σ, and furthermore, γ b (τ γb (s)) A <ε s 0, if γ b (0) A δ(ε). It follows that γ b (t) A = γb (τ γb (τ γ b (t))) A <ε, t 0, whenever γ b(0) A δ(ε). This shows that condition () of Definition 2.2 holds for Σ b, with the same function δ. Fix any r, ε > 0. Pick any x 0 with x 0 A <rand any d M D. Again let γ b (t) denote the corresponding trajectory of Σ b. Then Let γ b (t) A = γb (τ γb (τ γ b (t))) A <δ (r), t 0. L = sup{a f (ξ) : ξ A δ (r)}. Then one sees that τ(t) L, which implies that τ γb (t) Lt for all t 0. Note that for the given r, ε > 0, by the UGAS property for Σ, there exists T>0such that for every d M D, γ b (τ γb (s)) A <ε 22

23 whenever γ b (0) A <rand s T. This implies that γ b (t) A <ε whenever γ b (0) A <rand t τ γb (T ). Combining this with the fact that τ γb (t) Lt, one proves that for any d M D, it holds that γ b (t) A <ε whenever γ b (0) A <rand t LT. Hence we conclude that Σ b is UGAS. In Lemma 7.2, the assumption that A is compact is crucial. Without this assumption, the conclusion may fail as the following example shows. Example 7.3. Consider the following system Σ: (47) ẋ =(+y 2 ) tanh x, ẏ = y 4. (Here f is independent of d.) Let A = {(x, y) : x =0}. Clearly the system is UGAS with respect to A. For this system, a natural choice of a f is 2 + y 4. Thus, the corresponding Σ b is as follows: ẋ = (tanh x) +y2 2+y 4, ẏ = y4 2+y 4. However, the system Σ b is not UGAS with respect to A. This can be seen as follows. Assume that Σ b is UGAS. Then for ε =/2, there exists some T>0such that for any solution (x(t),y(t)) of Σ b with x(0) =, it holds that (48) x(t) <, t T. 2 Since +y2 0asy, it follows that there exists some y0 > 0 such that 2+y4 +y2 2+y 4 <, y y0. 3T Now consider the trajectory (x(t), y(t)) of Σ b with x(0) =,y(0) = y 0, where y 0 is as above. Clearly y(t) y 0 for all t 0, and thus, which implies that ẋ = (tanh x) +y2 (tanh x) 2+y4 3T 3T, x(t ) 3T T = 2 3. This contradicts (48). From here one sees that Σ b is not UGAS with respect to A. We now prove Theorem 2.9. The proof of the sufficiency part is the same as in the proof of Theorem 2.8. Observe that the fact that V (ξ) is nonincreasing along trajectories implies, by compactness of A, that trajectories are bounded, so x(t) is defined for all t 0. We now prove necessity. Let a f be a function for f as in Lemma 7., and let Σ b be the corresponding system. Then by Lemma 7.2, one knows that the system Σ b is UGAS. Applying Theorem 2.8 to the complete system Σ b, one knows that there exists a smooth Lyapunov function V for Σ b such that α ( ξ A ) V (ξ) α 2( ξ A ), ξ IR n, 23

24 and L fd V (ξ) α 3( ξ A ), ξ A, d D, for some K -functions α,α 2 and some positive definite function α 3, where f d (ξ) = f(ξ, d) a f (ξ). Since a f (ξ) everywhere, it follows that L fd V (ξ) α 3( ξ A ), ξ A, d D. Thus, one concludes that V is also a Lyapunov function of Σ. 8. An Example. In general, for a noncompact parameter value set D, the converse Lyapunov theorem will fail, even if the vector fields f(ξ, d) are locally Lipschitz uniformly on d on any compact subset of D (for instance, if f is smooth everywhere). To illustrate this fact, consider the common case of systems affine in controls: ẋ = f(x)+g(x)d, where for simplicity we consider only the unconstrained single-input case, that is, D = IR. Assume that there would exist a Lyapunov function V for this system in the sense of definition 2.6. Then, calculating Lie derivatives, we have that, in particular, L f V (ξ)+dl gv (ξ) < 0, ξ 0, d IR, which implies that L gv (ξ) =0, ξ 0. Thus V must be constant along all the trajectories of the differential equation ẋ = g(x). In general, such a property will contradict the properness or the positive definiteness of V, unless the vector field g is very special. As a way to construct counterexamples, consider the following property of a vector field g, which is motivated by the prolongation ideas in [28]. Consider the closure W (ξ 0) of the trajectory through ξ 0 with respect to the vector field g. Note that if ξ W (ξ 0), then the fact that V is constant on trajectories, coupled with continuity of V, implies that V (ξ )=V (ξ 0). Now assume that there is a chain ξ 0,ξ,ξ 2,... so that for each i =, 2,..., ξ i W (ξ i ). Then we conclude that V (ξ i )=V(ξ 0 ) for all i. If the sequence {ξ i } converges to zero (and ξ 0 0) or diverges to infinity, we contradict positive definiteness or properness of V respectively. For an example, take the following two dimensional system, which was used in [7] to show essentially the same fact. Let S be the spiral that describes the solution of the differential equation ẋ = x y, ẏ = x y, passing through the point (, 0). Explicitly, S can be parameterized as x = e t cos t, y = e t sin t, <t<. In polar coordinates, the spiral is given by r = e θ, <θ<. Let a(x, y) be any nonnegative smooth function which is zero exactly on the closure of the spiral S (that is, S plus the origin). (Such a function always 24