Converse Lyapunov Functions for Inclusions 2 Basic denitions Given a set A, A stands for the closure of A, A stands for the interior set of A, coa sta

A smooth Lyapunov function from a class-kl estimate involving two positive semidenite functions Andrew R. Teel y ECE Dept. University of California Santa Barbara, CA 93106 teel@ece.ucsb.edu Laurent Praly Centre Automatique et Systemes Ecole des Mines de Paris 35 rue St. Honore 77305 Fontainebleau cedex, FRANCE praly@cas.ensmp.fr Submitted to Journal of Dierential Equations, June 15, 1999 Abstract We consider dierential inclusions where a positive semidenite function of the solutions satises a class-kl estimate in terms of time and a second positive semidenite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-kl estimate, exists if and only if the class-kl estimate is robust, i.e., it holds for a larger, perturbed dierential inclusion. It remains an open question whether all class-kl estimates are robust. One sucient condition for robustness is that the original differential inclusion is locally Lipschitz. Another sucient condition is that the two positive semidenite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for dierential equations and dierential inclusions that have appeared in the literature. Corresponding author. y Research supported in part by the NSF under grant ECS-9896140 and by the AFOSR under grant F49620-98-1-0087. 1

Converse Lyapunov Functions for Inclusions 2 Basic denitions Given a set A, A stands for the closure of A, A stands for the interior set of A, coa stands for the closed convex hull of A and @A stands for the boundary of A. The notation x! @A 1 indicates a sequence of points x belonging to A converging to a point on the boundary of A or, if A is unbounded, having the property jxj! 1. Given a closed set A R n and a point x 2 R n, jxj A denotes the distance from x to A. A function : R0! R0 is said to belong to class-k ( 2 K) if it is continuous, zero at zero, and strictly increasing. It is said to belong to class-k 1 if, in addition, it is unbounded. A function : R0 R0! R0 is said to belong to class-kl if, for each t 0, (; t) is nondecreasing and lim s!0 + (s; t) = 0, and, for each s 0, (s; ) is nonincreasing and lim t!1 (s; t) = 0. The requirements imposed for a function to be of class-kl are slightly weaker than usual. In particular, (; t) is not required to be continuous or strictly increasing. See, also, Remark 2. 1 Introduction 1.1 Background In the same text [19] where Lyapunov introduced his famous sucient conditions for asymptotic stability of the origin of a dierential equation _ = f(; t) ; (1) we can nd the rst contribution [19, x20, Theorem II] to the converse question: what aspects of asymptotic stability and the function f guarantee the existence of a (smooth) function satisfying Lyapunov's sucient conditions for asymptotic stability? The answers have proved instrumental, over the years, in establishing robustness of various stability notions and have served as the starting point for many nonlinear control systems design concepts. One of the early important milestones in the pursuit of smooth converse Lyapunov functions was Massera's 1949 paper [21] that provided a semi-innite integral construction for time-invariant, continuously dierentiable systems with an asymptotically stable equilibrium. Later, in 1954, Malkin observed that Massera's construction worked even for time-varying systems as long as the asymptotic stability and the dierentiability of f with respect to the state were uniform in time [20]. Regarding stability Malkin assumed, in eect, the existence of a class-kl function such that the solutions (t; t ; ) of the system (1), issued from at time t, satisfy j(t; t ; )j (j j; t? t ) 8t t 0 (2)

Converse Lyapunov Functions for Inclusions 3 at least for initial conditions suciently small. In [5], Barbashin and Krasovskii generalized Malkin's result to the case where (2) holds for all initial conditions. Both Massera [22] and Kurzweil [15], independently in the mid-1950's, weakened the assumptions made by Malkin, and Barbashin and Krasovskii about the function f(; t). Kurzweil's contribution especially stands out because he was able to establish a converse theorem even when f(; t) is only continuous so that uniqueness of solutions is not guaranteed. In his work he made precise a notion of strong stability of the origin on an open neighborhood G of the origin which amounted to the existence of a function 2 KL and a locally Lipschitz, positive denite function! : G! R0, proper on G, such that all solutions of the system (1) satisfy!((t; t ; )) (!( ); t? t ) t t 0 : (3) Kurzweil showed that this strong stability and continuity of f(; t) imply the existence of a smooth converse Lyapunov function, i.e., a function whose derivative along solutions can be used to deduce (3). Much of the research in the 1960's focused on developing converse Lyapunov theorems for systems possessing asymptotically stable closed, not necessarily compact, sets. Taking this approach, the time-varying case can be subsumed into the time-invariant case by augmenting the state-space of (1) as: _x = d f(; p) = =: F (x) ; x dt p 1 = : (4) t (One disadvantage in treating time-varying systems as time-invariant ones is that it usually leads to imposing stronger than necessary conditions on the time-dependence of the righthand side, e.g., continuity where only measurability is needed. An example where a converse theorem is developed for systems with right-hand sides measurable in time, and for Lyapunov and Lagrange stability, is [4].) A closed set A for (4) is said to be uniformly asymptotically stable if there exists of a function 2 KL such that all solutions of (4) with jx j A suciently small exist for all forward time and satisfy j(t; x )j A (jx j A ; t) 8t 0 : (5) Some of the early results on converse Lyapunov functions for set stability are summarized in [39]. Particularly noteworthy are the result of Hoppensteadt [14] who generated a C 1 converse Lyapunov function for parameterized dierential equations and the result of Wilson [38] who provided a smooth converse Lyapunov function for uniform asymptotic stability of a set. Converse theorems for (compact) set stability for nonlinear dierence equations are also now standard (see, e.g., [32, Theorem 1.7.6]). Noting that every solution (t; x ) of (4) can be written as (t + t ; t (t; x ) = ; ) (6) t + t where (; t ; ) is a solution of (1), it follows that (2), (3) and (5) are all particular cases of the estimate!((t; x )) (!(x ); t) 8t 0 (7)

Converse Lyapunov Functions for Inclusions 4 where! is a continuous, positive semidenite function. In the 1970's, Lakshmikantham and coauthors [17] (see also [16, Section 3.4]) provided a Lipschitz converse Lyapunov function for Lipschitz ordinary dierential equations of the form (4) under an assumption essentially the same as: given two continuous, positive semidenite functions! 1 and! 2, there exists a function 2 KL such that, for all initial conditions x with! 2 (x ) suciently small, all solutions exist for all forward time and satisfy! 1 ((t; x )) (! 2 (x ); t) 8t 0 : (8) The stability concept described by (8), apparently rst introduced in [27] and often called stability with respect to two measures, generalizes (7) and thus includes the notions of local uniform asymptotic stability of a point, of a prescribed motion and of a closed set. It also covers the notion of local uniform partial asymptotic stability such as when, for x suciently small, jh((t; x ))j (jx j; t) 8t 0 (9) where y = h(x) is a continuous (output) function of the state. A smooth converse Lyapunov theorem for the global version of (9) was recently derived in [31, Theorem 2]. (See [37] for a survey on the partial stability problem.) Extensions of the above results to dierential inclusions started to appear in the late 1970's with some of the most general results appearing only recently. Some motivations for the study of dierential inclusions are: 1) they describe the solution set for ordinary dierential equations with arbitrary, measurable bounded disturbances (see Footnote 1 below), and 2) they describe important notions of solutions for control systems that use discontinuous feedbacks. (see [12, x8.3].) The results in [23] pertain to dierential inclusions of the form _x 2 F (x) := co fv 2 R n : v = f(x; d) ; d 2 Dg (10) where D is compact, f(x; d) is continuous and continuously dierentiable with respect to x, and asymptotic stability in the rst approximation is assumed, i.e., for the inclusion an estimate of the form _x 2 v 2 R n : v = @f @x (0; d)x ; d 2 D ; (11) jx(t)j kjx(0)j exp(?t) k > 0 ; > 0 (12) is assumed for all solutions starting from suciently small initial conditions. The result [23, Theorem 2] states that this assumption implies the existence of a smooth converse Lyapunov function for local exponential stability and asymptotic stability on the basin of attraction of the origin for the inclusion (10). Related results for inclusions of the type (11) can also be found in [24, 25, 26]. In [18], Lin and coauthors considered the dierential inclusion (10), relaxing the continuous dierentiability assumption of f with respect to x to a local Lipschitz assumption, and

Converse Lyapunov Functions for Inclusions 5 assumed the estimate (5), i.e., the estimate (8) with! 1 (x) =! 2 (x) = jxj A, for all initial conditions. 1 They showed, under the additional assumption that either A is compact or all solutions exist for all backward time, that the estimate (5) for the dierential inclusion (10) implies the existence of a smooth converse Lyapunov function. In [1], the authors combined the ideas of [18] with the idea of Kurzweil [15] establishing the existence of a smooth converse Lyapunov function for the dierential inclusion (10) in the case of the existence of a compact set A, a neighborhood G of A and function! : G! R0 that is locally Lipschitz, positive denite with respect to A and proper with respect to G and a function 2 KL such that, for all x 2 G, the solutions of (10) satisfy!((t; x )) (!(x ); t) 8t 0 : (13) The rst results on smooth converse theorems for dierential inclusions that are only upper semicontinuous (see Denition 1 below) appeared in [6]. In that work, Clarke and coauthors studied _x 2 F (x) (14) under the assumption that F (x) is nonempty, compact and convex for each x 2 R n and F (x) is upper semicontinuous. They assumed the estimate (8) with! 1 (x) =! 2 (x) = jxj, and showed that this implies the existence of a smooth converse Lyapunov function. A related result, for the case of uniform exponential stability for switching systems, can be found in [9]. Other interesting results on the existence of nondierentiable converse Lyapunov functions can be found in [2, Chapter 6] and [34, 35, 36]. 1.2 Contributions In this paper, we consider dierential inclusions _x 2 F (x) (15) that satisfy the conditions assumed in [6]. In particular, F (x) is a set-valued map from an open set G to subsets of R n that is upper semicontinuous on G (see Denition 1) and is such that F (x) is nonempty, compact and convex for each x 2 G. The stability property we will assume for (15) we will refer to as \KL-stability with respect to (! 1 ;! 2 ) on G". Namely, given two continuous functions! 1 : G! R0 and! 2 : G! R0, we assume the existence of 1 More precisely, [18] considers systems of the form _x = f(x; d(t)) where d(t) belongs to the set of measurable functions taking values in the compact set D. It is known (see [11] or [7, problem 3.7.20], for example) that the solutions of this system are exactly the solutions of the dierential inclusion _x 2 F (x) := fv 2 R n : v = f(x; d) ; d 2 Dg. Also, since this F (x) is a nonempty, compact, locally Lipschitz set-valued map under the assumptions of [18], it is known (see, for example, [3, Corollary 10.4.5]) that the closure of the solution set of _x 2 F (x) is exactly the solution set of _x 2 cof (x). Thus, the result [18, Proposition 5.1] shows that if the solutions of _x 2 F (x) exist for all forward time and satisfy (5) then the same is true for the solutions of _x 2 cof (x).

Converse Lyapunov Functions for Inclusions 6 a class-kl function such that all solutions of the dierential inclusion (15) starting in G remain in G for all forward time and satisfy! 1 ((t; x )) (! 2 (x ); t) 8t 0 : (16) (See also Denition 6). This is like the stability property considered in [17]. Our main result is (see Theorem 1): A smooth converse Lyapunov function for KL-stability with respect to (! 1 ;! 2 ) (see Denition 7) exists if and only if the KL-stability is robust, i.e., it holds for a larger, perturbed dierential inclusion (see Denition 8). This type of equivalence between robust stability and the existence of a Lyapunov function, reminiscent of the classical \total stability" results for ordinary dierential equations [13, Theorem 56.4], is already present in the proofs of Kurzweil [15] and Clarke, et al., [6]. It remains an open question whether KL-stability with respect to (! 1 ;! 2 ) is robust, in general. However, we will show (see Theorem 2): If the dierential inclusion (15) is locally Lipschitz on G (see Denition 3) then KL-stability with respect to (! 1 ;! 2 ) on G is robust. This is the case for the problems consider by Lakshmikantham, et al. [16], Lin, et. al. [18], and Sontag and Wang [31, Theorem 2]. We will also show (see Theorem 3): If the dierential inclusion (15) is backward completable by!-normalization (see Denition 9) then KL-stability with respect to (!;!) on G is robust. This condition holds for the problems considered by Kurzweil [15] (see Corollary 1) and Clarke, et al. [6] (see Corollary 2). It is also useful for generating smooth converse Lyapunov functions for compact, stable attractors. As an illustration, we provide a smooth converse Lyapunov function for nite time convergence to a compact set from a larger compact set (see Corollary 3). This result is useful for the problem of semiglobal practical asymptotic stabilization of nonlinear control systems as studied in [33], for example. Our converse Lyapunov function is constructed in the following steps : 1. We imbed the original dierential equation or dierential inclusion into a larger, locally Lipschitz dierential inclusion that still exhibits KL-stability with respect to (! 1 ;! 2 ). This idea is due to Kurzweil [15] for the case of ordinary dierential equations with continuous right-hand side under strong stability of the origin. It is due to Clarke and co-authors [6] for the case of nonempty, compact, convex, upper semicontinuous dierential inclusions and global asymptotic stability of the origin. In general, it is possible if and only if the KL-stability with respect to (! 1 ;! 2 ) is robust. 2. We nd class-k 1 functions e 1 and e 2 such that e 1 ((s; t)) e 2 (s)e?2t, where quanties KL-stability with respect (! 1 ;! 2 ) for the locally Lipschitz dierential inclusion constructed in step 1. A recent result by Sontag [30, Proposition 7] shows that this is always possible.

Converse Lyapunov Functions for Inclusions 7 3. We dene a trial Lyapunov function V 1 (x) as the supremum, over time and solutions (; x) of the locally Lipschitz dierential inclusion constructed in step 1, of the quantity e 1 (! 1 ((t; x)))e t where e 1 was constructed in step 2. This is a classical construction once the estimate in step 2 is available, at least for locally Lipschitz ordinary dierential equations where the supremum over solutions is not needed. (See, e.g., [39, x19].) We show, using many of the tools used in [6] and [18], that this trial Lyapunov function has all of the desired properties except smoothness. However, it is locally Lipschitz. It would only be upper semicontinuous, in general, if the supremum were taken over solutions of the original dierential inclusion. 4. We smooth the trial Lyapunov function using ideas that go back to Kurzweil [15] and that have been claried, generalized and used over the years by, for example, Wilson [38], Lin, et al., [18] and Clarke, et al. [6]. The rest of the paper is organized as follows: In Section 2 we present some denitions related to set-valued maps and some properties of solutions to dierential inclusions. These denitions are needed for understanding the statement of our main results. In Section 3 we give precise denitions of KL-stability with respect to (! 1 ;! 2 ) (Denition 6) as well as the robust version of this property (Denition 8), and of a smooth converse Lyapunov function for KL-stability with respect to (! 1 ;! 2 ) (Denition 7). Then we present our main results on the existence of a smooth converse Lyapunov function and relate these results to others that have appeared in the literature. Section 4 contains some technical prerequisites that are necessary for the proofs of our main results. We prove our main results in Section 5. Particularly noteworthy are Section 5.1.2, which contains the construction of our smooth converse Lyapunov function under the assumption of robust KL-stability, and Section 5.3 where we establish robust KLstability under the assumption of nominal KL-stability with respect to (!;!) plus a backward completability assumption. Section 6 contains the proofs of some propositions that are used to make connections to other results that have appeared in the literature. In an addendum, we include (elements of) the proofs of Lemmas 1, 8, 9, 16 and 17 and of Proposition 1. They can be found elsewhere in the literature, may be with some minor modications. The addendum is not intended to be part of the published version of the paper. In this paper, we have borrowed many ideas and technicalities from our predecessors. We try to make this point clear in bibliographical notes.

Converse Lyapunov Functions for Inclusions 8 2 Preliminaries Throughout this paper F (x) will be a set-valued map from G to subsets of R n where G is an open subset of R n. Also B denotes the open unit ball in R n and F (x) + "B := z 2 R n : jzj F (x) < " : (17) We review some denitions concerning set-valued maps (see also [12, x5.3, x7.2]): Denition 1 The set-valued map F is said to be upper semicontinuous on G if, given x 2 G, for each " > 0 there exists > 0 such that, for all 2 G satisfying jx? j < we have F () F (x) + "B. Denition 2 The set-valued map F is said to satisfy the basic conditions on G if it is upper semicontinuous on G and, for each x 2 G, F (x) is nonempty, compact and convex. We will need the following fact: Lemma 1 If the set-valued map F satises the basic conditions on G and : G! R0 is a continuous function such that for all x 2 G, we have then the set-valued map co fxg + (x)b G ; (18) 0 @ [ 2fxg+(x)B 1 F () A + (x)b ; which we denote by cof (x + (x)b) + (x)b ; satises the basic conditions on G. Denition 3 Let O be an open subset of G. The set-valued map F is said to be locally Lipschitz on O if, for each x 2 O, there exists a neighborhood U O of x and a positive real number L such that x 1 ; x 2 2 U =) F (x 1 ) F (x 2 ) + Ljx 1? x 2 jb : (19) Given a set-valued map F (x), we can dene a solution of the dierential inclusion _x 2 F (x) : (20) Denition 4 A function x : [0; T ]! G (T > 0) is said to be a solution of the dierential inclusion (20) if it is absolutely continuous and satises, for almost all t 2 [0; T ], z { _ x(t) 2 F (x(t)) : (21) A function x : [0; T )! G (0 < T 1) is said to be a maximal solution of the dierential inclusion (20) if it does not have an extension which is a solution belonging to G, i.e., either T = 1 or there does not exist a solution y : [0; T + ]! G with T + > T such that y(t) = x(t) for all t 2 [0; T ).

Converse Lyapunov Functions for Inclusions 9 The following basic fact about the existence of maximal solutions is a combination of [12, x7, Theorem 1] and [29, Propositions 1 and 2]. Lemma 2 If F (x) satises the basic conditions on G then for each x 2 G there exist solutions of (20) for suciently small T > 0 satisfying x(0) = x. In addition, every solution can be extended into a maximal solution. Moreover, if a maximal solution x() is dened on a bounded interval [0; T ) then x(t)! @G 1 as t! T. Henceforth, we will use (; x) to denote a solution of (20) starting at x and we will denote by S(x) or S(C) (respectively, S[0; T ](x) or S[0; T ](C)) the set of maximal solutions (respectively, solutions dened on [0; T ]) of the dierential inclusion (20) starting at x or in the compact set C. Note that with 1 2 S[0; T 1 ](x) and 2 2 S[0; T 2 ]( 1 (T 1 ; x)) and dening 3 (t; x) = 1 (t; x) if 0 t T 1 ; = 2 (t? T 1 ; 1 (T 1 ; x)) if T 1 t T 1 + T 2 : (22) we have 3 2 S[0; T 1 + T 2 ](x). Denition 5 The dierential inclusion (20) is said to be forward complete on G if, for all x 2 G, all solutions 2 S(x) are dened (and remain in G) for all t 0. The dierential inclusion (20) is said to be backward complete on G if the dierential inclusion _x 2?F (x) is forward complete on G. 3 Main results 3.1 General statements The stability concept we work with in this paper is called KL-stability with respect to (! 1 ;! 2 ) where! 1 and! 2 are continuous, positive semidenite functions. This concept is dened as follows: Denition 6 Let! i : G! R0, i = 1; 2, be continuous. The dierential inclusion _x 2 F (x) is said to be KL-stable with respect to (! 1 ;! 2 ) on G if it is forward complete on G and there exists 2 KL such that, for each x 2 G, all solutions 2 S(x) satisfy! 1 ((t; x)) (! 2 (x); t) 8t 0 : (23) As mentioned in the introduction, this stability concept was introduced in [27] and considered in [17] and [16]. It is often referred to as stability with respect to two measures. It covers standard stability notions like uniform global asymptotic stability of a closed set and partial asymptotic stability. In the case where A is a closed set,! 1 (x) =! 2 (x) = jxj A, and G = R n, it has been shown in [18, Proposition 2.5] that KL-stability is equivalent to the set A being uniformly globally stable and uniformly globally attractive. The technique used to prove [18, Proposition 2.5] is used to prove the following generalization for KL-stability with respect to (! 1 ;! 2 ):

Converse Lyapunov Functions for Inclusions 10 Proposition 1 Let! i : G! R0, i = 1; 2, be continuous. The following are equivalent: 1. The dierential inclusion _x 2 F (x) is KL-stable with respect to (! 1 ;! 2 ) on G. 2. All of the following hold: (a) The dierential inclusion _x 2 F (x) is forward complete on G. (b) (Uniform stability and global boundedness): There exists a class-k 1 function such that, for each x 2 G, all solutions 2 S(x) satisfy! 1 ((t; x)) (! 2 (x)) 8t 0 : (24) (c) (Uniform global attractivity): For each r > 0 and " > 0, there exists T (r; ") > 0 such that, for each x 2 G, all solutions 2 S(x) satisfy! 2 (x) r ; t T =)! 1 ((t; x)) " : (25) KL-stability with respect to (! 1 ;! 2 ) can be characterized in innitesimal (with respect to time) terms via the existence of a smooth Lyapunov function: Denition 7 Let! i : G! R0, i = 1; 2, be continuous. A function V : G! R0 is said to be a smooth converse Lyapunov function for KL-stability with respect to (! 1 ;! 2 ) on G for F (x) if V (x) is smooth on G and there exist class-k 1 functions 1, 2 such that, for all x 2 G, 1 (! 1 (x)) V (x) 2 (! 2 (x)) (26) and max hrv (x); wi?v (x) : (27) w2f (x) The motivation for this denition is that (27) guarantees the derivative of V (x) along solutions, denoted _V ((t; x)), satises _V ((t; x))?v ((t; x)) (28) for almost all t in the interval where (t; x) exists and belongs to G. It follows that V ((t; x)) V (x)e?t (29) on this interval and then, using (26) and assuming forward completeness on G, we can deduce KL-stability with respect to (! 1 ;! 2 ) on G. (By relying on a result like [18, Lemma 4.4], it is possible to deduce KL-stability with respect to (! 1 ;! 2 ) on G when V (x) on the right-hand side of (27) is replaced by any class-k 1 function of V (x).) We are interested in whether KL-stability with respect to (! 1 ;! 2 ) implies the existence of a smooth converse Lyapunov function for KL-stability with respect to (! 1 ;! 2 ). This is still an open question, in general. What we will show here is that a smooth converse Lyapunov function exists if and only if the KL-stability with respect to (! 1 ;! 2 ) is robust; that is KL-stability with respect to (! 1 ;! 2 ) still holds for a set of dierential inclusions given by supersets of F. This concept, which is present in the work of Kurzweil [15] and Clarke, et al. [6], is dened more precisely as follows:

Converse Lyapunov Functions for Inclusions 11 Denition 8 Let! i : G! R0, i = 1; 2, be continuous. The dierential inclusion _x 2 F (x) is said to be robustly KL-stable with respect to (! 1 ;! 2 ) on G if there exists a continuous function : G! R0 such that 1. fxg + (x)b G; 2. the dierential inclusion is KL-stable with respect to (! 1 ;! 2 ) on G; _x 2 F (x) (x) := cof (x + (x)b) + (x)b (30) 3. (x) > 0 for all x 2 GnA where A := ( 2 G : sup! 1 ((t; )) = 0 t0;2s () ) (31) and where S () represents the set of maximal solutions to (30). The main feature of the dierential inclusion (30) is that its solution set includes the solution set of the dierential inclusion _x 2 F (x) since F (x) F (x) (x). Note that even for ordinary dierential equations, robust stability will be expressed in terms of stability for a corresponding dierential inclusion. The following theorem emphasizes that robust KL-stability with respect to (! 1 ;! 2 ) is the key property for getting a smooth converse Lyapunov function. Theorem 1 Let! i : G! R0, i = 1; 2, be continuous and let F (x) satisfy the basic conditions on G. The following statements are equivalent: 1. The dierential inclusion _x 2 F (x) is forward complete on G and there exists a smooth converse Lyapunov function for KL-stability with respect to (! 1 ;! 2 ) on G for F (x). 2. The dierential inclusion _x 2 F (x) is robustly KL-stable with respect to (! 1 ;! 2 ) on G. Proof. See Section 5.1. We now specify cases where robust KL-stability is guaranteed. The rst case is when the dierential inclusion is locally Lipschitz, at least on a specic subset of G: Theorem 2 Let! i : G! R0, i = 1; 2, be continuous and let F (x) satisfy the basic conditions on G. If the dierential inclusion _x 2 F (x) is KL-stable with respect to (! 1 ;! 2 ) on G and F (x) is locally Lipschitz on an open set containing GnA where A := ( 2 G : sup! 1 ((t; )) = 0 t0;2s() ) : (32) then the dierential inclusion _x 2 F (x) is robustly KL-stable with respect to (! 1 ;! 2 ) on G.

Converse Lyapunov Functions for Inclusions 12 Proof. See Section 5.2. With the combination of Theorems 1 and 2 we recover the converse Lyapunov function results of [18] and [31, Theorem 2]. We also obtain a smooth, global version of [16, Theorem 3.4.1]. Several important converse Lyapunov function results, like those due to Kurzweil [15] and Clarke, et al. [6] are not covered by Theorem 2. However, they will be covered (see, respectively, Corollary 1 of Section 3.2 and Corollary 2 of Section 3.3) by our next set of sucient conditions for robust KL-stability. We will show that KL-stability implies robust KL-stability in the case where! 1 (x) =! 2 (x) =:!(x) and the dierential inclusion is backward completable by!-normalization. The latter is dened as follows: Denition 9 Let! : G! R0 be continuous. The dierential inclusion _x 2 F (x) is said to be backward completable by!-normalization if there exists a continuous function : G! [1; 1), a class-k function and a positive real number c such that and the dierential inclusion is backward complete on G. (x) (!(x)) + c (33) _x 2 1 (x) F (x) =: F N(x) (34) In this denition, the existence of (x) making (34) backward complete on G is always guaranteed. Indeed, from [12, x5, Lemma 15] or [3, Theorem 1.4.16], sup v2f (x) jvj can be upper bounded by a function (x) that is continuous on G. So, for instance, in the case where G = R n, by picking (x) = (x) we get sup v2fn (x) jvj 1 which implies that the dierential inclusion (34) is backward complete on R n. The diculty comes from requiring that (x) simultaneously satises (33). However, when!(x) is proper on G this diculty disappears. More generally, when G is the Cartesian product G := G 1 R n 2, with G 1 an open subset of R n 1 and by writing _x1 _x = 2 F (x) ; (35) _x 2 we have : Proposition 2 If 1. F satises the basic conditions on G; 2. we have lim x 1!@G 1 1 inf!(x 1 ; x 2 ) = 1 ; (36) x 2 2R n 2 3. there exist positive real numbers ` and b such that v = v1 v 2 2 F (x) =) jv 2 j `jx 2 j + b (37)

Converse Lyapunov Functions for Inclusions 13 then the dierential inclusion (35) is backward completable by!-normalization. Proof. See Section 6.1. With backward completability, KL-stability with respect to (!;!) implies robust KLstability with respect to (!;!): Theorem 3 Let! : G! R0 be continuous and let F (x) satisfy the basic conditions on G. If the dierential inclusion _x 2 F (x) is backward completable by!-normalization and KL-stable with respect to (!;!) on G then it is robustly KL-stable with respect to (!;!) on G. Proof. See Section 5.3. 3.2 Results specialized to ordinary dierential equations To illustrate our results, we specialize them to ordinary dierential equations _x = F (x) (38) where F : G! R n is continuous. As noted in the introduction, this covers the case of time-varying systems _ = f(; t) (39) where f is continuous in both variables (compare with [4]) by taking f(; p) x = ; F (x) = : (40) p 1 A direct consequence of Theorem 2 or 3 and Theorem 1 is the following: Corollary 1 If either or I. the function F (x) is locally Lipschitz on G, II. i.) F (x) is continuous on G, ii.)! 1 =! 2 =!, and iii.) the dierential equation _x = F (x) is backward completable by!-normalization then the following two statements are equivalent: 1. _x = F (x) is KL-stable with respect to (! 1 ;! 2 ) on G. 2. _x = F (x) is forward complete on G and there exists a smooth converse Lyapunov function for KL-stability with respect to (! 1 ;! 2 ) on G for F (x).

Converse Lyapunov Functions for Inclusions 14 With condition (II) of this corollary, we recover Kurzweil's main result [15, Theorem 7] which applies to the case described by (40) and Proposition 2, with when G 1 = R n?1 and, otherwise,! 1 (x) =! 2 (x) = max! 1 (x) =! 2 (x) = jj (41) jj; 3.3 Results for compact attractors 1 jj R n?1 ng 1? 2 j0j R n?1 ng 1 : (42) The main result of this section is that KL-stability with respect to (!;!), where! is a type of indicator for a compact set A, is equivalent to (local) stability of A plus attractivity. Uniform boundedness and uniform attractivity are guaranteed by the fact that the attractor is compact. Various applications of this observation are made including a corollary that recovers [6, Theorem 1.2]. Denition 10 Given a compact subset A of an open set G, a function! : G! R0 is said to be a proper indicator for A on G if! is continuous,!(x) = 0 if and only if x 2 A, and lim!(x) = 1. x!@g1 Remark 1 For each open set G and each compact set A G, there exists a proper indicator function. When G = R n we can take!(x) = jxj A. Otherwise, we can take, for example, 1 2!(x) = max jxj A ;? : (43) jxj Rn ng dist(a; R n n G) Note that the right-hand side of (42) is a function that is a proper indicator for the origin (in R n?1 ) on G 1. The properties of a proper indicator,!(x), for A on G enforce that KL-stability with respect to (!;!) on G implies that the set A is stable and all trajectories starting in G converge to A. The rst result of this subsection, which is similar to [15, Theorem 12], shows that the opposite is also true. Namely, for dierential inclusions with right-hand side satisfying the basic conditions, the basin of attraction G for a stable, compact attractor A is open and, for each function! that is a proper indicator for A on G, the dierential inclusion is KL-stable with respect to (!;!) on G. Proposition 3 Let F (x) satisfy the basic conditions on an open set O and let A O be compact. If the set A is stable and the set of points G from which A is strongly attractive contains a neighborhood of A, i.e., 1. Stability: for each " > 0 there exists > 0 such that, for each x 2 O T? A + B, each solution 2 S(x) is dened and belongs to O for all t 0 and satises j(t; x)j A " 8t 0 ; (44)

Converse Lyapunov Functions for Inclusions 15 2. Attractivity: the set G of points x 2 O such that each solution 2 S(x) is dened and belongs to O for all t 0 and satises lim t!1 j(t; )j A = 0 contains a neighborhood of A, then the set G is open and, for each function! that is a proper indicator for A on G, the dierential inclusion _x 2 F (x) is KL-stable with respect to (!;!) on G. Proof. See Section 6.2. By combining Theorems 1, 3 and Proposition 2 with Proposition 3, we recover [6, Theorem 1.2]: Corollary 2 Suppose F (x) satises the basic conditions on R n and the origin of the dierential inclusion _x 2 F (x) is globally asymptotically stable, i.e., for each " > 0 there exists > 0 such that jxj ; 2 S(x) =) j(t; x)j " 8t 0 ; (45) for each x 2 R n, all solutions 2 S(x) are dened for all t 0 and satisfy lim j(t; x)j = 0 : t!1 Then, taking!(x) = jxj, there exists a smooth converse Lyapunov function for KL-stability with respect to (!;!) on R n for F (x). Nontrivial compact attractors arise in various ways. One situation, which is commonly encountered in the semiglobal practical asymptotic stabilization of nonlinear control systems (see, for example, [33]), is when: Assumption 1 There exist two compact sets C 1, C 2, two strictly positive real numbers, T and an open set O such that C 1 + B C 2 O, F (x) satises the basic conditions on O and is Lipschitz on C 1 + B, for all x 2 C 2, all solutions 2 S(x) are dened and belong to O for all t 0 and belong to C 1 for t T. It can be shown that: Proposition 4 Under Assumption 1 the set A := f 2 C 1 : (t; ) 2 C 1 ; 8 2 S() ; 8t 0g (46) is a nonempty, compact stable attractor with basin of attraction containing C 2.

Converse Lyapunov Functions for Inclusions 16 Proof. See Section 6.3. As a consequence, Proposition 3 applies for this set A. Also, for each function! that is a proper indicator for A on its strong domain of attraction, Proposition 2 allows us to apply Theorem 3 and then Theorem 1. So we can state the following converse Lyapunov function theorem, for nite-time convergence to a compact set from a larger compact set : Corollary 3 Under Assumption 1, there exist a compact set A C 1 and an open set G C 2 such that, for each function! : G! R0 that is a proper indicator for A on G, there exists a smooth converse Lyapunov function for KL-stability with respect to (!;!) on G for F (x). 3.4 Bibliographical Notes Our main result, Theorem 1, is inspired by the observations of Kurzweil [15] and Clarke, et al. [6] who recognized that robust stability, in the context of their specic problems, allowed them to construct a smooth converse Lyapunov function. The results of Section 3.3 for compact attractors are based on similar results in the special cases considered by Kurzweil [15] and Clarke, et al., [6]. 4 Technical prerequisites The proofs of our main results will be based on several technical lemmas. They concern: 1. Sontag's lemma on KL-estimates. 2. Solutions to dierential inclusions satisfying the basic conditions. 3. Solutions to locally Lipschitz dierential inclusions. 4. Derivatives of locally Lipschitz functions. 5. Smoothing continuous and locally Lipschitz functions. 4.1 Sontag's lemma on KL-estimates We recall a recent result of Sontag [30, Proposition 7] that is one of the keys in our converse Lyapunov function construction. The lemma is a global version of a particular aspect of the well-known Massera Lemma [21, Section 12] (cf. [16, Lemma 3.4.1]). We provide an alternative proof. Lemma 3 For each class-kl function and each number > 0, there exist functions e 1 2 K 1 and e 2 2 K 1 such that e 1 (s) is locally Lipschitz and e 1 ((s; t)) e 2 (s)e?t 8(s; t) 2 R0 R0 : (47)

Converse Lyapunov Functions for Inclusions 17 Proof. First we pick 2 K 1 and a function : R0! R >0 continuous and strictly decreasing with lim t!1 (t) = 0 such that, for all t 0, we have ((t); t) (t) : (48) To see that such functions exist, let f" k g 1 k=1 be a sequence of strictly positive real numbers decreasing to zero. Since 2 KL, there exists a sequence ft k g 1 k=1 of strictly positive real numbers strictly increasing to innity such that (k + 1; t k ) " k. Dene t 0 = 0 and " 0 = max f(1; 0); 2" 1 g. Then, choosing to be any K 1 function upper bounded by the piecewise constant curve p 1 (t) = j + 1 for t 2 [t j ; t j+1 ) and choosing to be any continuous, strictly decreasing to zero function that is lower bounded by the piecewise constant curve p 2 (t) = " j for j 2 [t j ; t j+1 ) and using that 2 KL, we have, for each integer j 0 and each t 2 [t j ; t j+1 ), ((t); t) (j + 1; t j ) " j = p 2 (t) (t). Next, let?1 be the inverse of, which is dened and continuous on (0; (0)]. It is also strictly decreasing with lim s!0?1 (s) = +1. It follows that the function e?2?1 (s) is welldened, continuous, positive and strictly increasing on (0; (0)]. Then we can nd e 1 2 K 1, locally Lipschitz and such that, for all s 2 (0; (0)], With (48), it follows, for all t 0, Now, using (50) and the fact that 2 KL, e 1 (s) e?2?1 (s) : (49) e 1 (((t); t))e 2t e 1 ((t))e 2t 1 : (50) 0 < s (t) ) e 1 ((s; t))e t = p e 1 ((s; 0)) s e1 ((s; t)) p e1 ((s; t))e e 1 ((s; 0)) 2t ;(51) p e 1 ((s; 0)) p e 1 (((t); t))e 2t ; (52) p e 1 ((s; 0)) : (53) (t) s ) e 1 ((s; t))e t e 1 ((s; 0))e?1 (s) : (54) So (47) holds by taking e 2 2 K 1 such that e 2 (s) maxnp e1 ((s; 0)) ; e 1 ((s; 0))e?1 (s) o : (55) Remark 2 Even though we are assuming minimal continuity properties for KL functions, the preceding result shows that any KL function can always be upper bounded by a continuous KL function. In particular, (47) can be rewritten as (s; t)?1 1? 2 (s)e?t (56) where the right-hand side is of class-kl and is also continuous in (s; t).

Converse Lyapunov Functions for Inclusions 18 4.2 Solutions to inclusions satisfying the basic conditions For the dierential inclusion _x 2 F (x) we denote the set of points reachable from a compact set C G in time T > 0 as R T (C) := f 2 R n : = (t; x) ; t 2 [0; T ] ; x 2 C ; 2 S(x)g : (57) The following comes from [12, x7, Theorem 3] or [10, Theorem 7.1]: Lemma 4 Let F (x) satisfy the basic conditions on G and suppose the compact set C G and the strictly positive real number T > 0 are such that all solutions 2 S(x) are dened and belong to G for all t 2 [0; T ]. Then the set R T (C) is a compact subset of G and the set S[0; T ](C) is a compact set in the metric of uniform convergence. A consequence of Lemma 4 is the following: Lemma 5 Let F (x) satisfy the basic conditions on G and suppose x 2 G is such that all solutions 2 S(x) are dened and belong to G for all t 0. Then each sequence f n g 1 n=1 of solutions in S(x) has a subsequence converging to a function 2 S(x) and the convergence is uniform on each compact time interval. Proof. From Lemma 4, we know that for each integer k, the set S[0; k](x) is a compact set in the metric of uniform convergence. Since for all n and k, n is in S[0; k](x), it follows that f n g 1 n=1 has a subsequence f 1m g 1 m=1 converging uniformly on [0; 1] to a function 1 2 S(x). Similarly f 1m g 1 m=1 has a subsequence f 2m g 1 m=1 converging uniformly on [0; 2] to a function 2 2 S(x). And so on. The result follows by taking the subsequence given by the diagonal elements mm. The next result is on \continuity" of solutions with respect to initial conditions and perturbations of the right-hand side. See [12, x8, Corollary 2]. Lemma 6 Suppose _x 2 F (x) is forward complete on G, F (x) satises the basic conditions on G, and! : G! R0 is continuous on G. For each triple (T; "; C) where T > 0, " > 0 and C G compact there exists > 0 such that every maximal solution (t; x ) of _x 2 F (x) := cof (x + B) + B ; (58) with x 2 C+B, remains in G for all t 2 [0; T ] and there exists a solution (t; x) of _x 2 F (x) with x 2 C and jx? x j " such that, for all t 2 [0; T ], j!( (t; x ))?!((t; x))j " : (59) Remark 3 A useful remark is that if Lemma 6 holds for > 0 then it holds for all 2 (0; ] since, in this case, C + B C + B and F (x) F (x). Lemmas 4 and 6 can be used to show:

Converse Lyapunov Functions for Inclusions 19 Lemma 7 Suppose F (x) satises the basic conditions on G and the dierential inclusion _x 2 F (x) (60) is forward complete on G. Then there exists a continuous function : G! (0; 1) such that the dierential inclusion fxg + (x)b G ; (61) _x 2 F (x) (x) := cof (x + (x)b) + (x)b (62) is forward complete on G and F (x) (x) satises the basic conditions on G. Proof. In the following, let R 1 (C), respectively R (x) 1 (C), denote the reachable set in time t = 1 from the compact set C G for the dierential inclusion (60), respectively the dierential inclusion (62). According to Lemma 4, the forward completeness assumption on (60) implies that the reachable set in each nite time from each compact subset of G for the system (60) is a compact subset of G. So, we can nd a compact, countable covering C i, i = 1; 2; : : : ; of G such that, for each i, there exists " i > 0 satisfying C i R 1 (C i ) + " i B C i+1 G : (63) By applying Lemma 6 with the triple (1; " i ; C i ) and!(x) = x we get, for each i, the existence of i 2 (0; " i ] (64) such that R i 1 (C i) C i+1 : (65) Without loss of generality, from Remark 3, we can assume that the sequence f i g 1 i=1 is nonincreasing. Dene i(x) := inf i (x) := i(x) fx 2 C i g (x) := inf [() + jx? j] : 2G As in the proof of [7, Theorem 1.5.1], one can check that the function is well-dened and is Lipschitz on G with Lipschitz constant 1. We have also, for all x 2 G, (66) (x) i(x) ; (x) > 0 (67) and, from (63) and (64), fxg + (x)b G : (68)

Converse Lyapunov Functions for Inclusions 20 We will show that R (x) 1 (C i) C i+1 8i : (69) From the denition of i(x) and the fact that the sequence f i g 1 i=1 is nonincreasing, it follows that x 2 @C i [ (GnCi ) =) (x) i : (70) To establish a contradiction, suppose the existence of an integer j and 2 R (x) (C 1 j) such that =2 C j+1, i.e., there exists a solution of the dierential inclusion (62) starting from a point x 2 C j that reaches, at time t 1, the point =2 C j+1. Then, exploiting time-invariance, continuity of this solution with respect to time S and compactness of C j, there exists t 2 [0; t) such that (t ; x) 2 @C j, and (t; x) 2 @C j (GnCj ) for all t 2 [t ; t]. It follows from (70) that ((t; x)) j for all t 2 [t ; t]. Therefore, for t 2 [t ; t], (t; x) is a solution of the dierential inclusion (62) with (x) = j. But then, using (65) and the fact that t? t 1 and (t ; x) 2 C j, it is impossible to have (t; x) =2 C j+1. This contradiction establishes that (69) holds. Now, suppose that (62) is not forward complete. Then there exist an integer j, a point x 2 C j, a solution of (62) starting at x and t < 1 such that for each integer m > j there exists t m < t such that (t m ; x) =2 C m. On the other hand, (69) implies R (x) m?j (C j) C m (71) which implies that m? j < t m < t for all m > j. This is impossible. The nal result of this subsection provides a link to the next subsection where solutions of locally Lipschitz dierential inclusions are considered. The result is slight modication of [6, Proposition 3.5]. Lemma 8 Let A be such that GnA is open and : G! R0 be bounded away from zero on compact subsets of GnA and such that fxg + (x)b G. If F (x) satises the basic conditions on G then there exists an dierential inclusion _x 2 F L (x) satisfying the basic conditions on G, locally Lipschitz on GnA and such that F (x) F L (x) F (x) (x) := cof (x + (x)b) + (x)b : (72) 4.3 Solutions to locally Lipschitz dierential inclusions We start with the following fact (see [7, Exercise 4.3.3.a]): Lemma 9 Let O be an open subset of G. If the set-valued map F is locally Lipschitz on O, then for any compact set K O there exists a positive real number L K such that for any x 1 and x 2 in K, we have F (x 1 ) F (x 2 ) + L K jx 1? x 2 jb : (73) The next result, on solutions to locally Lipschitz dierential inclusions, is similar to [10, Lemma 8.3], [7, Lemma 4.3.11] and [3, Theorem 10.4.1]. (See also [6, Proof of Lemma 4.9]). We provide a proof for completeness.

Converse Lyapunov Functions for Inclusions 21 Lemma 10 Let F (x) satisfy the basic conditions on G and be locally Lipschitz on the open set O G. For each T > 0 and each compact set C O, there exist L and > 0 such that, for each x 2 C, each 2 S(x) and each satisfying jx? j, there exists 2 S() with the property j(t; x)? (t; )j Ljx? j 8t 2 [0; T x ] ; (74) where T x 2 [0; T ] is such that (t; x) 2 C for all t 2 [0; T x ]. Proof. Preparation step : For each pair (z; v) 2 G R n, dene g(z; v) to be the unique (since F (z) is closed and convex; see [12, x5, Lemma 2]) closest point in F (z) to v. Since F (z) is locally Lipschitz, closed and convex it follows from [12, x6, Lemma 8] that g(z; v) is continuous in z for each xed v. Since F (z) is closed and convex, g(z; v) is continuous in v for each xed z. From [12, x5, Lemma 15], for each compact subset X of G there exists a constant m such that jg(z; v)j m for all (z; v) 2 X R n. Below, we will pick x in G and in S(x) dened on [0; T x1 ) and will dene w(t) := z _ { (t; x) 2 F ((t; x)) for almost all t 2 [0; T x1 ). (The function w(t) can be dened arbitrarily z _ { for those t 2 [0; T x1 ) where (t; x) is not dened.) Since (t; x) is absolutely continuous, w(t) is measurable. Then we will dene g x (z; t) = g(z; w(t)). Since w(t) is measurable and g has the properties given above, g x (z; t), dened on G [0; T x1 ), satises the Caratheodory conditions for existence of solutions for ordinary dierential equations. Core of the proof : Let C O and T > 0 be given. Since C is a compact subset of O which is open, there exists " > 0 so that C + "B is a compact subset of O. Using Lemma 9, let K be a Lipschitz constant for F on C + "B. We choose L = exp(kt ) ; = " 2L : (75) Let x, 2 S(x), T x and T x1 be such that T x 2 (0; T ], (t; x) 2 C for all t 2 [0; T x ] and (t; x) is right maximally dened on [0; T x1 ). Necessarily T x1 > T x. (If T x = 0 then, since L 1, there is nothing to prove.) Let g x be as above, dened on G [0; T x1 ). Pick satisfying jx? j. Then is an interior point of C + "B. Let (t; ) be a solution with values in G of z { _ (t; ) = g x ( (t; ); t) ; (0; ) = (76) right maximally dened on [0; T ). We have T T x1 and from the denition of g x, (t; ) is a solution on [0; T ) of the dierential inclusion _x 2 F (x). Now, either T = T x1 or there exists t 2 [0; T ) such that (t ; ) =2 C + "B. We dene t := sup t 2 [0; T x ] : (s; ) 2 C + "B ; 8s 2 [0; t] : (77) We must have that t < T since in the case where T = T x1 we have t T x < T x1 = T and in the case where T < T x1 we have t t < T. Thus (t; ) is well-dened on [0; t] and,

Converse Lyapunov Functions for Inclusions 22 by the continuity of (; ) and since is an interior point of C + "B and T x > 0, we have t > 0 and t < T x =) (t; ) 2 @(C + "B) : (78) Then, from the denition of g x we have, for almost all t 2 [0; t], d dt j(t; x)? (t; )j jw(t)? g x( (t; ); t)j (79) = jw(t)? g( (t; ); w(t))j (80) Then, since we have, from the Lipschitz property : = inf jw(t)? vj : (81) v2f ( (t;)) w(t) 2 F ((t; x)) F ( (t; )) + K j(t; x)? (t; )j B ; (82) we conclude : d j(t; x)? (t; )j Kj(t; x)? (t; )j : (83) dt Invoking a comparison theorem, we get, for all t 2 [0; t], j(t; x)? (t; )j exp(kt )jx? j = Ljx? j L = " 2 : (84) Now if t = T x we are done. Suppose t < T x. Since we have that (t; x) 2 C for all t 2 [0; T x ], (84) implies that (t; ) is in the interior of C +"B, for all t 2 [0; t]. But the latter contradicts (78). So we must have t = T x. Lemma 11 Let F satisfy the basic conditions on G and let F be locally Lipschitz on a neighborhood of x 2 G. Then for each v 2 F (x) there exists a solution 2 S(x) satisfying (t; x) = x + t(v + r(t)) 8t 2 [0; T ) (85) for some T > 0 and for some function r(t) that is continuous on [0; T ) and satises lim t!0 + r(t) = 0. Proof. (See also [6, proof of Lemma 4.8]). As in the previous proof, for each in a neighborhood of x, let g() 2 F () be the unique closest point in the compact convex set F () to the vector v. Again, the function g is well-dened and continuous on a neighborhood of x since F is locally Lipschitz on a neighborhood of x. Let (t; x) be a solution to the dierential equation _ = g() (86) starting at x dened on [0; T ). Since g() 2 F (), is also a solution of _x 2 F (x). Since g(x) = v, the result follows.

Converse Lyapunov Functions for Inclusions 23 4.4 Derivatives of locally Lipschitz functions First we recall the denition of the Dini subderivate of a function V : O! R (O open), at a point x 2 O in the direction v 2 R n : DV (x; v) := From [7, Exercise 3.4.1]), we have : V (x + "w)? V (x) lim inf w!v;"!0 + " : (87) Lemma 12 If V is a locally Lipschitz function on an open set O of R n, then we have DV (x; v) = hrv (x); vi (88) for all x 2 O such that the gradient of V, denoted rv (x), exists. The set of points where the gradient exists is characterized by Rademacher's Theorem (see [28, Denition VIII.3.2, Corollary VIII.3.1] and [7, p.147]) : Lemma 13 If V is a locally Lipschitz function on an open set O of R n, it has a gradient rv at almost all points x 2 O. Finally, like in [6], to establish the Lipschitz property we use the following result from [8, Corollary 3.7] : Lemma 14 Let the function V : O! (?1; 1] be lower semicontinuous. Let U O be open and convex. The function V is Lipschitz with Lipschitz constant M on U if and only if DV (x; v) Mjvj 8x 2 U ; 8v 2 R n : (89) 4.5 Smoothing continuous and locally Lipschitz functions A standard approximation result is the following: Lemma 15 Let O R n be open and let : O! (0; 1) be continuous. Suppose V : O! R is continuous. Then there exists a smooth function V s : O! R such that, for all x 2 O, jv (x)? V s (x)j (x) : (90) The next result is similar to [6, Lemma 5.1] which is based on similar results in [15], [38] and [18, Theorem B.1]. Lemma 16 Let O R n be open and let the three functions : O! R and ; : O! (0; 1) be continuous. Suppose V : O! R is locally Lipschitz on O, and the set-valued map F satises the basic conditions on O and is locally Lipschitz on O, and for almost all x 2 O, max hrv (x); wi (x) : (91) w2f (x)