On the representation of switched systems with inputs by perturbed control systems

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1 Nonlinear Analysis 60 (2005) On the representation of switched systems with inputs by perturbed control systems J.L. Mancilla-Aguilar a, R. García b, E. Sontag c,,1, Y. Wang d,2 a Department of Mathematics, Faculty of Engineering, University of Buenos Aires, Argentina b Department of Physics & Mathematics of the Instituto Tecnológico de Buenos Aires; and the Faculty of Engineering of the University of Buenos Aires, Argentina c Department of Mathematics, Rutgers University, New Brunswick, NJ, USA d Department of Mathematical Sciences, Florida Atlantic University, FL, USA Received 19 July 2004; accepted 11 October 2004 Abstract This paper provides representations of switched systems described by controlled differential inclusions, in terms of perturbed control systems. The control systems have dynamics given by differential equations, and their inputs consist of the original controls together with disturbances that evolve in compact sets; their sets of maximal trajectories contain, as a dense subset, the set of maximal trajectories of the original system. Several applications to control theory, dealing with properties of stability with respect to inputs and of detectability, are derived as a consequence of the representation theorem Elsevier Ltd. All rights reserved. Keywords: Switched systems; Differential inclusions; Relaxation theorems; Input-to-state stability; Lyapunov method Corresponding author. Department of Mathematics, Rutgers, The State University of New Jersey, Hill Center, 110 Frelinghuysen Rd, Piscataway, NJ , USA. Tel.: ; fax: addresses: jmancil@fi.uba.ar (J.L. Mancilla-Aguilar), ragarcia@itba.edu.ar (R. García), sontag@math.rutgers.edu (E. Sontag), ywang@math.fau.edu (Y. Wang). 1 Work partially supported by US Air Force Grant F and NSF Grant CCR Work partially supported by NSF Grant DMS and Chinese National Natural Science Foundation Grant X/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi: /j.na

2 1112 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) Introduction In the last decade, the study of the properties of switched systems described by ẋ(t) = f σ(t) (x(t), u(t)), y(t) = h(x(t)), (1) with σ :[0, + ) Γ an arbitrary switching signal and Γ an index set, has received a great deal of attention, mainly motivated by the rapidly development of the area of intelligent control (see [11] and references therein for details). In fact, switched systems (1) enable us, for example, to model the continuous portion of a hybrid system (see [4,14]). Different stability properties for system (1) were studied and characterized in terms of Lyapunov functions (see [11,12,16,17]). In [16] in particular, it was proved that, under suitable hypotheses, the set of maximal trajectories of system (1) is dense in the set of maximal trajectories of an associated (non-switched) system with controls and perturbations. This amounts to a representation of the switched system by a system with controls and perturbations. By using this fact, different results about perturbed control systems described by differential equations could be extended to switched systems (1) ([8,16,17]). On the other hand, although under mild regularity conditions, the differential equation (1) provides, for each initial condition and each switching signal, a complete description of the time evolution of the state x( ), a more robust model of behavior should take into account uncertainties caused by modeling errors and disturbances that are inevitable in any real-world control problem. This leads one to consider a more general model: switched systems described by forced differential inclusions ẋ(t) F σ(t) (x(t), u(t)), y(t) = h(x(t)). (2) In this work we obtain representations of switched systems (2) (assuming locally Lipschitz right-hand sides) by means of perturbed control systems described by ordinary differential equations, driven by inputs consisting of the controls of the original system and perturbations that evolve in compact sets. The representations obtained are characterized by the facts that every maximal trajectory of a system (2) is also a maximal trajectory of the representing system, and that the set of trajectories of (2) is dense (see Remark 2.4 for the precise meaning) in the set of trajectories of the representing system. The latter statement is closely related to the relaxation theorems of differential inclusions which assert that, under suitable conditions, relaxed trajectories can be approximated by (regular) trajectories (see [7,9]). These approximation results allow one to convert the analysis of an inclusion to its relaxation or vice versa. An interesting application can be found in the recent work [15]. Our representation results in the current paper allow us to achieve two theoretical and conceptual simplifications: (1) all uncertainty about the system can be summarized into just one disturbance input, and (2) switching, which is in principle very hard to study (because spaces of switching signals do not have any completeness properties), can be understood in terms of arbitrary Lebesgue-measurable disturbances with values on a compact space. As immediate applications of our results on representations, we extend previous results on Lyapunov characterizations of input/output stability and detectability properties for systems

3 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) of differential equations to switched systems defined by differential inclusions. (Further corollaries will be explored in other papers; for instance, it is shown in [18] that Lyapunov characterizations can also be developed for systems whose switching functions are governed by digraphs.) We wish to emphasize that many of the results that we present are new even in the very special cases of un-controlled differential inclusions (and even ordinary differential equations) with switching, or of differential inclusions without switching. The paper is organized as follows. In Section 2 we present the basic notation, the class of switched systems that we address and the main results. Section 3 presents results on forced differential inclusions that are instrumental for the proof of the main result. In Section 4, a result on parametrization of set valued maps is presented and the main results are proved. In Section 5, we develop Lyapunov characterizations of input output-to-state stability and several input-to-output stability properties. Finally, in Section 6, conclusions are given. An appendix contains several additional results as well as some technical lemmas needed in the main text. 2. Switched systems We first introduce some notations and definitions that will be used in the sequel. We use to denote the Euclidean norm for any given R q and with B q we denote the closed unit ball in R q. Given a metric space E, we denote by M(E) the set of Lebesgue measurable functions η :[0, + ) E that are locally essentially bounded. In case E = R m we write U instead of M(R m ). We say that a sequence {η n,n N} M(E) is locally equibounded if for each compact interval J [0, + ) there exists a compact subset K E such that for all n N, η n (t) K for almost all t J. For a measurable function θ :[0, ) R l, we denote by θ the (possibly infinite) L l -norm of θ and, for any t 0, θ [0,t] stands for the L l -norm of θ restricted to the interval [0,t]. Let X be a metric space. We denote the distance from a point ξ X to a set A X by dist (ξ,a). The Hausdorff distance between two nonempty closed subsets of X, A and B,is defined as d H (A, B) := max{sup ξ B dist (ξ, A), sup η A dist (η,b)}. We let K(X) be the set of nonempty compact subsets of X and we recall that the Hausdorff distance d H is a metric on K(X) and that K(X) is compact in the metric d H if X is compact (see [19, p. 279]). For a normed space X, we still use to denote the norm on X. For a subset A of X,coA and cl A stand for the convex hull and the closure of A respectively. We define A :=sup{ a :a A}. Given another metric space Z, we say that a set-valued map G : Z K(X) is continuous (Lipschitz) if it is continuous (Lipschitz) when the Hausdorff distance is considered in K(X). WeuseC(Z, K(X)) to denote the class of continuous maps from Z to K(X) equipped with the topology of uniform convergence on compact sets. It must be remarked that this topological space is metrizable when Z is a finite dimensional vector space. In order to see it, consider any norm on Z and let C r denote the closed ball centered at 0 with radius r in Z. Then C r is compact. Let d r be the metric for C(C r, K(X)) given by

4 1114 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) d r (f, g) = sup x Cr d H (f (x), g(x)). Then the metric d on C(Z, K(X)) defined by d(f,g) = j=1 d j (f, g) 1 + d j (f, g) 2 j, induces the topology of uniform convergence on compact sets. As usual, by a K-function we mean a function α : R 0 R 0 that is strictly increasing and continuous, and satisfies α(0)=0, by a K -function one that is in addition unbounded, and we let KL be the class of functions R 0 R 0 R 0 which are of class K on the first argument and decrease to zero on the second argument. Before we introduce the class of switched systems with which we deal in this work, it is convenient to describe the class of functions that we will take as switching signals: Definition 2.1. Given a nonempty set C, we say that a function g :[0, + ) C is a switching signal if g verifies one of the two following conditions: 1. g is a piecewise constant function (i.e. the set of points where the function g has jumps is finite in each compact subinterval of [0, + ), and g is constant between jumps) continuous from the right; 2. there exist a sequence of real numbers {t k : k N 0 } with 0 = t 0 <t 1 < <t k < and lim k + t k = T g < +, a sequence {c k C : k N 0 } with c k = c k+1 for all k 0 and a point c C such that g(t) = c k for all t k t<t k+1 and g(t) = c for all t T g. In what follows, we will denote by S(C) the family of all C-valued switching signals and with S pc (C) the subfamily of all C-valued piecewise constant switching signals. Remark 2.2. The definition of switching signal that we use here is slightly more general than that usually considered in the literature on switched systems, where a switching signal means an element in S pc (C). Switching signals that do not belong to S pc (C) are usually related with the so-called Zeno-behavior of a hybrid system (see [26]) and due to this reason they will be here referred to as Zeno-switching signals. As pointed out above, in this work we consider switched systems whose subsystems are described by forced differential inclusions. More precisely, given a family of locally Lipschitz set-valued maps P ={F γ C(R n R m, K(R n )) : γ Γ}, (3) where Γ is an index set and, without loss of generality, F γ = F γ if γ = γ, we consider the switched system with inputs ẋ F σ (x, u), where x takes values in R n, u U, and σ S(Γ). Given an input u U and a switching signal σ S(Γ), we say that a locally absolutely continuous function x : I R n where I =[0,T] or [0,T) with 0 <T + is a (4)

5 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) trajectory of (4) corresponding to u U and to σ S(Γ) if ẋ(t) F σ(t) (x(t), u(t)) for almost all t I. Observe that, due to the assumptions about F γ, for each ξ R n, each u U and each σ S(Γ) there always exists a trajectory x corresponding to u and to σ that verifies x(0) = ξ and that is defined on an interval [0,T)for some small T>0. A trajectory x :[0,T) R n corresponding to u U and to σ S(Γ) is called maximal if it does not have an extension which is a solution corresponding to u and to σ, i.e., either T =+ or there does not exist a trajectory z :[0,T ) R n corresponding to u and to σ with T >T so that z(t) = x(t) for all t [0,T). Given a maximal trajectory x corresponding to u U and to σ S(Γ), we denote its domain by [0,T x ). We write just T x for simplicity, even though it is T x,u,σ. For any ξ R n,anyu U and any σ S(Γ), we denote by T s (ξ,u,σ) the collection of all the maximal trajectories x of (4) corresponding to u and to σ that satisfy x(0) = ξ Main results In what follows we will establish one of the main results of this work, which asserts that under suitable hypotheses on the family P the set of maximal trajectories of system (4) is dense, in a sense that we will make precise, in the set of maximal trajectories of a system with inputs described by a system of differential equations: ż = f(z,u,d,ω), (5) where z takes values in R n,u U,d M(D) with D a compact metric space, and ω M(B n ). In order to establish the precise result, we consider for the family P the following hypotheses: C1: The family P is uniformly locally Lipschitz, i.e., for each N N, there exists l N 0 such that d H (F γ (ξ, μ), F γ (ξ, μ )) l N ( ξ ξ + μ μ ), for all ξ, ξ NB n and all μ, μ NB m (where we have used rb q to denote the closed ball centered at 0 of radius r in R q ). C2: The family P is pointwise equibounded, i.e., for each (ξ, μ) R n R m there exists M (ξ,μ) 0 such that F γ (ξ, μ) M (ξ,μ) for all γ Γ. One of the main results of the paper is the following: Theorem 1. Suppose that P verifies C1 and C2. Then there exist a compact metric space D, an injective function ι : Γ D, anda continuous function f : R n R m D B n R n such that 1. The set ι(γ) is dense in D.

6 1116 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) The map f(,, ν, ) is locally Lipschitz uniformly on ν D, i.e., for each compact subset K of R n R m there is some constant c K so that f(ξ, μ, ν, ζ) f(ξ, μ, ν, ζ ) c K ( ξ ξ + μ μ + ζ ζ ), for all (ξ, μ), (ξ, μ ) K, all ζ, ζ B n andall ν D. 3. For each ξ R n, μ R m and γ Γ, co F γ (ξ, μ) ={f(ξ, μ, ι(γ), ζ) : ζ B n }. 4. Given u U, σ S(Γ) anda maximal trajectory x of (4) corresponding to u and σ, there exists ω M(B n ) such that x is a maximal trajectory of (5) corresponding to u, d σ = ι σ and ω. 5. The trajectories of (4) are dense in the set of trajectories of (5) in the two following senses: (a) Given ε > 0 anda trajectory z :[0,T] R n of (5) corresponding to u U, d M(D) and ω M(B n ), there exist σ S pc (Γ) anda trajectory x of (4) corresponding to u and σ with x(0) = z(0) such that z(t) x(t) < ε, t [0,T]. (b) Given a trajectory z :[0,T) R n of (5) with T corresponding to u U, d M(D) and ω M(B n ) anda continuous function r :[0,T) R >0, there exist σ S(Γ) anda trajectory x of (4) corresponding to u and σ such that z(t) x(t) <r(t), t [0,T). In the case when T =, the switching function σ can be chosen in S pc (Γ). Remark 2.3. Parts 1 3 of Theorem 1 contain as a particular case a result obtained in [17] for switched systems described by differential equations (namely, Theorem 3.2 of [17]). In fact, if a family of functions {f γ C(R n R m, R n ) : γ Γ} verifies the hypotheses of Theorem 3.2 of [17], then P={F γ C(R n R m, K(R n )) : γ Γ} with F γ (, )={f γ (, )} verifies C1 and C2. Consequently, there exist a compact metric space D, an injective function ι : Γ D and a continuous function f : R n R m D B n R n that verify 1 3 of Theorem 1. From statement 3, it easily follows that f(ξ, μ, ν, ζ) = f(ξ, μ, ν, ζ ) for all ζ, ζ B n, that is, the function f(ξ, μ, ν, ) is constant for any given ξ, μ and ν. Hence, f can be considered as a function from R n R m D to R n, and we recover Theorem 3.2 of [17]. Remark 2.4. Part 5(a) of Theorem 1 asserts that for fixed T>0, u U and ξ R n, the set of trajectories of (4) corresponding to u and the initial state ξ with σ S pc (Γ) that are defined on [0,T] is dense in the set of trajectories of (5) corresponding to u and the initial state ξ with d M(D) and ω M(B n ) that are defined on [0,T], when the topology in consideration is the topology of uniform convergence on the interval [0,T]. On the other hand, part 5(b) states that for fixed 0 <T<+ (T = respectively) and u U, the set of trajectories of (4) corresponding to u with σ S(Γ) (σ S pc (Γ)

7 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) respectively) that are defined on [0,T)is dense in the set of trajectories of (5) corresponding to u with d M(D) and ω M(B n ) that are defined on [0,T), when the topology under consideration is the C 0 Whitney topology. We point out that part 5(b) does not hold if we consider only piecewise constant switching signals (see Section 3.1). That is the reason why we must also consider Zeno-switching signals. As a corollary of Theorem 1, we will also obtain the following representation theorem: Theorem 2. Suppose that P verifies C1 and C2. Then there exists a locally Lipschitz function f : R n R m B n R n such that 1. For each ξ R n and μ R m, co cl F γ (ξ, μ) ={f (ξ, μ, ζ) : ζ B n }. γ Γ 2. Given u U, σ S(Γ) anda maximal trajectory x of (4) corresponding to u and σ, there exists ω M(B n ) such that x is a maximal trajectory of ẋ = f (x, u, ω). (6) 3. The trajectories of (4) are dense in the set of trajectories of (6) in the two following senses: (a) Given ε > 0 anda trajectory z :[0,T] R n of (6) corresponding to u U and ω M(B n ) there exist σ S pc (Γ) anda trajectory x of (4) corresponding to u and σ such that x(0) = z(0) and z(t) x(t) < ε, t [0,T]. (b) Given a trajectory z :[0,T) R n of (6), with T, corresponding to u U and ω M(B n ) anda continuous function r :[0,T) R >0, there exist σ S(Γ) anda trajectory x of (4) corresponding to u and σ such that z(t) x(t) <r(t), t [0,T). In the case when T =, the switching signal σ can be choosen in S pc (Γ). The two results complement each other: while Theorem 1 provides a representation of the given switched differential inclusion in terms of a switched system of differential equations (driven by the same switching signal, provided that we identify σ with ι σ), Theorem 2 is more abstract, and it provides a representation in terms of a non-switched system of differential equations. This latter form, which looses the information about the switching signal, is of use for theoretical purposes, since it allows reducing questions about switched differential inclusions into questions about ordinary differential equations; examples are given later in the paper.

8 1118 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) Next, we establish an association between the switched system (4) and a forced differential inclusion with two inputs. In order to do so, we will use the following result. Lemma 2.5. Suppose that P verifies C1 and C2. Then there exist a compact metric space D, an injective function ι : Γ D anda continuous set-valuedmap F : R n R m D K(R n ) such that 1. ι(γ) is dense in D. 2. F(,, ν) is locally Lipschitz uniformly with respect to ν D. 3. F(ξ, μ, ι(γ)) = F γ (ξ, μ) for all ξ R n, all μ R m andall γ Γ. Proof. Consider P as a subset of C(R n R m, K(R n )) equipped with the topology of the uniform convergence on compact sets. Condition C2 implies that for each (ξ, μ) R n R m there exists a compact set K (ξ,μ) R n such that F γ (ξ, μ) K (ξ,μ) for all γ Γ. Thus, for each pair (ξ, μ), F γ (ξ, μ) K(K (ξ,μ) ) for all γ Γ. Since K(K (ξ,μ) ) is compact, {F γ (ξ, μ) : γ Γ} has compact closure in K(R n ). Since condition C1 implies that the family P is also equicontinuous, it follows from the Arzelá Ascoli Theorem (see [19, p. 290]) that P, the closure of P in C(R n R m, K(R n )), is compact.as C(R n R m, K(R n )) is metrizable, we have that D := P is a compact metric space. Now we define, for each γ Γ, ι(γ) := F γ and, for (ξ, μ, ν) R n R m D, F(ξ, μ, ν) := ν(ξ, μ). Clearly statements 1 and 3 of the lemma are verified. As for the continuity of F, note that this function is the restriction to R n R m D of the evaluation map e v : R n R m C(R n R m, K(R n )) K(R n ), and that this last map is continuous (see [19, p. 287]). In order to prove assertion 2, it is sufficient to show that for each N N, F is Lipschitz on NB n NB m uniformly with respect to ν D. Let N N, ν D and (ξ, μ), (ξ, μ ) NB n NB m. Then there exists a sequence {ν k = ι(γ k ), γ k Γ,k N} such that ν k ν and, due to C1 and to the continuity of F, d H (F (ξ, μ, ν), F (ξ, μ, ν)) = lim k d H (F γk (ξ, μ), F γk (ξ, μ )) l N ( ξ ξ + μ μ ). It follows that F(,, ν) is Lipschitz on K = NB n NB m uniformly with respect to ν D with constant c K = l N. Now we associate with the switched system (4) the following system with two inputs: ẋ F (x, u, d), (7) where x takes values in R n, u U, d M(D) and where F and D are as in Lemma 2.5. Given ξ R n, u U and d M(D), we denote with T(ξ,u,d)the collection of all the maximal solutions x of (7), corresponding to the inputs u and d, that satisfy x(0) = ξ.

9 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) Remark 2.6. The following facts readily follow from Lemma 2.5: (i) For each σ S(Γ) (S pc (Γ) respectively), if d σ = ι σ, then d σ S(D) (S pc (D) respectively) and T s (ξ,u,σ) = T(ξ,u,d σ ). (ii) For each d S(ι(Γ)) (S pc (ι(γ)) respectively), there exists a unique switching signal σ d S(Γ) (S pc (Γ) respectively) such that ι σ d = d. In this manner, any switched system (4) corresponding to a family P that verifies C1 and C2 can be viewed as a system described by the forced differential inclusion (7) with two inputs: one of them is the input to the switched system, and the other one is a switching signal that takes values in a dense subset of a compact metric space D. 3. Some results about forced differential inclusions In what follows we consider the time-varying system with inputs described by ẋ G(t, x, d), where t 0, x takes values in R n and d M(D) with D a separable metric space. We also consider the relaxation of (8) ẋ co G(t, x, d). Throughout the rest of this section we will suppose that the set-valued map G : R 0 R n D K(R n ) verifies the following hypotheses: (H 1 )G(, ξ, ν) is measurable for each (ξ, ν) R n D; (H 2 ) G(t,, ) is continuous for each t 0; (H 3 ) for each compact subset K R n D there exists L K L 1 loc (R 0, R) such that for each (ξ, ν), (ξ, ν) K, d H (G(t, ξ, ν), G(t, ξ, ν)) L K (t) ξ ξ, t 0; (H 4 ) for each compact subset K R n D there exists α K L 1 loc (R 0, R) such that for each (ξ, ν) K G(t, ξ, ν) α K (t), t 0. (8) (9) Next, we will show that the set of trajectories of (8) generated by switching signals that take values in a dense subset D of D are dense in the whole set of trajectories of its relaxation (9) when either one of the two following topologies are considered: The topology of the uniform convergence on a compact interval [0,T] with T>0; the C 0 Whitney topology on a interval [0,T)with 0 <T +. Theorem 3. Suppose that the set-valuedmap G: R 0 R n D K(R n ), where D is a separable metric space, satisfies the hypotheses (H 1 ) (H 4 ). Let D be a dense subset of

10 1120 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) D. Then, given a maximal trajectory z :[0,T z ) R n of (9) corresponding to d M(D), the following hold: 1. Given 0 <T<T z and ε > 0, there exist a piecewise constant switching signal d S pc (D ) anda maximal trajectory x of (8) corresponding to d such that x(0) = z(0) and x(t) z(t) < ε, t [0,T]. (10) 2. Given 0 <T T z anda continuous function r :[0,T) R >0, there exist a switching signal d S(D ) (d S pc (D ) if T =+ ) anda maximal trajectory x of (8) corresponding to d such that x(t) z(t) <r(t), t [0,T). (11) Remark 3.1. Theorem 3 (whose proof is given in Sections ) remains valid, with the same proof, when the domain of the first variable of G is a finite interval I of the form [0,a] or [0,a)with R 0 replaced by I in hypotheses (H 1 ) (H 4 ). Remark 3.2. Parts 1 and 2 of Theorem 3 can be considered extensions of the Filippov Wa zewski Theorem and Theorem 1 of [9], respectively, since they are particular cases of Theorem 3. To see it, just consider the case when D in Theorem 3 is a singleton A result on forwardcompleteness of time-varying forceddifferential inclusions In this short section we present a result about the forward completeness of the timevarying differential inclusion (8), which is a simple consequence of Theorem 3 part 2. We also show that part 2 of that theorem does not hold if we consider only piecewise constant switching signals. Given a subclass of inputs A M(D), we will say that system (8) is forward complete with respect to A if every maximal trajectory of (8) which corresponds to an input d A and starts at time t 0 = 0, is defined for all t 0. In case A = M(D) we will say that the system is forward complete. The following result follows readily from Theorem 3: Theorem 4. Suppose that the set-valuedmap G : R 0 R n D K(R n ), with D a separable metric space, satisfies the hypotheses (H 1 ) (H 4 ). Let D be a dense subset of D. Then, the following statements are equivalent. 1. System (8) is forwardcomplete. 2. System (8) is forwardcomplete with respect to S(D ). The implication is trivial. To prove the implication 2. 1., we quote an existence result for differential inclusions (e.g. [6], Theorem 7 on p. 85). Lemma 3.3. Let Ω R R n be an open set containing (0,x 0 ), andlet G be a set-valued map so that G(t, x) is nonempty andclosedfor (t, x) Ω. Suppose that, for (t, x) Ω,

11 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) G(t, x) m(t) for some m( ) L 1 loc (R 0, R), andthat G is continuous in x, measurable in t. Then there exist T>0 anda solution x( ) of ẋ G(t, x), x(0) = x 0 defined on the interval [0,T]. Proof of Theorem If system (8) is not forward complete, there exist an input d M(D) and a trajectory z( ) corresponding to d defined on a maximal interval [0,T) with T<. On the other hand, consider r :[0,T) R > 0 defined by r(t) = (t T) 2. By Theorem 3 part 2, there exist an input d S(D ) and a maximal trajectory x( ) of (8) defined on [0, ) corresponding to d such that z(t) x(t) <r(t) for all t [0,T). Consequently, lim t T z(t) = x(t ). By Lemma 3.3, there exists some δ > 0 such that z(t) can be extended to [0,T + δ], which contradicts the maximality of T. Remark 3.4. From the proof of Theorem 4 it is clear that the statements of Theorem 4 still hold if we replace S(D ) by any subclass of inputs A M(D) for which part 2 of Theorem 3 holds. Now, we show that part 2 of Theorem 3 does not hold if we consider only inputs that belong to S pc (D ) instead of the whole S(D ). Due to Remark 3.4, a system that is forward complete with respect to S pc (D) but not forward complete serves as a counterexample. Example 3.5. Consider the system with (d 1,d 2 ) as inputs ẋ = d 1 f 1 (x) + d 2 f 2 (x), (12) where f i : R 2 R 2 (i = 1, 2) is given by f 1 (ξ) = (1 + ξ 2 )A 1 ξ,f 2 (ξ) = (1 + ξ 2 )A 2 ξ, with [ ] [ ] A 1 =, A = 2 0 and where d = (d 1,d 2 ) M(D) with D = D ={(1, 0), (0, 1)}. Observe that G(t, ξ, ν) := {ν 1 f 1 (ξ) + ν 2 f 2 (ξ)} verifies hypotheses (H 1 ) (H 4 ). Since the vector fields f 1 and f 2 are both globally asymptotically stable and therefore forward complete, it follows that the system (12) is forward complete with respect to S pc (D). On the other hand, with the input d = (d 1,d 2 ) defined by the feedback rule d(t) = { (1, 0) if x1 (t)x 2 (t) 0, (0, 1) if x 1 (t)x 2 (t) < 0, (13) there are trajectories with finite escape time. To see this, observe that the system (12) has the same phase portrait as the linear system ż = d 1 A 1 z + d 2 A 2 z. (14)

12 1122 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) By explicitly solving the linear system (14), it can be seen that with the input give by (13), the trajectories of (14) are spirals running clockwise. Consider the trajectory starting at (0,c)for some c = 0. Let 0 = t 0 <t 1 <t 2 <t 3 < be the switching times for d(t) given by (13) (i.e., the time when the trajectory crosses the coordinate axes). Then, for t [0,t 1 ], the trajectory of (14) with the initial state z(0) = (0,c)is given by z 1 (t) = 2c b e at sin bt, z 2 (t) = ac b e at sin bt + ce at cos bt, where a = 0.05,b = 1 (1/400). From this it can be calculated that t 1 =[arctan(b/a)]/b π/2, z 1 (t 1 ) 1.5 c (and z 2 (t 1 ) = 0). Consequently, z(t 1 ) 1.5 z(t 0 ). By symmetry (or by the same calculation), it can be shown that for each k>1, t k t k 1 = t 1, z(t k ) 1.5 z(t k 1 ). Furthermore, with r 0 := min 0 t t1 e A1t 1 (=min t1 t t 2 e A 2(t t 1 ) 1 ), one has z(t) r 0 z(t k 1 ) for all t [t k 1,t k ]. We now consider the trajectory x(t) of (12) starting at (0, 1). By the uniqueness property of the trajectories, one has x(t) = z(φ(t)), where φ( ) is the solution of φ = 1 + z(φ) 2, φ(0) = 0. Let 0 = τ 0 < τ 1 < τ 2 < be the crossing times of the trajectory x( ) with the coordinate axes. Then t k = φ(τ k ). Since for t [τ k 1, τ k ], φ(t) = (1 + z(φ(t)) 2 ) 1 + r 2 0 z(t k 1) 2 r (k 1), it follows that τ k τ k 1 t k t k (k 1) r0 2 = 1.5 2k ˆτ, where ˆτ = t 1 /r0 2. Let T k = τ 1 + τ 2 + +τ k. Then T k ˆT for some ˆT <. As x(τ k ) 1.5 k, it follows that limsup t ˆT x(t) =. This shows that the system (12) is not forward complete. Remark 3.6. Note that system (14) also serves as an example to show that a system that switches between two globally asymptotically stable linear systems may fail to be asymptotically stable for some choices of the switching signals in S pc (D) Proof of Theorem 3 The following lemma is needed in the proof of Theorem 3. Lemma 3.7. Let G be as in Theorem 3 and x :[0,T] R n be a solution of (8) corresponding to d M(D). Let {d k,k N} M(D) be a locally equibounded sequence such that lim k d k (t) = d(t) a.e. on [0, + ). Then, there exists a sequence {x k } k 1, with x k a maximal trajectory of (8) corresponding to d k that verifies x k (0) = x(0), such that x k is defined on [0,T] for k large enough and in addition {x k } converges uniformly to x on [0,T].

13 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) Proof. Let K R n be a compact set such that x(t) K for all t [0,T]. Pick a smooth function θ : R n R such that 0 θ(ξ) 1 for all ξ R n ; θ(ξ) = 1 for all ξ K + B n ; θ(ξ) = 0, for all ξ R n \(K + 2B n ) (where we have used K + rb n to denote the set {η R n : d(η,k) r}), and define the set-valued map G :[0,T] R n D K(R n ) by G (t, ξ, ν) := θ(ξ)g(t, ξ, ν). It easily follows that G verifies (a) G (, ξ, ν) is measurable for each ξ R n and each ν D; (b) G (t,, ) is continuous for each t [0,T]; (c) for each compact subset Δ D there exists β Δ L 1 ([0,T], R) such that for all ξ, ξ R n and all ν Δ, d H (G (t, ξ, ν), G (t, ξ, ν)) β Δ (t) ξ ξ, t [0,T]; (d) for each compact subset Δ D there exists α Δ L 1 ([0,T], R) such that for all ξ R n and all ν Δ, G (t, ξ, ν) α Δ (t), t [0,T]. (15) Then, if we consider for each k N, Ĝ k :[0,T] R n K(R n ) defined by Ĝ k (t, ξ) := G (t, ξ,d k (t)), we have that Ĝ k is measurable in t. In addition, from the local equiboundedness of {d k } and (c), it follows that for all ξ, ξ R n, d H (Ĝ k (t, ξ), Ĝ k (t, ξ )) κ(t) ξ ξ, t [0,T], for a suitable κ L 1 ([0,T], R). Let ρ k (t) = dist (ẋ(t),ĝ k (t, x(t))). We have, for almost all t [0,T], ρ k (t) = dist (ẋ(t),ĝ k (t, x(t))) dist (ẋ(t), G(t, x(t), d(t))) + d H (G(t, x(t), d(t)), Ĝ k (t, x(t))) = d H (G (t, x(t), d(t)), G (t, x(t), d k (t))). (16) Thus, due to the continuity of G with respect to its third argument, lim d H (G (t, x(t), d(t)), G (t, x(t), d k (t))) = 0, a.e. t [0,T]. k Consequently, lim k + ρ k (t) = 0 for almost all t [0,T]. On the other hand, from (16) it follows that for almost all t [0,T], ρ k (t) G (t, x(t), d(t)) + G (t, x(t), d k (t)). Taking into account the equiboundedness of {d k } and (15), we conclude that there exists α L 1 ([0,T], R) such that for each k N, ρ k (t) α(t) for almost all t [0,T].

14 1124 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) From Lemma 8.3 of [5] we have that for each k N there exists a solution x k of the initial value problem ẋ k Ĝ k (t, x k ), x k (0) = x(0) such that x(t) x k (t) φ k (t), t [0,T], where φ k is the solution of the initial value problem φ k (t) = κ(t)φ k (t) + ρ k (t), φ k (0) = 0. As ρ k (t) 0 almost everywhere and {ρ k } is majorized by an integrable function, φ k 0 uniformly on [0,T]. Consequently, {x k } converges uniformly to x on [0,T]. The proof concludes by noticing that x k is a solution of (8) corresponding to d k for k large enough. In fact, there exists N N such that for all k N, x k (t) x(t) 1 for all t [0,T]. Thus, if k N, for almost all t [0,T] we have that ẋ k (t) Ĝ k (t, x k (t)) = θ(x k (t))g(t, x k (t), d k (t)) = G(t, x k (t), d k (t)), and hence x k is a solution of (8) corresponding to d k Proof of part 1 of Theorem 3 From the Filippov Wa zewski Relaxation Theorem (cf. [5]) we can easily deduce the existence of a trajectory x of (8) corresponding to d so that x(0) = z(0) and z(t) x(t) < ε 2, t [0,T]. On the other hand, due to the density of D in D, there exists a locally equibounded sequence {d k } in S pc (D ) such that d k d almost everywhere on [0, + ). (First of all, since D is separable, by Remark C.1.2 of [21], there exists a locally equibounded sequence { dˆ k } in S pc (D) so that ˆd k d almost everywhere on [0, + ). Next, for each k N, we choose a piecewise constant switching signal d k S pc (D ) such that the distance from d k (t) to ˆd k (t) is less than 1/k for all t R 0. The equiboundedness of {d k,k N} can be proved by using arguments similar to those of Remark C.1.3 of [21]). By Lemma 3.7, there exists a sequence {x k }, with x k a maximal trajectory of (8) corresponding to d k that verifies x k (0)=x(0), such that x k is also defined on [0,T] for k large enough and in addition {x k } converges uniformly to x on [0,T]. In consequence, for some k we have that x(t) x k (t) < ε 2, t [0,T], and, a posteriori, that z(t) x k (t) < ε, t [0,T]. The proof of part 1 is completed by taking d = d k Proof of part 2 of Theorem 3 The proof that we give here is similar to the proofs of Lemma III 2 in [22] and its generalization in Theorem 1 in [9] and it uses similar techniques to those used in these papers.

15 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) Let {T k } k=0 be a strictly increasing sequence of times such that T 0 = 0 and lim k + T k = T. We claim that there exist a sequence {d k } k=1 S pc(d ), a sequence {δ k } k=0 of positive real numbers and, for each nonnegative integer k, a sequence of points {ξ k j } j=0 which satisfy the following: (i) For each k 0, 0 < δ k min{r(t) : t [T k,t k+1 ]}; (ii) for each k 0, ξ k j V k := z(t k ) + δ k B n for all j 0, where for ξ R n and δ > 0, we use ξ + δb n to denote the closed ball centered at ξ with radius δ; (iii) for any k 1, if a subsequence {ξ k j l } l=1 converges, say to ξk, then the subsequence {ξ k 1 j l } l=1 also converges, say to ξk 1, and there is a solution x k :[0,T k T k 1 ] R n of the initial value problem ẋ G(T k 1 + t,x,d k ), x(0) = ξ k 1, (17) that verifies x k (T k T k 1 ) = ξ k and x k (t) z(t k 1 + t) <r(t k 1 + t), t [0,T k T k 1 ]. (18) Proof of the Claim. Let {r k } k=1 be the sequence of positive numbers defined by r k = min{r(t) : t [T k 1,T k ]}. First, we will construct by induction a sequence of positive numbers {δ k } k=0, a sequence of sets {V k } k=0, with V k = z(t k ) + δ k B n, and a sequence of piecewise switching signals {d k } k=1 S pc(d ) such that the following hold for each k 1: δ k 1 r k ; there exists a continuous function x k :[0,T k T k 1 ] V k R n which satisfies the following: (a) for each η V k, x k (, η) is a solution of ẋ G k (t, x), (19) with G k : [0,T k T k 1 ] R n K(R n ) defined by G k (t, ξ) = G(T k t,ξ,d k (T k T k 1 t)), that verifies x k (0, η) = η, and z(t k t) x k (t, η) r k, t [0,T k T k 1 ]; (b) x k (T k T k 1,V k ) V k 1. We start the induction procedure by setting δ 0 = r 1 and V 0 = z(t 0 ) + δ 0 B n. Assuming that we have already constructed δ k 1 for some positive k, we will construct δ k, d k, and x k. First note that if we consider the forced differential inclusion ẋ Ĝ k (t,x,d), (20)

16 1126 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) with ˆ G k :[0,T k T k 1 ] R n D K(R n ) defined by Ĝ k (t, ξ, ν) = G(T k t,ξ, ν) and its relaxation ẋ co Ĝ k (t,x,d), (21) we have that z k :[0,T k T k 1 ] R n defined by z k (t) = z(t k t) is a solution of (21) corresponding to the input dk (t) = d (T k t). Note that Ĝ k verifies (H 1 ) (H 4 ) (with [0,T k T k 1 ] instead of R 0 ). Then, from the first part of Theorem 3 (which holds if we replace R 0 by [0,T k T k 1 ]), there exist a piecewise constant switching signal ˆd k S pc (D ) and a trajectory xk of (20) corresponding to ˆd k that verifies xk (0) = z k(0) = z(t k ) and xk (t) z k(t) < ε k 2, t [0,T k T k 1 ], where ε k := min{δ k 1,r k+1 }. Let d k S pc (D ) be the piecewise constant switching signal defined by d k (t) = ˆd k (T k T k 1 t) for t [0,T k T k 1 ] and d k (t) = ˆd k (0) for t T k T k 1. Then xk is a solution of (19) which corresponds to d k and verifies xk (0) = z(t k). By applying Lemma 3.1 of [9] to system (19) (with xk and ε k/2 instead of z and ε respectively), it follows the existence of a0< δ k < ε k and a continuous function x k :[0,T k T k 1 ] V k R n such that for each η V k, x k (, η) is a solution of (19) with x k (0, η) = η and xk (t) x k(t, η) ε k 2, t [0,T k T k 1 ]. In consequence, for every η V k and all t [0,T k T k 1 ], z(t k t) x k (t, η) = z k (t) x k (t, η) z k (t) xk (t) + x k (t) x k(t, η) < ε k 2 + ε k 2 = ε k r k+1. In particular, taking t = T k T k 1, we have that z(t k 1 ) x k (T k T k 1, η) < ε k δ k 1. In other words, {x k (T k T k 1, η) : η V k } V k 1. Next, we construct a sequence of points {ξ k j Rn : k 0,j 0} as follows. We set, for j 0 and k j, ξ k j = z(t k). For j 0 and 0 k<j, we obtain ξ k j by the recursive formula ξ k j = x k+1(t k+1 T k, ξ k+1 j ). By construction, each ξ k j V k.

17 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) We note that {δ k } k=0 and {ξk j : k 0,j 0} verify (i) and (ii). It remains to verify that this construction satisfies (iii). Suppose that for some k 1, {ξ k j l } l=1 converges to ξk. Taking into account that, by definition, ξ k 1 j = x k (T k T k 1, ξ k j ), k j and that x k is continuous, it follows that lim l ξk 1 j l = x k (T k T k 1, ξ k ) := ξ k 1. Now, consider x k :[0,T k T k 1 ] R n defined by x k (t) = x k (T k T k 1 t,ξ k ). Note that x k is a solution of the initial value problem (17) which verifies x k (T k T k 1 ) = ξ k and (18). The claim is thus proved. We are now ready to show the existence of a switching signal d S(D ) and a trajectory x of (8) corresponding to d that verify (11). Pick any c D and define d :[0, + ) D by d(t) = d k (t T k 1 ) for all t [T k 1,T k ) and d(t) = c for all t T. Clearly d S(D ) and, in the case when T =+, d S pc (Γ). Since {ξ k j } j=0 V k and V k is compact for each k 0, by using a diagonalization process, we deduce the existence of a subsequence {j l } l=1 of the sequence of the nonnegative integers such that {ξ k j l } converges, say to ξ k, for each k 0. Then, from statement (iii) in the previous claim, there exists a solution x k :[0,T k T k 1 ] R n of (17) that verifies x k (0) = ξ k 1, x k (T k T k 1 ) = ξ k and that satisfies (18). Therefore, the function x :[0,T) R n given by x(t) = x k (t T k 1 ) when t [T k 1,T k ), is a trajectory of (8) corresponding to d. In addition, it follows from (18) that x(t) z(t) <r(t), t [0,T). 4. Proofs of Theorems 1 and 2 In this section we give the proofs of the main results Theorems 1 and 2. For this purpose, we first study the existence of parametrizations for set-valued maps that take nonempty convex compact values Parametrizations of set-valuedmaps The following parametrization result for set-valued maps, whose proof is based on a construction developed in [20] for globally Lipschitz set-valued maps (see also [2]), asserts that under suitable conditions a set-valued map F admits a parametrization which is as regular as F.

18 1128 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) Theorem 5. Consider two metric spaces E, D anda continuous set-valuedmap F : E D K(R n ) which takes convex values. Assume that for each compact subset K D, F (, ν) is locally Lipschitz uniformly in ν K. Then there exists a function f : E D B n R n such that 1. f is continuous; 2. for each compact subset K D, f(, ν, ) is locally Lipschitz uniformly in ν K; 3. f(ξ, ν, B n ) = F (ξ, ν) e E, ν D. Proof. Let K c (R n ) denote the family of all nonempty compact convex subsets of R n.for K K c (R n ), let s n (K) be the Steiner point of K. It follows from Theorem of [2] that s n (K) K for all K K c (R n ) and that s n (K) s n (L) nd H (K, L), K,L K c (R n ). (22) Let P : R n K c (R n ) K c (R n ) be the map defined by P(y,K)= K (y + 2 dist (y, K)B n ). (It can be seen indeed that P(y,K) K c (R n ) for any y E and any K K c (R n ).) Observe that P(y,K)={y} if and only if y K. From Lemma of [2] (see also Lemma 1 of [20]), we have that P is a Lipschitz map, more precisely, d H (P (y, K), P (x, L)) 5(d H (K, L) + y x ), x,y R n, K,L K c (R n ). (23) Let f : E D B n R n be defined by f(ξ, ν,y)= s n (P ( F (ξ, ν) y,f (ξ, ν))). Proof of 3. First note that P( F (ξ, ν) y,f (ξ, ν)) F (ξ, ν) for all ξ E and ν D. Since s n (K) K for all K K c (R n ), it follows that f(ξ, ν,y) F (ξ, ν) for all ξ E, all ν D and all y B n. This implies that f(ξ, ν, B n ) F (ξ, ν) for all ξ E and all ν D. On the other hand, for each ζ F (ξ, ν) there exists y 0 B n such that ζ= F (ξ, ν) y 0. In consequence, f(ξ, ν,y 0 ) = s n (P ( F (ξ, ν) y 0,F (ξ, ν))) = s n (P (ζ,f (ξ, ν))) = s n ({ζ}) = ζ. Thus F (ξ, ν) f(ξ, ν, B n ) and statement 3 follows.

19 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) Proof of 1 and 2. From (22) and (23) we easily deduce that f(ξ, ν,y) f(ξ, ν,y ) = s n (P ( F (ξ, ν) y,f (ξ, ν))) s n (P ( F (ξ, ν ) y,f (ξ, ν ))) nd H (P ( F (ξ, ν) y,f (ξ, ν)), P ( F (ξ, ν ) y,f (ξ, ν ))) 5n[d H (F (ξ, ν), F (ξ, ν )) + F (ξ, ν) F (ξ, ν ) y + F (ξ, ν) y y ]. As F (ξ, ν) F (ξ, ν ) + d H (F (ξ, ν), F (ξ, ν ))B n, it follows that F (ξ, ν) F (ξ, ν ) d H (F (ξ, ν), F (ξ, ν )). Then, by symmetry, F (ξ, ν) F (ξ, ν ) d H (F (ξ, ν), F (ξ, ν )). In consequence, we have that for all ξ, ξ E, all ν, ν D and all y,y B n, f(ξ, ν,y) f(ξ, ν,y ) 5n[(1 + y )d H (F (ξ, ν), F (ξ, ν )) + F (ξ, ν) y y ]. (24) Now, from (24), it easily follows that f is continuous and that f(, ν, ) is locally Lipschitz uniformly with respect to ν K when K is a compact subset of D. Now, we are ready to prove Theorem 1 and afterwards Theorem Proof of Theorem 1 Consider a compact metric space D, a set-valued map F : R n R m D K(R n ) and an injective function ι : Γ D as in Lemma 2.5. As ι(γ) = D is dense in D, part1of Theorem 1 holds. Let F : R n R m D K(R n ) be defined by F (ξ, μ, ν) = co F(ξ, μ, ν). Since F is compact valued, F takes values in K c (R n ). Also observe that F has the same regularity as F, that is, F is continuous and locally Lipschitz in (ξ, μ) uniformly on ν D. According to Theorem 5 (applied with ξ = (ξ, μ)), there exists a continuous function f : R n R m D B n R n so that f(,, ν, ) is locally Lipschitz uniformly with respect to ν D and for all ξ R n, all μ R m and all ν D it holds that f(ξ, μ, ν, B n ) = F (ξ, μ, ν). (25) As for each ξ R n, μ R m and γ Γ, F γ (ξ, μ) = F(ξ, μ, ι(γ)), we have for all ξ R n, all μ R m and all γ Γ that co F γ (ξ, μ) = co F(ξ, μ, ι(γ)) = F (ξ, μ, ι(γ)) = f(ξ, μ, ι(γ), B n ). Parts 2 and 3 of the Theorem 1 are thus proven.

20 1130 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) Before we prove the remaining statements of the theorem, it is convenient to note that, from (25) and Filippov s Lemma, it easily follows that the set of maximal trajectories of the relaxation of (7) ẋ F (x,u,d) (26) coincides with the set of maximal trajectories of (5). More precisely, given u U and d M(D), we have that x is a maximal trajectory of (26) corresponding to u and d if and only if there exists ω M(B n ) such that x is a maximal trajectory of (5) corresponding to u, d and ω. In order to prove part 4, let x be a maximal trajectory of (4) corresponding to u U and σ S(Γ). Then, according to Remark 2.6, x is a maximal trajectory of (7) and therefore of (26), corresponding to u and d σ =ι σ and hence, as was said above, there exists ω M(B n ) such that x is a maximal trajectory of (5) corresponding to u, d σ and ω. As for part 5(a), let u U, d M(D), ω M(B n ) and let z :[0,T] R n be a trajectory of (5) corresponding to u, d and ω. Pick ε > 0 and consider the set valued map G : R 0 R n D K(R n ) defined by G(t, ξ, ν) = F(ξ, u(t), ν). Observe that G satisfies (H 1 ) (H 4 ). As z is a trajectory of (26) corresponding to u and d, z is a trajectory of (9) corresponding to d. From Theorem 3 part 1, there exist a piecewise constant switching signal d S pc (D ) and a maximal trajectory x of (8) corresponding to d such that x(0) = z(0) and z(t) x(t) < ε, t [0,T]. On the other hand, from the definition of G, we have that x is a maximal trajectory of (7) corresponding to u and d. The proof of this part concludes by noticing that, from Remark 2.6, we know that x is a maximal trajectory of (4) corresponding to u and a certain piecewise constant switching signal σ d S pc (Γ). The proof of part 5(b) is the same as the proof of part 5(a) if we replace the interval [0,T] and ε by the interval [0,T) and r(t) respectively, and use part 2 of Theorem 3 instead of part 1 of that theorem Proof of Theorem 2 Let D, ι : Γ D, and f : R n R m D B n R n be as in Theorem 1 and define ˆF : R n R m K(R n ) by ˆF(ξ, μ) = f(ξ, μ,d,b n ) ={f(ξ, μ, ν, ζ) : ν D,ζ B n }. We will consider the following differential inclusions with u as input: ẋ ˆF(x,u) (27) and ẋ co ˆF (x, u). (28)

21 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) Since f satisfies statement 2 of Theorem 1, it follows that ˆF, and hence the convex hull co ˆF of ˆF, are both locally Lipschitz. Therefore, from Theorem 5, there exists a locally Lipschitz function f : R n R m B n R n, so that for all ξ R n and all μ R m it holds that co ˆF(ξ, μ) = f (ξ, μ, B n ) ={f (ξ, μ, ζ) : ζ B n }. (29) We claim that f satisfies part 1 of Theorem 2, or equivalently, for all ξ R n and all μ R m, co cl F γ (ξ, μ) = co ˆF(ξ, μ). (30) γ Γ Let ξ R n and μ R m. By part 3 of Theorem 1, co F γ (ξ, μ) ={f(ξ, μ, ν, η) : ν D, η B n }, γ Γ where D = ι(γ) is dense in D. By the continuity of f and the compactness of D and B n, one has cl co F γ (ξ, μ) ={f(ξ, μ, ν, η) : ν D, η B n }= ˆF(ξ, μ). γ Γ Therefore, co cl co F γ (ξ, μ) = co ˆF(ξ, μ). γ Γ Taking into account that co cl co F γ (ξ, μ) = cl co co F γ (ξ, μ) = cl co F γ (ξ, μ) γ Γ γ Γ γ Γ = co cl F γ (ξ, μ), γ Γ we get (30). Part 1 of the theorem thus follows. Part 2 is a straightforward consequence of part 1 and Filippov s Lemma. Let x be a maximal trajectory of (4) corresponding to u U and to certain σ S(Γ). Then, from (30), x is a maximal trajectory of (28) corresponding to u. From this fact, (29) and Filippov s Lemma, there exists ω M(B n ) so that x is a maximal trajectory of (6). Finally we prove part 3. We note first that for a fixed u U and due to Filippov s Lemma, the maximal trajectories of (5) and those of (27) coincide, and that the same holds for the maximal trajectories of systems (6) and (28). That is, x is a maximal trajectory of (27) (resp. (28)) corresponding to u if and only if x is maximal trajectory of (5) (resp. (6)) corresponding

22 1132 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) to u and certain d M(D) and ω M(B n ) (resp. ω M(B n )). Combining this fact with parts 5(a) (b) of Theorem 1, it will be enough to prove the following statements: (i) Given a trajectory z :[0,T] R n of (28) corresponding to u U and ε > 0, there exists a trajectory x of (27) corresponding to u satisfying x(0) = z(0) such that z(t) x(t) < ε, t [0,T]; (ii) given a trajectory z :[0,T) R n of (28), with T +, corresponding to u U and a continuous function r :[0,T) R >0, there exists a trajectory x of (27) corresponding to u such that z(t) x(t) <r(t), t [0,T). Part (i) readily follows from the Filippov Wa zewski Theorem (or Theorem 3 part 1; see Remark 3.2). Part (ii) can be easily deduced from Theorem 1 of [9] (or part 2 of Theorem 3). 5. Stability properties of switched systems In this section, we consider several stability properties for switched systems with input and output as in the following: ẋ(t) F σ(t) (x(t), u(t)), y = h(x(t)), (31) where x,u, σ and F γ are still as in Section 2, the output map h : R n R p is locally Lipschitz, and h(0)=0. The results on parametrization given in Section 2 allow the extension of several important previous results for systems given by differential equations too switched systems defined by differential inclusions. (Many of these extensions are novel even for the very special cases of switched systems of differential equations, or for non-switched differential inclusions.) In this section, we treat the notion of input output-to-state stability and a few notions on input-to-output stability. Throughout this section, we assume that the collection P (see (3)) satisfies assumptions C1 and C Uniform input output-to-state stability The following definition is based on the work [10]: Definition 5.1. Given a subclass S of S(Γ), we say that the system (31) is uniformly input output-to-state stable with respect to S (uioss w.r.t. S ) if there exist β KL, θ K and γ K such that for all ξ R n, all u U, all σ S and all x T s (ξ,u,σ), x(t) β( ξ,t)+ θ( y [0,t] ) + γ( u ), t [0,T x ). (32) Observe that it results in the same definition if one replaces u in (32) by u [0,t].

23 J.L. Mancilla-Aguilar et al. / Nonlinear Analysis 60 (2005) When applying the uioss definition to systems with zero output map (i.e., h 0), one recovers the standard input-to-state stability notion, but extended to switched systems defined by differential inclusions: Definition 5.2. The system (31) is uniformly input-to-state stable with respect to S (uiss w.r.t. S ) if there exist β KL and γ K such that for all ξ R n, all u U, all σ S and all x T s (ξ,u,σ), x(t) β( ξ,t)+ γ( u ), t [0,T x ). (33) For system (31), let f : R n R m B n R n be as in Theorem 2, and consider the corresponding parametrization of (31): ẋ(t) = f (x(t), u(t), ω(t)), y(t) = h(x(t)). (34) For such a system, the uioss property means that an estimate as in (32) holds for any maximal trajectory starting at ξ with any u U and any ω M(B n ) over the maximal interval (c.f. [10]). Lemma 5.3. The system (31) is uioss w.r.t. S pc (Γ) if andonly if the corresponding system (34) is uioss. Proof. The statement that uioss property of (34) implies the uioss property of (31) w.r.t. S pc (Γ) follows immediately from statement 2 of Theorem 2. Suppose system (31) is uioss w.r.t. S pc (Γ) with the decay estimate (32). Let z : [0,T z ) R n be a solution of (34) with some u U and some ω M(B n ). Pick any ε > 0 and 0 <T<T z. Let r = z [0,T ], and let δ > 0 be such that h(ξ) h(ζ) < ε for all ξ (r + 1)B n and all ζ (r + 1)B n such that ξ ζ < δ. Without loss of generality, we assume that δ max{1, ε}. By 3(a) of Theorem 2, there exists some trajectory x( ) of (31) corresponding to u and some σ S pc (Γ) with x(0) = z(0) such that z(t) x(t) < δ, t [0,T]. (35) By the choice of δ, one has h(x(t)) h(z(t)) < ε, t [0,T]. (36) With the uioss estimate (32) for (31), we get x(t) β( x(0),t)+ θ( h(x) [0,t] ) + γ( u ), Since z(0) = x(0), it follows from (35) (36) that z(t) β( z(0),t)+ θ( h(z) [0,t] + ε) + γ( u ) + ε, Since T<T z and ε > 0 can be chosen arbitrarily, we get t [0,T]. z(t) β( z(0),t)+ θ( h(z) [0,t] ) + γ( u ), t [0,T z ). This shows that the system (34) is uioss. t [0,T].