control-lyapunov functions for time-varying set ... - Semantic Scholar

CONTROL-LYAPUNOV FUNCTIONS FOR TIME-VARYING SET STABILIZATION

Francesca ALBERTINIy , Dipartimento di Matematica, Universita di Padova, Via Belzoni 7, 35100, Italy, fax: +49-8758596, E-mail: [email protected], and Eduardo D. SONTAG Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. E-mail: [email protected]. Keywords. Nonlinear control, Stability, Control Lya- A classical technique for stabilization is to look for Lyapunov-type functions, which play the role of abstract punov function. \energy" or \cost" functions that can be made to decrease in directions corresponding to possible controls, as long as the state is away from A. For smooth such functions This paper shows that, for time varying systems, global V , and taking for simplicity the case when V and f are asymptotic controllability to a given closed subset of the time-invariant, and A = f0g, this amounts to the requirestate space is equivalent to the existence of a continuous ment that, for each nonzero state x, there must be some control value u = ux so that rV (x):f (x; u) < 0. One control-Lyapunov function with respect to the set. uses the generic term \control-Lyapunov function" (clf) for such a function. The clf approach, assuming a V has been found, allows the search for stabilizing inputs by iteratively solving a static nonlinear programming problem: We will study continuous-time systems with dynamics when at state x, nd u such that this inequality holds. Recent expositions of results about (smooth) clf's can be given by dierential equations of the type: found in the textbooks [5] and [6]. An obvious question is x_ (t) = f (t; x(t); u(t)); (1) whether the existence of a continuously dierentiable clf is equivalent to the possibility of driving every state asympwhere x(t) 2 Rn represents the state variable and u(t) 2 U totically to the set A (zero in this particular case).m The for instance, if controls are in R , and is the control variable. (Technical assumptions on f and answer is negative; f (x; u) = f0 + Pmi=1 uifi (x) is ane in u, the existence U are described below.) We are interested in questions of stabilization relative of a (smooth) V would imply that there is some feedback to a subset A of the state space Rn. For example, the set law u=k(x) so that the origin is a globally asymptotically system x_ = f (x; k(x)) and A may be just an equilibrium point, or it may represent stable state for the closed-loop a target subset of a dierent kind. This target set might k is continuous on Rn n f0g. This was proved by Artstein be a desired periodic orbit, or, in the context of designing in [2]. But continuous feedback may fail to exist, even for observers, the equations (1) might represent a composite very simple controllable systems (see e.g. [9], Section 4.8). state x = (x1; x2), consisting of the state x1 of the original Thus smooth clf's do not always exist, even if the system system together with the state x2 of an observer; in that is asymptotically controllable. case, A would be the set of states x for which identi cation However, it was shown in [8] that continuous clf's do has been achieved, that is, the set consisting of those x exist. Of course, one must modify the statement of the clf for which x1 = x2. (Note that in this last example, the condition, since the gradient is not well-de ned if V is not set A is not bounded.) There are several motivations for dierentiable. This modi cation can be done in various considering time-varying dynamics. For instance, in this ways. The rst possibility is to ask that for each state manner one may encompass problems of tracking, in which x 6= 0 (or rather, for each state not in A, in the general the dierence between some variables and a signal to be case) there exist a trajectory which is de ned on a small followed evolves according to a dierential equation which interval of time and which decreases the value of V . This depends explicitly in t. The purpose of this paper is to is the modi cation that was used in [8] (for time-invariant generalize the results of the paper [8], which dealt only systems and the special case A = f0g), and which we with A = f0g and time-invariant systems, to the general also employ in this paper. A second possibility, much more attractive, is to use generalized gradients of various model (1). types. We refer the reader to [10] and [4] for descriptions  This research was supported in part by US Air Force Grant of this second approach, including, in the second citation, F49620-95-1-0101 an application of these ideas to the construction of disy This research was done while visiting the Rutgers Center for continuous stabilizing feedback. The two approaches are Systems and Control (SYCON)

Abstract

1 Introduction

equivalent, but providing technical details is impossible in this short paper, due to space restrictions; thus, we limit ourselves to the rst possibility, and leave the equivalence for a fuller journal version. The proofs are in general based on the ideas in [8], but when dealing with time varying systems, and especially with possibly non-compact sets A, many technical complications arise. (Of course, one must allow now for timedependent V , and the de nition of asymptotic controllability must be in some sense uniform on time, in order to obtain a necessary and sucient result.) We also use some ideas from the paper [7], which dealt with set stability but only for systems with no controls (which allows constructing smooth V 's).

De nitions and Main Result

We take U to be a locally compact metric space, with a distinguished \0" element. We denote Ur = fu j d(u; 0)  rg. The map f is supposed to be measurable in t, and locally Lipschitz in (x; u) uniformly for t in a bounded interval. The set of control maps u 2 U are measurable essentially bounded functions u : [t0 ; 1) ! U , with t0 2 R. We denote by kuk the essential supremum of the map u. Notice that with these assumptions we guarantee the local existence and uniqueness of solutions of (1). For a given (t0 ; x0) 2 R  Rn, and a control map u : [t0; 1), we denote by x(t; t0; x0; u) the maximal solution of (1) with initial condition x(t0 ; t0; x0; u) = x0 . In general this solution will be de ned on an interval of the form [t0; t). A function : R0 ! R0 is called a K-function if it is continuous, strictly increasing, and (0) = 0; a K-map is called a K1-function if lims!1 (s) = 1. A function : R0  R0 ! R0 is said to be a KL-function if for each xed t  0 the map (
; t) is a K-function, and if for each xed s the map (s;
) is decreasing to zero as t ! 1. Given a non-empty closed set A  Rn, we denote by jxjA the distance from x to A, and we say that A is weakly invariant if, for all x0 2 A and all t0 2 R, there exists a control map u0 such that x(t; t0; x0; u0) is de ned and lies in A for all t  t0 . De nition 1.1 Let A  Rn be a closed, weakly invariant, and nonempty set. We say that (1) is globally asymptotically controllable (gac) to A if there exist a KLfunction (
;
), and a continuous, positive and increasing function (
) such that: for each (t0 ; x0) 2 R  Rn there exists a control function u : [t0; 1) ! U , with kuk  (jx0jA), such that the corresponding trajectory x(t) = x(t; t0; x0; u) exists for all t  t0 and satis es: jx(t)jA  (jx0 jA ; t ? t0 ) for all t  t0: (2) An alternative way to de ne gac to a set A is as follows. The proof, which we omit for reasons of space, consists of a routine argument involving comparison functions, and is patterned along the lines of the completely analogous result in [7], Proposition 2.5. Lemma 1.2 Let A  Rn be a closed, weakly invariant, and nonempty set. The system (1) is gac to A if and

only if there exists a continuous, positive and increasing function 1 such that: 1. there exists a K1 -map 2 such that for all  > 0, if jx0jA  2 () and t0 2 R, then there exists u : [t0; +1) ! U , with kuk  1 (jx0 jA ) such that jx(t; t0; x0; u)jA   for all t  t0; 2. for any r;  there exists T 2 R such that given (t0 ; x0) 2 R Rn, with jx0jA  r, then there exists u : [t0; 1) ! U with kuk  1 (jx0jA ) such that jx(t; t0; x0; u)jA   for all t  T + t0 We now recall some standard notions regarding relaxed controls. The set of relaxed controls W is the set of measurable functions w : [t0; 1) ! P (U ) where P (U ) is the set of probability measures on U equipped with the weak topology. We let Wr denote the set of those relaxed controls w such that w(t) has support on Ur for almost all t. An ordinary control u 2 U can be seen as a relaxed one if u(t) is identi ed with the Dirac measure concentrated in u(t). One de nes a topology on W charby weak convergence: wk ! w if and only if R R Racterized R T U g(t; u)dwk (u)dt ! T U g(t; u)dw(u)dt for all functions g : T  U ! R which are continuous in u, measurable in t, and such that maxfjg(t; u)j; u 2 U g is integrable on T , where T is a bounded interval. With this topology Ur is dense in Wr , and Wr is sequentially compact. We let kwk = inf fr j w(t) 2 P (Ur ) for a.e. tg (notice that for ordinary controls this is the essential supremum). Given w 2 P (Ur ), oneR extends f to relaxed controls by de ning f (t; x; w) := U f (t; x; s)dw(s). As for ordinary controls, given (t0 ; x0) 2 R  Rn, and a relaxed control w : [t0; 1) ! P (U ), we denote by x(t; t0; x0; w) the solution of (1) with initial condition x(t0; t0; x0; w) = x0. For more details on relaxed controls see [1]. De nition 1.3 A continuous function V : R Rn ! R0 is a Lyapunov function for the model (1) with respect to the closed, weakly invariant, and non-empty set A if the two following properties hold. 1. There exist two K1 -maps 1, 2 such that, for all (t; x) 2 R  Rn: 1(jxjA )  V (t; x)  2 (jxjA ): 2. There exist a continuous, positive and increasing map , and a K-function 3, such that for each (t0 ; x0) 2 R  Rn, there exist t1 > t0, and w : [t0; t1) ! P (U ) with kwk  (jx0jA) and: lim inft!t+0 V (t;x(t))t??tV0 (t0 ;x0 )  ?3(jx0jA); where x(t) = x(t; t0; x0; w). Remark 1.4 From the proof of Theorem 1 it will be clear that in the de nition of Lyapunov function, we could have used the lim sup instead of the lim inf, and the same result will hold. In that case, however, the suciency part of the result becomes weaker. De nition 1.5 Given a closed and non-empty set A, we say that the function f (t; x; u) satis es the boundedness assumption with respect to A if it holds that, for each r1; r2 > 0: jf (t; x; u)j  Mr1 ;r2 a.e. t 2 R: (3) sup jxjA  r1 d(u; 0)  r2 r

Notice that if f is independent of t and continuous and A (i ; Ti )  2 ; we have: i+1  2 8 i  1; and thus is compact then (3) holds. lim  = 0: (4) i!+1 i Theorem 1 Let  be a given model of type (1), and A  Rn be a closed, weakly invariant, and non-empty Let g?1 and g0 be two constants such that g?1 > g0 > 2g1, set. Assume that the model satis es the boundedness as- and, for i  0, let Pi be the point (Ti ; gi?1). Let lP P (
) +1 sumption with respect to A (De nition 1.5). Then (1) is be ( T ) = g the linear function such that l i i ? 1 , and P P +1 gac to A if and only if there exists a Lyapunov function l P P +1 (Ti+1 ) = gi . It is easy to see that, by choosing g?1 V for (1) with respect to A. suciently large, we may assume: The suciency part is much easier to establish than (5) (x; t) < lP0 P1 (t) 8 t 2 [T0; T1]; 8 x < 0 : the necessity part. One way to prove that the existence of a Lyapunov function is enough to guarantee gac is by Now let: g(t) = lP P (t) for t 2 [Ti ; Ti+1]: Then +1 adapting the arguments used in [8] to this setting. This clearly the map g(
) is continuous and strictly decreasing. needs some attention, since we are in a non-autonomous Moreover, by equation (4), we have limt!+1 g(t) = 0 case and the set A is possibly not compact. (thus i) holds). Now we establish iii). Let p 2 R, and An alternative way to prove the suciency result is assume p < i . If i = 0 and t 2 [T0; T1], then property iii) through the use of recent results on set-valued maps and holds by construction (equation (5)). So we may non-smooth analysis (see [3]). By using Theorem 2.1 (in t  T1 . Let t  Ti , then t 2 [Tj ; Tj +1] with j assume i  1. the non-autonomous setting) and Theorem 7.1 of [3] one Then we have: g(t)  gj = j =2  (j ; Tj ): On the rst proves that if V is a Lyapunov function for (1) with other hand, if x < i and t 2 [Tj ; Tj +1 ], we also have: n respect to the set A, then for each (t0; x0) 2 R  R , there (x; t) < (i ; t)  (j ; t)  (j ; Tj )  gj  g(t); so iii) exists a relaxed control w0 such that kw0 k   (jx0jA ), holds. and the corresponding trajectory, x(t) = x(t; t0; x0; w0 ), From now on, we assume given a system  and a closed, exists R for all t  t0 and satis es: V (t; x(t)) ? V (t0 ; x0)  weakly invariant, and non-empty set A  Rn. Moreover ? tt0 3 (jx(s)jA )ds: Then, by using the previous equation, we assume that  satis es the boundedness assumption it is easy to see that properties 1. and 2. of Lemma 1.2 with respect to A (De nition (1.5)), and it is gac to A. follow, and so one concludes that the model is gac to A. Our aim is to construct a Lyapunov function for  with The next section is devoted to proving the necessity part respect to the set A. The idea of the construction is similar of Theorem 1. to the one given in [8]. Let (
) and (
;
) be respectively the continuous, positive and increasing, and KL maps given by De nition 1.1. Choose any 0 > 0, and let figi0 be any strictly increasing sequence such that: lim  = +1 and i+1 > minf (i ; 0); (i )g: The next proposition establishes a technical property of i!+1 i KL-functions. Now let fTi gi0, and g(
) be the sequence and the funcProposition 2.1 Let (
;
) be KL-function, and choose tion given by Proposition 2.1. Since the map g is strictly a strictly increasing sequence of positive real numbers, decreasing, and g : R0 ! (0; g(0)], we may let: figi0, such that limi!+1 i = +1. Then there exists  ?1 a continuous, strictly decreasing function g : R0 ! R0, if p 2 (0; g(0)]; h(p) = g 0(p) and a strictly increasing sequence fTi gi0, with T0 = 0, if p > g(0): such that: [i)] limt!+1 g(t) = 0; [ii)] limi!+1 Ti = +1; [iii)] if p < i then (p; t)  g(t) for all t  Ti . Notice that h : R>0 ! R0 is decreasing, continuous, and lim p!0+ h(p) = +1. Now we de ne: Proof. Let T0 = 0 and, for i  1, de ne, inductively on i:  0 if p = 0; i = (i?1 ; Ti?1 + 1=i); N (p) = p expf? (6) h(p)g if p > 0: and Ti be such that (i ; Ti)  i=2 = gi: Notice that: [i)] N is a continuous and strictly increasNote that such a Ti exists since (s;
) is decreasing to zero ing map; [ii)] limp!+1 N (p) = +1; [iii)] if p   then N (p)  . For each i  0, let Mi = M (i+2; i+2), where as t ! 1. M (i+2; i+2) are the constants given by the boundedness Since (i ; Ti?1 + 1=i) > i , it holds that: assumption, i.e.: i X sup jf (t; x; u)j  Mi for a.e. t 2 R: Ti > Ti?1 + 1=i ) Ti > 1=j; for i  1: jxjA  i+2 j =1 d(u; 0)  i+2 Thus the sequence fTi gi0 is strictly increasing and sat1 ) < Clearly Mi  Mi+1 . is es ii). Moreover, since: i+1 = (i ; Ti + i+1 i

i i

i

i

i

i

i

2 Lyapunov Function Construction

i

i

i

Now let M : R0 ! R0 and  : R0 ! R0 be any two continuous and increasing maps such that:  if p < 20 ; 0 M (p)  M (7) Mi if 2?1  p < 2 ;  (p)  (p) for all p  0; (8) and, for all i  0,    (i+2)  2i N 2i +  ( (0 )): (9) Moreover, for i  1, we require that, if i+1 < p  i+2 , then:  (p) >  ( (i )) + (10) M ( (i ; 0)) [N ( (i ; 0))Ti + (i ; 0)] and, for i  0,: i

M (i+1 ) >



2Mi 1 N (i+1 ) i+2 ?i+1

n

i

M ( (i ; 0)) o

[N ( (i ; 0))Ti + (i ; 0)] +  ( (i )) :

(11)

Notice that, since minf (i ; 0); (i )g < i+1, one shows easily that two continuous and increasing functions satisfying inequalities (7), (8), (9), (10), and (11) exist. Finally, we let: F (p) = M (p)N (p) = M (p)p expf?h(p)g: Fix (t0 ; x0) 2 R  Rn. Then let i0 be the rst index such that jx0 jA < i0 . For each control w, denote by x(t) = x(t; t0; x0; w) and let:

Q(t0; x0; w) =

R +1 t0 F (jx(t)jA )dt +

(12) + maxf (kwk) ?  ( (0 )); 0g; if x(t) exists for all t  t0 and is such that jx(t)jA  i0 +2 for all t  t0 . In all the other cases we let Q(t0 ; x0; w) = +1. Then we let V (t0; x0) = inf w2W Q(t0; x0; w): We want to prove that the function V de ned above is a Lyapunov function for the model (1) with respect to the set A. Lemma 2.2 Let (t0; x0) 2 R Rn, and i  0 be the rst index such that jx0 jA < i . Then there exists an ordinary control u, with kuk  (jx0jA), such that: Q(t0 ; x0; u) 

F ( (jx0 jA ; 0))Ti + M ( (jx0jA ; 0)) (jx0jA ; 0) + (13) + maxf ( (jx0 jA )) ?  ( (0 )); 0g: Proof. Given (t0 ; x0) 2 RRn with jx0 jA < i , let u be the control function given by the gac property. Then kuk 

(jx0 jA ), moreover the corresponding trajectory x(t) is de ned for all t  t0 and satis es jx(t)jA  (jx0jA ; t ? t0 ) < (i ; 0) < i+2 : Moreover, for t  Ti + t0, we have: jx(t)jA  (jx0jA ; t ? t0) < (i ; t ? t0 ) < g(t ? t0 ):

Thus, for all t  Ti + t0, h(jx(t)jRA)  h(g(t ? t0)) = tR ? t0. So we have: Q(t0 ; x0; u) = tt00+T F (jx(t)jA )dt + +1 t0 +T F (jx(t)jA )dt + maxf (kuk) ?  ( (0 )); 0g   F ( (jx0jA ; 0))Ti + M ( (jx0 jA ; 0)) (jx0jA ; 0))+ maxf ( (jx0 jA ))? ( (0 )); 0g: Thus the Lemma is proved. Remark 2.3 Notice that, from the previous lemma, we have in particular: (a) if jx0jA < 0 , then T0 = 0 and the max is also zero, so: V (t0 ; x0)  M ( (jx0 jA ; 0)) (jx0jA ; 0); (b) if ei?1  jx0jA < i , with i  1, then V (t0; x0)   ( (i )) ?  ( (0 ))+ i

i

+ M ( (i ; 0)) [N ( (i ; 0))Ti + (i ; 0)] :

Lemma 2.4 There exists a K1-map 2(
) such that, for all (t0; x0) 2 R Rn, it holds that: V (t0 ; x0)  2(jx0jA): Proof. For i?1  p < i (?1 = 0), let ~ 2(p) = F ( (p; 0))Ti + M ( (p; 0)) (p; 0)) + maxf ( (p)) ?  ( (0 )); 0g: Then ~ 2(
) is increasing and ~ 2(0) = 0. Now let 2(
) be any K1 -function such that ~ 2(p)  2(p). For each (t0; x0) 2 RRn, let u be the control map given by Lemma 2.2. Then, from equation (13), it follows that: V (t0 ; x0)  Q(t0; x0; u)  ~2 (jx0 jA )  2 (jx0 jA ): Lemma 2.5 There exists a K1-map 1(
) such that, for all (t0; x0) 2 R Rn, it holds that: 1(jx0jA)  V (t0; x0): Proof. Fix any (t0 ; x0) 2 R  Rn, and let i  0 be the rst index such that jx0jA < i . Let w be any control function such that the corresponding trajectory is de ned for all t  t0 and satis es jx(t)jA  ei+2 . First, assume that kwk  ei+2 , and let: x0 = j2xM0 jA : Fact. If t 2 [t0; t0 + x0 ), then jx(t)jA > jx0jA =2. Proof of Fact. If the conclusion does not hold, then, by continuity, there exists a t 2 [t0; t0 + x0 ), such that jx(t)jA = jx0jA =2. However, we have: jx(t) ? x0j  Mi (t ? t0 ) < Mi x0 = jx0 jA =2; which implies: jx0jA  jx(t)jA + jx(t) ? x0j < jx02jA + jx0 jA = jx j : So the fact is established. Now we 0 A 2 R have: Q(t0; x0; w)  tt00 + 0 M (jx(t)jA )N (jx(t)jA )dt M (jx0jA =2)N (jx0jA =2) j2xM0 jA : Next, by combining equation (7) with the previous inequality, one gets: Q(t0; x0; w)  jx02jA N (jx0 jA =2): If kwk > ei+2 , then: Q(t0; x0; w)   ( (i+2 )) ?  ( (0 ))  2 N (i =2); where to get this last inequality, we have used inequality (9). So, in any case, we conclude: V (t0 ; x0)  jx02jA N (jx0 jA =2): Thus, by letting 1(p) = p=2N (p=2), the conclusion follows. Thus the function V (
;
) satis es property 1 of De nition 1.3. It remains to show that V is continuous and it satis es also property 2 of the same de nition. First, we prove continuity at the points (t0; x0) 2 R  A. i

x

i

i

Lemma 2.6 The function V (
;
) is continuous at (t ; x ) 2 R  A. Moreover it holds that for each  > 0 there exists a  > 0 such that if jxjA   then jV (t; x)j   8 t 2 R: (14) 0

0

Proof. It is clear that it is suces to prove (14), since

V (t0 ; x0) = 0 for all (t0 ; x0) 2 R A. Given  > 0, choose 0 <  < 0 such that M ( (p; 0)) (p; 0)   for all p   . Now if jxjA   , then by Remark 2.3, property (a), we have: V (t; x)  M ( (jxjA ; 0)) (jxjA ; 0)  ; as desired. Lemma 2.7 For each (t0; x0) 2 R  Rn, let i  0 be the rst index such that jx0 jA < i , and let u0 be the control map given by Lemma 2.2. If w is any control such that: Q(t0 ; x0; w)  Q(t0 ; x0; u0); then kwk  i+2 , and the corresponding trajectory x(t) is such that: jx(t)jA < i+2 : (15) Proof. Assume that kwk > i+2 . Then it holds that: Q(t0 ; x0; w)   (kwk) ?  ( (0 ))   (i+2 ) ?  ( (0 )) > F ( (i ; 0))Ti + M ( (i ; 0)) (i ; 0)) + maxf ( (i )) ?  ( (0 )); 0g; where, to get the last inequality, we have used (10). Notice that this last inequality contradicts the assumption Q(t0 ; x0; w)  Q(t0 ; x0; u0). Now, assume that kwk  i+2 , but that the inequality (15) is not satis ed. Then there exists T > t0 such that jx(t)jA = i+2. Let ? i+1 : T~ = T + i+22M i Then, for all t 2 [T; T~], we have: jx(t) ? x(T )j  Mi (t ? T )  1=2(i+2 ?i+1 ); from which it follows that jx(t)jA  R i+1 . So, we have: Q(t0 ; x0; w)  TT~ F (jx(t)jA)dt   M (i+1 )N (i+1)  +22M? +1 F ( (i ; 0))Ti +M ( (i ; 0)) (i ; 0)+ ( (i )); where the last inequality holds be equation (11). Thus, the assumption that Q(t0; x0; w)  Q(t0 ; x0; u0): is again contradicted. Lemma 2.8 Let tn ! t0, xn ! x0, and wn ! w0 , be such that kwn k  r1, and all the solutions xn(t) = x(t; tn; xn; wn) exist for all t  tn and satisfy jxn (t)jA  r2, where r1; r2 are two positive real constants. Then the solution x(t) = x(t; t0; x0; w0 ) exists for all t  t0 , and it is such that: xn (t) ! x(t) ( as n ! 1); and jx(t)jA  r2; again for all t  t0. Proof. Since xn ! x0 , we have jx0 jA  r2 . Given w0 , by local existence of solutions, there exists a maximal interval [t0; t), in which x(t) is de ned. Moreover, we may assume that xn 2 clos B (x0 ; 1), and tn 2 [t0 ? 1; t0 + 1] for all n  1. For each t > t0 , we let: A(t) = f xn( ) j  2 [tn; t] g  Rn: Then A(t) is bounded, in fact,: i

i

i

jxn( )j  jxnj +

Zt tn

jf (s; xn(s); u(s))jds 

 jxnj + Mr1 r2 ( ? t)  jx j + 1 + Mr1 r2  (t); 0

where Mr1 r2 = sup jf (t; x; u)j for jxjA  r2 , and kuk  r1, and  (t) = (t ? t0 + 1). First we prove the following fact. Fact. If x(t) exists for all t 2 [t0; t0], then xn (t) ! x(t) as n ! 1 for all t 2 [t0; t0 ]. Proof of Fact. Let: K = clos A(t0 ) [ f x(t) j t 2 [t0 ; t0] g; then K is a compact set. Let LK;r1 , and M~ be the Lipschitz constant and a bound for f with x 2 K , kuk  r1 , and t 2 [t0 ? 1; t0]. Then one gets: R 0jxn(t) ? x(t)j  jxn ? x0j + M~ jtn ? t0 j+ Lk;r1 tt0 _t kwn (s) ? w0 (s)k + jxn(s) ? x(s)jds; where t0 _ tn indicates maxft0; tng. Using theBellman-Gronwall inequality, one gets: jxn(t)?x(t)j  jxn ?x0 j+M~ jtn ?t0 j+  R 0 Lk;r1 tt0 _t kwn (s) ? w0 (s)k ds eL 1 (t0 ?t0_t ) : From this last inequality the fact easily follows. Now we prove that x(t) exists for all t  t0. Assume that x(t) exists only for t 2 [t0; t). Since, for t 2 [t0; t), we have xn(t) ! x(t) (from the previous fact), it holds that: x(t) 2 clos A(t): Since clos A(t) is a compact set, this contradicts the fact that x(t) does not exist for t = t. Lemma 2.9 Let (tn ; xn) ! (t0 ; x0), and wn ! w0 . Let i  0 be the rst index such that jx0jA < i. Assume that all Q(tn ; xn; wn) are nite, and that kwn k  i+2. Then: Q(t0; x0; w0 )  lim n!+inf 1 Q(tn ; xn; wn): n

k;r

n

n

Proof. For n suciently large, we must have jxnjA  i, and jxn(t)jA  i+2 . Thus, from Lemma 2.8, it holds that x0(t) is de ned for all t  t0 and jx(t)jA  i+2 . Moreover it is also true that: F (jxn(t)jA )R ! F (jx0 (t)jA): Given  > 0 there exists  > 0 such that: tt00+ F (jx0 (t)jA )dt < : By Fatou's Lemma, we get (notice that for n large tn  t0 + ): Z1

t0+

F (jx0 (t)jA )dt  lim inf n!1

Z1

t0 +

F (jxn(t)jA )dt:

On the other hand, one easily sees that: maxf (kw0 k) ?  ( (0 )); 0g  lim inf maxf (kwn k) ?  ( (0 )); 0g: n!+1 Summing up, we conclude: Z1

Q(t0 ; x0; w0 )   + F (jx0(t)jA)dt t0 + + maxf (kw0 k) ?  ( (0 )); 0g; which is upper bounded by Z1

 + lim inf F (jxn(t)jA )dt n!1 t0 + + lim inf maxf (kwn k) ?  ( (0 )); 0g n!+1

and is itself bounded by  + lim infn!+1 Q(tn ; xn; wn): From which the conclusion follows since  was arbitrary. The arguments used in the proofs of next lemmas are similar to the one used in [8] to prove the corresponding results.

Lemma 2.10 For each (t ; x ) 2 R  Rn there exists w such that: V (t ; x ) = Q(t ; x ; w ): Proof. Assume that i  0 is the rst index such that jx jA < i . Let wn be a minimizing sequence, then, by Lemma 2.7, it must holds that kwn k  i . Thus, by sequential compactness of W +2 , we may assume that wn ! w (possibly extracting a subsequence). From Lemma 2.9, we have: Q(t ; x ; w )  lim infn!1 Q(t ; x ; wn ) = = V (t ; x )  Q(t ; x ; w ): Lemma 2.11 The function V (
;
) is lower semicontinu0

0

0

0

0

0

0

0

0

+2

i

0

0

0

0

0

0

0

0

0

0

0

w0 be such that V (t0; x0) = Q(t0; x0; w0 ). Let 1 be any continuous, positive and increasing map such that 1 (p)  i+2 for i?1  p < i , for i  0 (set e?1 = 0). Denote by x(t) = x(t; t0; x0; w0 ), then we know that jx(t)jA < i+2. For t suciently close to t0 we have that jx(t)jA < i. Moreover, if we denote by w00 the translation of w0 by (?t), we have: V (t; x(t)) 

Z1 t

F (jx(s)jA )ds+

+ maxf (kw00 k) ?  ( (0 )); 0g; thus (notice that kw00 k  kw0 k): Proof. Let (tn ; xn) ! (t0 ; x0). Then if i  0 is the rst inV (t; x(t)) ? V (t0 ; x0)  dex such that jx0jA < i, we may also assume that jxnjA < lim inf + t ? t0 t!t0 i . Let wn be such that V (tn ; xn) = Q(tn; xn; wn ), then, by Lemma 2.7, kwn k  i+2. Since W +2 is  Zt  1 sequentially compact, we may assume that wn ! w0 . ? F (jx(s)jA )ds = ?F (jx0 jA ): lim inf By applying Lemma 2.8, we know that the trajectory t!t+0 t ? t0 t0 x0(t) = x(t; t0; x0; w0 ) exists for all t  t0 and it is the So also property 2. holds with  = F , and  = . 3 1 limiting trajectory. Moreover by Lemma 2.9, we have: V (t0 ; x0)  Q(t0; x0; w0 )  lim infn!1 Q(tn ; xn; wn) = lim infn!1 V (tn ; xn): Lemma 2.12 The function V (
;
) is continuous. [1] Z. Arstein, Relaxed controls and dynamics of control systems, SIAM J. Control and Optimization, 16 Proof. We need only to prove upper semicontinuity. Fix (1978), pp. 689-701.  > 0, and let 0 <  < 0 be such that if jxjA   then V (t; x)  =3 for all t 2 R (use Lemma 2.6). Fix [2] Z. Arstein, Stabilization with relaxed controls Nonl. any (t0 ; x0) 2 R  Rn. Let i  0 be the rst index Anal.,TMA 7 (1983), pp. 1163-1173. such that jx0jA < i, and w0 be such that V (t0 ; x0) = [3] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern, and P.R. Q(t0 ; x0; w0 ). Then kw0 k  i+2 (Lemma 2.7). Denote Wolenski, Qualitative properties of trajectories of conby x(t) = x(t; t0; x0; w0), then, again by Lemma 2.7, we trol systems: A Survey, Journal of Dynamical and know that jx(t)jA < i+2 , and that there exists T such Control Systems, Vol. 1 , No. 1, 1995, pp. 1-48. that x(T ) <  . By continuity, there exist a neighborhood [4] F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag, and A.I. J  H of (t0; x0) such that, for all (t0 ; z ) 2 J  H , jz jA < i , Subbotin, Asymptotic controllability and feedback staz (t) = x(t; t0; z; w0) exists for all t 2 [t0; T ], jz (t)jA < i+2 , bilization, in Proc. Conf. on Information Sciences and jz (T )jA <  , and Systems (CISS 96), Princeton, NJ, 1996, pp. 12321237. ZT ZT F ( j z ( t ) j ) < F ( j x ( t ) j ) + = 3 : [5] A. Isidori, Nonlinear Control Systems, Third Edition , A A t0 t0 Springer-Verlag, London, 1989. [6] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, NonNow we have: linear and Adaptive Control Design, John Wiley & ZT Sons, New York, 1995. V (t0 ; z )  =3 + 0 F (jz (t)jA)+ [7] Y. Lin, E.D. Sontag, and Y. Wang, Recent results on t ous.

i

References

maxf (kw0 k) ?  ( (0 )); 0g