.Vooniinear Analwu. Tkeory. Pnnred in Great Bnrain. .Uerho& & Applicanonc. Vol. 7. No. 11. pp. 1163-1173. 1983. STABILIZATION WITH RELAXED 0362-546Xg3 $3.00 + .OO @ 1983 Pergamon Press Ltd. CONTROLS* Zvr ARTSTEIN? Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel (Received Key words and phrases: Stabilization, in revised form 1 December 1982) relaxed controls, Lyapunov functions. 1. INTRODUCTION NONLINEAR systems of the form i = f(x, u) (1.1) cannot in general be stabilized using a continuous closed loop control U(X), even if each state separately can be driven asymptotically to the origin. (An example is analyzed in Section 2.) In this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control. We find, indeed, that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls. An even larger class is obtained if the continuity of the closed loop at the origin is not required. The latter class includes all one dimensional systems for which states can be driven asymptotically to the origin. This result does not hold in two dimensional systems and we provide a counter-example. It should be pointed that relaxed control-type stabilization is used both in theory and in practice; the method is known as dither. We shall comment on the similarities. Lyapunov functions for the system (1) help us in the construction of the continuous closed loop stabilizers. In fact, we find that the existence of a smooth Lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin; an additional condition on the Lyapunov function implies the continuity at the origin as well. We present these results in Section 4, after a brief introduction of closed loop relaxed controls, notations and terminology in Section 3. Prior to that, in Section 2, we discuss an example illustrating the power of relaxed controls. In the particular case of systems linear in the controls, relaxed controls can be replaced by ordinary controls, this is discussed in Section 5. The role of Lyapunov functions in the stability and stabilization theories is of course well known. Examples of systems with Lyapunov functions are available in the literature. We display some in Section 6, along with general comments on the construction, applications and counterexamples, including one which cannot be continuously stabilized, yet possesses a nonsmooth Lyapunov function. Closed loop stabilization with ordinary controls is analyzed extensively in the literature, see Sontag [8], Sussmann [ll] and references therein. Lyapunov functions techniques in stabill The research was supported by the National Science Foundation under Grant MCS8102079. : Part of this work was done at the Department of Mathematics, Northeastern University, Boston, MA 02115, U.S.A. 1163 ZCI ARTSTED 1164 ization and regulation can be found in Aizerman and Gantmacher [l] and Barbashin [2]. The theory of relaxed control, primarily in connection with optimization, is developed in Warga [12]. Dither techniques are analyzed in Zames and Shneydor [ 131. 2. AN EXAMPLE This section is aimed primarily at those unfamiliar with relaxed controls and their role. We construct a one dimensional control system which cannot be stabilized with a continuous closed loop of ordinary controls, yet a relaxed control stabilizer does exist. Systems which cannot be continuously stabilized with ordinary controls are provided by Sontag and Sussmann [lo], where such one dimensional systems are characterized. Example 2.1. Let x and u be scalars and x = g(u) (2.1) where g(u) = 1 - u if u 2 1, g(u) = -1 - u if u < -1 and g(u) = 0 otherwise. Each state can be driven asymptotically to 0, even by using a closed loop, e.g. u(x) = x A 1 if x > 0 and u(x) = x - 1 if x < 0. The discontinuity at 0 cannot be overcome, since u(x) must be greater than 1 for x > 0 and less than -1 for x < 0. We allow now the use of the following, seemingly peculiar, behavior of the control. The control is allowed to oscillate, or chatter, very rapidly between the values. say, 2 and -2. Furthermore. the proportion of time spent at each of these two values might vary and depend, in a closed loop form, on the state x. (Such a control law seems terribly discontinuous, but it is not, in a sense which we explain later.) If in a certain instance the control assumes the value 2 with the portion p (0 s p < 1) of its time, and the portion 1 -p is spent at the value -2, the right hand side of (2.1) in that instance becomes pg(2) + (1 - p)g( -2) = 1 - 2p. We let p depend on the state x according to the following rule: The differential p(x) = l/2 + x if 1x( S l/2 P(X) = 1 ifx 2 l/2 P(X) = 0 ifx < -l/2. equation generated by this closed loop is i=-?J if jx/ s l/2 i = -x/Ix] if 1x12 l/2 and it is clearly globally asymptotically stable. Notice that only two values of the controls participated in the process. In particular, no use was made of the fact that g(uo) = 0 for a certain ordinary control UO. We see that the stabilizing closed loop suggested above is continuous. even Lipschitz continuous, in the parameter p(x). The point is that in many practical cases it is not difficult to generate a highly oscillatory control signal; then the continuity of the parameter p(x) reflects actual continuity of the closed loop. As was mentioned before, such techniques, known as dither, are used in practice, see Zames and Shneydor [13], [14] and references therein. Stabilization 3. CLOSED LOOPS with relaxed WITH controls RELAXED 1165 COKTROLS Relaxed controls were developed primarily within the framework of optimal control theory, see Warga [12]. Here we use them in stabilization. In this section we briefly review the concepts and display the notations and terminology. The state variable x in (1) belongs to the n-dimensional euclidean space R”. Its norm is denoted 1x1 and x. y denotes the scalar product of n and y. A dot above a variable denotes differentiation with respect to time. The gradient of a function V is denoted gradV. The control variable u in (1) belongs to a metric space CJ’. We assume throughout that f(x, u): R” X U+ R” is a continuous function. A relaxed control is a probability measure, say u, on the space U. The family of relaxed controls is denoted by U,Q.Two topologies will be considered on UR. The first is associated with weak convergence of measures, i.e. uk-$ 00 if Ih dVk-+ Ih duo for any continuous and bounded h:U- R. The second is the norm topology, generated by the variational norm IlUll=SUp{~lU(E,)l:Ei form a finite partition of U} (Ei C U are always assumed Borel.) For details see Billingsley [4] or Warga [12]. If at a certain state x a relaxed control u is used, the right hand side of (1) assumes the value (3.1) The ordinary control u can then be identified as the relaxed control having full measure assigned to the singleton {u}. We do not distinguish between the two in the sequel. Unfortunately the mapping f(.~, u) might not be defined for every pair (x. v). and when defined it might not be continuous in either of the variables, even with respect to the norm topology. Joint continuity in (x, u) is actually the desired property, and additional conditions (e.g. compactness of U or local compactness of U plus boundedness off on sets B x U with B bounded) would result in a well defined (3.1) and joint continuity off, with respect to weak convergence on UR. We are interested in closed loop schemes of the form and the resulting differential u(x): R”+ UR (3.2) =f(x, u(x)). (3.3) equation i As was indicated, even if u(x) is smooth the equation (3.3) might have a discontinuous right hand side, unless further conditions are imposed. In this paper, rather than imposing one of the aforementioned conditions, we will consider only those closed loops U(X) Lvhich are continuous (with respect to one of the topologies on UR) and which produce a continuous right hand side f(x, U(X)). H owever, in general, the continuity of both u(x) andf(s. U(X)) will not be required at x = 0, for a reason which we now explain. An ordinary differential equation i = h(x) is asymptotically stable if there is a neighborhood W of 0, a continuous function b(r): [0, =)* [0, 3~) with b(0) = 0 and a continuous nondecreasing function T(&):(O, 30) ---, (0, =) such that whenever x(t) is a solution and s(O) E IV then lx(t)\ s b(lx(O)l) for TV 0 and ]x(T(&))I s E. The equation is globally asymptotically stable if it is asymptotically stable and the neighborhood W can be chosen to be any bounded set in R”. Zvr ARTSTEIN 1166 If X = h(x) is asymptotically stable and h(x) is continuous except possibly at x = 0, yet h(O) = 0, then solutions satisfying x(0) E W are defined for all f 2 0 and are continuously differentiable except possibly at the first time t for which x(t) = 0. Namely, in case of asymptotic stability, discontinuity of h(x) at x = 0 (with h(O) = 0) produces no problem in the definition, existence and continuation of solutions. 4. STABILIZATION AND LYAPUNOV FUKCTIO.US In what follows and throughout, we say that (1) is stabilizable by a closed loop control if a mapping u(x) from a neighborhood of 0 into UR exists, continuous with to the weak convergence except possibly at x = 0, such that f(x, u(x)) is continuous possibly at x = 0 and such that (3.3) is asymptotically stable. Global srubilizarion is similarly. We assume throughout that f(0, 00) = 0 for a certain relaxed control ug. relaxed respect except defined THEOREM 4.1. The system (1) is stabilizable by a closed loop relaxed control if and only if there is a C’ function V: W-+ R, defined on a neighborhood W of 0 such that V(0) = 0. V(x) > 0 if x f 0 and for x + 0 inf gradV(x) .f(x, u) < 0. UEU (4.1) The system is globally stabilizable by a closed loop relaxed control if and only if W in the previous statement can be chosen R” with V(x) + r: as 1x1--, =. Furthermore, in either case, V can be chosen C” and the closed loop u(x) can be chosen piecewise linear on bounded sets bounded away from 0. Proof. Suppose first that (3.3) is asymptotically stable or globally asymptotically stable, with f(x, u(x)) continuous except at x = 0. If global asymptotic stability holds then f(x, u(x)) # 0 for all x, if asymptotic stability holds thenf(x, u(x)) # 0 for x near 0; otherwise there is a rest point of (3.3) within the basin of attraction of 0. Let a(x) be a scalar function defined for x # 0, positive and continuous and such that (~(x)f(x, u(x)) - 0 as x+ 0; e.g. 4x) = Ix / lf(x, u(x)) I-‘. The differential equation x = @>f(x, u(x)> (4.2) has now a continuous right hand side. Since (u(x) > 0 it follows that (4.2) and (3.3) share the same phase portrait; in particular (4.2) is asymptotically stable or globally asymptotically stable in accordance with (3.3). The converse Lyapunov function as constructed in Kurzweil [7] provides a C” function V: W +- R (with W = R” in case of global asymptotic stability) with V(0) = 0, V(x) > 0 if x # 0 (V(x)+ m as 1x1-+ = in case of global asymptotic stability) and such that gradV(x) . a(x)f(x, u(x)) < 0 for x # 0. (Unfortunately, from the introduction to Kurzweil [7] one can get the impression that the solutions of the equation are assumed to be uniquely determined by initial conditions. This is not correct and in fact, as indicated in the paper itself, one of the goals of Kurzweil’s beautiful construction is to avoid such an assumption.) Since a*(x) is positive it follows from the displayed inequality that already gradV(x) .f(x, u(x)) < 0. Stabilization with relaxed controls 1167 Since f(x, u(x)) is defined as an integral, see (3.1), it follows that at least for one u E U gradV(x) .f(x, u) < 0 and (4.1) is checked. This completes one direction of the statement. Suppose now that a CLfunction V: W ---, R satisfying the conditions of the statement is given. For x # 0 in W we denote by F(x) the collection of all relaxed controls u for which f(x, u) is defined and gradV(x) *f(x, u) < 0. Then F(x) is convex and contains at least one ordinary control which we denote by u(x). Since V is C’ and f(x, u) is continuous it follows that for x fixed u(x) E F(y) if y is close enough to x, say if y is in the open ball B(x) around x. Since w\(O) is locally compact it follows that a sequence Bi , Bz, . . . of such balls exists which covers the set W\(O) such that each point x belongs to only a finite number of them. An elementary consideration would then show that a continuous positive function d(x) exists on W\{O} such that the ball {y : Iy - XI G d(x)} is included in one of the balls Bi. Let zl, zz, . . be a denumerable number of points in W\(O) which generate a triangulation of W\(O) (see e.g. Hocking and Young [5]) with the property that each simplex in the triangulation which contains Zj has a diameter less than d(zj)/2. It is not hard to construct such a triangulation. Recall that each Bi is the ball B(xi) such that pi = u(x,) belongs to F(y) if y E Bi. We assign now to each vertex zj one of the ordinary controls ui for which a ball of radius d(zj) around z, is included in Bi. The definition of d(x) implies that such Ui exists. Suppose now that a point x belongs to a simplex in the triangulation generated by z,, , . . . , Zjks The function u(x) is defined to be the measure with weight pjL assigned to Uj, when pi, , . . . , pi,, are the barycentric coordinates of x in the simplex. Then u(x) is piecewise linear on compact sets bounded away from zero. The continuity of f(x, u) implies that f(x, u(x)) 1s . continuous except possibly at x = 0. Also, since Ix -zj,l =S d(zj,)/2 it follows that rlj, E F(x) and the convexity of the latter implies that V is a Lyapunov function for (3.3). A u(x) E F(x). In particular gradV(x) *f(x, u(x)) < 0 I.e. . standard argument shows that iff(0, UO)= 0 for a certain 00 then (3.3) is asymptotically stable, with region of attraction that contains at least the set {x:V(x) < ao} with no = min{Vcv):y in the boundary of W); and (3.3) is globally asymptotically stable if W = R”. This completes the proof. Continuity of the closed loop at the origin might not hold even for seemingly simple systems. Here is an example. Example 4.2. Let x be scalar and U = (0,1). Let f(x, 0) = -x if x 2 0 and f(x. 0) = 3x if x G 0. Let f(x, 1) = -f(-X, 0). The system is stabilizable with a closed loop relaxed control, e.g. u(x) = 0 if x L 0 and u(x) = 1 if x < 0. The discontinuity at 0 cannot be overcome since unless the measure given by u(x) to 0 for x > 0 is at least l/4 and for x < 0 is at most l/4 the system would not be stabilized. Extending f(x, U) to u E [0, l] by f(x, n) = cuf(xt 1) + (1 cr)f(x, 0) yields the same consequences. We shall provide conditions for the existence of a stabilizing closed loop which is continuous at x = 0 with respect to a given metric on UR. We say that a metric or a semi-metric p(. , -) on UR is conuex if the sets {u:p(u, UO)< E}, for u. fixed, are convex. An example is the norm metric, which, as it is easy to see, is finer than any convex semi-metric. Another example is the Prohorov metric p(uo, ul) defined as the minimal E B 0 for which uo(A) s ul(A ‘) - E and Q(A) c uo(A “) + E for all measurable A C U, and when A’ denotes the s-neighborhood of A in U. The Prohorov metric is equivalent to the metric of weak convergence whenever the 1168 ZvI ARTSTEIN latter is metrizable, e.g. if U is separable. For details see Billingsley [4, appendis III]. Other metrics or semi-metrics might have relevance to the system. THEOREM 4.3. Suppose that lJ is separable and complete and let p( ., .) be a convex semimetric on iJR. There exists a stabilizing (or globally stabilizing) closed loop u(x) which is p continuous at x = 0 with f(x, u(x)) continuous if and only if there is a C’ function V satisfying the conditions of theorem 4.1 and in addition the following property holds: Au0 E I/R exists with f(0. uo) = 0 and such that for every E > 0 there is a 6 > 0 such that Ix/ < 6 implies the existence of ul with p(uO, ui) < E and ]f(x, ui)i < E and gradV(x) *f(x, oi) < 0. Furthermore V can be chosen C’ and u(x) can be chosen piecewise linear at s f 0. Proof. The construction of the Lyapunov function in the proof of theorem 1.1 covers the present case, with the additional conditions on V being implied by the p-continuity of u(x) and the continuity of f(x, u(x)). Suppose now that a function V is provided satisfying the conditions of the theorem. As in the proof of theorem 4.1 let F(x) denote the set of relaxed controls L: for which gradV(x) .f(x, u) < 0. Then F(x) is convex, nonempty and contains an element u,(x) such thatx-, Othenp(ui(x), ~a)-+ Oandf(x, ui(x)) + 0. This follows from the additional condition on V. Furthermore, u,(x) can be chosen to have a compact support. This follows from the separability and completeness of U, which makes every measure tight (see Billingsley [4 appendix III]), so that even if ui(x) does not have, originally. a compact support, its restriction to a large enough compact set will satisfy the requirements. From now on we proceed exactly as in the proof of theorem 4.1, with ul(x) replacing the ordinary controls u(x). The compact support of each u,(x) implies indeed that u,(x) E F(y) if y is close enough to x. so the balls B(x) are well defined. The piecewise linear closed loop u(x) then stabilizes (3.3) and the choice of ui(x) together with the convexity of the semi-metric p( a, .) imply that p(u(x). UO)- 0 and f(x, u(x)) -+ 0 as x--, 0. This completes the proof. Remark 4.4. If the existence of ul with p(ul, uo) < E in the preceding result is dropped, yet ]f(x, ut) 1< E is maintained then the closed loop u(x) might be discontinuous but the resulting differential equation f(x, u(x)) 1s continuous. Similarly, if p(ui, uo) < E can be obtained but without If(x, ul)] < E then the closed loop can be chosen continuous when f(x. u(x)) might not. Remark 4.5. The stabilizing rule that we have constructed has the form u(x) = Z:p,(x)ui where pi(x) form a partition of unity with respect to the balls B,. Note that any partition of unity would suit, in particular one which replaces the piecewise linear pi(x) by C” functions, which might be desired property. (I owe this comment to W. Fleming.) 5. SYSTEMS LINEAR IN THE CONTROL A major role in the previous results was played by the linearity of f(x, u) in the relaxed control u. If the system is linear in the control to begin with, then there is no need to refer to relaxed controls, as the results of this section show. (Notice, however, that these results are not included in the previous case, since the closed loops u(x) of relaxed control were norm continuous and an ordinary control closed loop n(x) is, unless constant, never continuous with respect to the variational norm.) Stabilization with relaxed controls 1169 In the following, U is a convex set in a Banach space (e.g. in R”) and the system has the form x = fl(X) + fi(X)U with fi :R” + U to R” (if continuous. are, clearly, (5.1) R” continuous and fi: R” - B(U), the latter denotes the linear functionals from UC Rm then fi(x) is an n x m matrix). We only require that (x, u) + fJx)u be We assume that fi(0) +f2(0)u0 = 0 for a certain KO E U. All bilinear systems included. 5.1. There exists a closed loop u(x):R + U, continuous except possibly at x = 0, and which makes (5.1) asymptotically stable if and only if a neighborhood W of 0 exists and a C’ function V: W-, R with V(0) = 0. V(x) > 0 if x # 0 and gradV(x) *f(x, rt) < 0 for at least one u E U if x # 0. Such a global stabilizer exists if and only if W in the previous statement can be chosen R” and V(x)+ 3~ as 1x1-P CJC.In either of the cases V can be chosen C” and U(X) can be chosen piecewise linear on compact sets bounded away from 0. THEOREM Proof. We can either follow the proof of theorem 4.1 when the linear set U replaces the (always) linear set UR, or note that if u(x) is the closed loop relaxed control guaranteed by theorem 4.1 then U(X) = Iudu(x)(u) is the desired ordinary control loop. This takes care of the “if” part. The “only if” direction is actually included in theorem 4.1 since u(x) continuous as an ordinary control implies u(x) continuous with respect to the weak convergence as a relaxed control, also, then, fr(x)u(x) is continuous when u(x) is. 5.2. The closed loop U(X) in theorem 5.1 can be made continuous at x = 0 if and only if the Lyapunov function V can be chosen with the following additional property: a uo E U exists with f(0, ~0) = 0 and with the property that for every E > 0, a 6 > 0 exists such that whenever 1x1< 6 the inequality gradV(x) .f( x, ul) < 0 holds for a certain u1 with 1~1 - uo] < E. (Here (uI - uoI stands for the distance in U.) THEOREM Proof. Again we can either imitate the proof of theorem 4.3 or note that if u(x) is the relaxed closed loop guaranteed by it when u,-,= uo then u(x) = Iudu(x)(LO is the desired ordinary closed loop. Remark. The referee brought to my attention that results similar to those of this section are provided in Jacobson [6, theorems 2.5.1 and 2.5.21. Jacobson works with equations of the form i = B(x)u, with u E R” unrestricted. The sufficient condition is then the existence of a C’ function V with V(0) = 0, V(x) > 0 if x # 0 and such that grad V(x) * B(x) does not vanish for x # 0. In this setting a stabilizing control is simply u(x) = -gradV(x) . B(x). Such a control might not be allowed in our framework. (Also, Jacobson presents the necessary condition for only smooth equations, but this can be easily corrected by applying to Kurzweil’s results on the existence of Lyapunov functions rather than to the results referred to by Jacobson.) 6. CO.MMENTS, Remark EXA.MPLES AND COUNTEREXAMPLES 6.1. The closed loop controls u(x) in the constructions of the theorems are piecewise linear (in UR or U) therefore locally Lipschitz with respect to the norm (on UR or respectively U). This does not mean that the resulting differential equation (3.3) has a Lipschitz right hand 1170 ZVI 6iRTSTEI.U side. As a counterexample consider an equation i = f(x, ~0) with a singleton I/ = {rca}and a non-Lipschitz function f. However, if f(x, u) is Lipschitz then the closed loops of our theorems provide a locally Lipschitz f(x, u(x)). Remark 6.2. Notice that theorem 4.1 provides a closed loop u(x) such that every measure is concentrated at most on n + 1 points, n being the dimensionality of the state space. The same phenomenon occurs in applications of relaxed control to optimization (Berkovitz [3] develops the entire theory of relaxed controls with only such controls). This is not the case in theorem 4.2, unless the particular control ug is known to be supported at m points; then the approximations u1 can also be picked to have this property and v(x) is supported at (n + 1)m points. Counterexample 6.3. The system analyzed in Section 2 can be stabilized with a closed loop ordinary control u(x) which is discontinuous only at x = 0. Take U(X) = 2 if I > 0 and u(x) = -2 if x < 0. This is not always possible, even if such stabilizing closed loops with relaxed control exist. Here is the counterexample. Let x and u be scalars, let f(x, u) = -x(sin(x-‘))*n if u 2 0 andf(x, u) = -x(cos(x-l)) ‘I( if ~1 s 0. With the aid of theorem 4.3 and the Lyapunov function u(x) = x2 it can easily be seen that a continuous closed loop relaxed control which stabilizes the equation exists, with u(O) = 0. Furthermore (see remark 6.2) each relaxed control u(x) would be concentrated at two points. However, any closed loop of ordinary controls U(X) which is continuous at x f 0 would produce rest points (i.e. points x with f(x, U(X)) = 0) arbitrarily close to 0, hence it cannot stabilize. The constructive argument of the preceding remark can be carried over to all one dimensional state systems, as follows. (By a piecewise continuous function we mean one for which the one sided limits exist at points of discontinuity.) PROPOSITION6.4. Let x be scalar and suppose that for any state x0 in a neighborhood of 0 there is a piecewise continuous control function u(r) such that X: =f(x, u(t)) has a unique solution x(t) with x(0) = x0 and x(r) + 0 as r+ =. Then there is a stabilizing closed loop with relaxed controls, continuous except possibly at x = 0. Global stabilization occurs if W = R. Proof. We claim that V(x) = x’ is a Lyapunov function as required in theorem 4.1. Indeed, if for xl E W and x1 # 0 the relation 2xlf(xi, rc) 2 0 for all U, then x&x,, et(t)) 2 0 for all n(t) which drives x0 asymptotically to 0 with jxol > (x1(. But then i = f(x, u(t)) has more than one solution, the bifurcation point being at xi, A contradiction. The uniqueness demand in the previous result is needed, since it is not hard to construct an example i =f(x) which has rest points arbitrarily close to 0 yet there are solutions x(r) + 0 with x(0) = x0 arbitrary. The regularity requirement on u(t) is also needed since, in general, the solutions of i = f(x, u(t)), with u(t) measurable are differentiable and satisfy the equation only almost everywhere. It is therefore possible to construct phatological examples where 2xif(xi, u) > 0 for a certain x1 and all U, yet the solution x(t) (which necessarily does not satisfy the equation at xl) tends to 0 as f+ 3~. The consequence of proposition 6.4 does not hold when x is two dimensional, as shown in the following example. 6.5. Let x = (5, n) be in R’. Consider in R’ the family of circles by the equations f’ + (q - a)’ = a’ for a # 0, (i.e. with radius Jaj and center at Counterexample determined Stabilization with relaxed controls 1171 (0, a)). (See Fig. 1.) Let T(x) be a unit vector, tangent at x to the circle to which x belongs and such that T(x) is continuous on R’ except at x = 0 (at x = (r, 0) the vector T(x) is parallel to the E-axis). See the arrows in Fig. 1. We define the system for u E [- 1, l] by f(x, U) = exp( - /x I-?) T(x)u. (6.1) Fig. 1. Thenfis continuous, even smooth. Any given state can be driven asymptotically to 0. by even using a closed loop discontinuous control which stabilizes the system. This is obtained for instance by taking u(x) = -1 if E> 0 and u(x) = 1 if t s 0. A continuous (even except at x = 0) stabilizing closed loop, relaxed or ordinary, does not exist. Indeed, since evev circle is invariant under any equation of the form X =f(x, u(x)), the stability of such an equation implies the existence of a rest point other than 0 on any of the circles, hence arbitrarily close to 0. Our characterization (theorem 4.1) implies that the system (6.1) does not possess a C’ Lyapunov function satisfying (4.1). Interestingly enough, a Lipschitz-continuous Lyapunov function does exist, whose typical level surface is indicated by the dashed line in Fig. 1. (The condition gradV(x) *f(x, u) < 0 should be replaced by the obvious analog.) We see therefore that in control systems nonsmooth Lyapunov functions might not be smoothened, as the case of ordinary differential equations is, see Kurzweil [7 theorem 31. Note. E. Sontag has kindly shown me a draft of [9]. There the existence of a Lyapunov function for systems of the form (1) is established under the condition that each initial point can be driven asymptotically to the origin. Sontag does not look for smooth Lyapunov functions and indeed our counterexample 6.5 shows that there might not exist a smooth Lyapunov function. The existence of a Lipschitz Lyapunov function in counterexample 6.5 is: however, not a coincidence, but guaranteed by Sontag’s result. Example 6.6. The system a! = xu2 - U’ for x and u scalars can be stabilized by a continuous closed loop ordinary control. For instance if U(X) = sgn(x)lx]” the resulting equation is asymptotically stable. A stabilizing continuous closed loop which is Lipschitz continuous does not exist, as then xu’(x) will dominate U’(X) for U(X) small, and since u(O) = 0 is a necessary 1172 Zvr ARTSTEN condition, the equation would not be stable. (See Sontag and Sussmann [lo] for a discussion concerning Lipschitz closed loops.) Within the class of relaxed controls it is rather easy to find a Lipschitz closed loop stabilizer, along the idea of Section 2. Example 6.7. We wish to illustrate the sufficiency of the Lyapunov technique. Consider the Lienard-type equation 1 + UX - 11’= 0 when both the friction and the restoring forces are partially controlled, with the controls being lumped together. The associated system in R’ is .r = y j = -KY + (6.2) K3. The goal is to stabilize the system. Consider a typical Lyapunov equations, e.g. function for Lienard-type V(x, y) = fx’ + f(x + ,.)! (6.3) Then, it is easy to check, gradV(x,y) .f(x,y,K) =X(2y - Ky f 1~:)+-J'(J- Ky + Kj). (6.4) It is immediate therefore that if x and y have the same sign, i.e. xy 2 0. then (6.4) can be made negative with the proper choice of K (large /K / with K having the opposite sign of x or y). If x > 0 and y < 0 the control K for which 2y - ICY- 113= 0 will make (6.4) negative and if x < 0 and y > 0 the solution of y - uy f 113= 0 will make (6.4) negative. In view of theorem 4.1 the system (6.2) is stabilizable by a piecewise linear (except possibly at 0) closed loop relaxed control. In order to achieve continuity at the origin consider the relaxed control ug equally concentrated on 1 and - 1. Then f(0, 0, UO)= 0. If u. is perturbed and the weights of 1 and - 1 are p and (1 - p) respectively, then the expression in (6.4) becomes 2P((X + Y) - Y(X + Y)) - ((x + f) - Y(2Y + 3x)). (6.5) If x and y have the same sign or if /x] G 1/2/y] then the leading term in (6.5) is 2p(x + y) (x + y) and for x and y small it can be made negative by a choice of p close to l/2. If xy < 0 and Ix] L 1/2]y] then a direct computation shows that (6.5) is negative for p = l/2. We can conclude from theorem 4.3 that the stabilizing closed loop of relaxed controls can be made continuous at x = 0 with u(O) = uo and the continuity being with respect to the variational norm. Moreover, the controls will chatter only between the values 1 and - 1. Simple considerations show that for x and y small there is an ordinary control K which makes (6.4) negative and 1~1 is small. Therefore a stabilizing closed loop relaxed control exists with u(0) = 0 and which is continuous at 0 with respect to the weak convergence. REFERENCES 1. AIZERMANM. S. & GAXIMACHERF. R., Absolute Stability of Regulator Systems, Holden-Day, San Francisco (1961). (English translation). 2. BARBASHINE. A., Inrroducrion to the Theory of Srability, Walters-Noordhoff, Groningen (1970). (English translation). 3. BERKOVITZL. D., Optimal Control Theory, Springer, New York (1971). 4. 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