On Convexity of Reachable Sets for Nonlinear Control Systems

Transcription

1 Proceedings o the European Control Conerence 27 Kos, Greece, July 2-5, 27 WeC5.2 On Convexity o Reachable Sets or Nonlinear Control Systems Vadim Azhmyakov, Dietrich Flockerzi and Jörg Raisch Abstract In this paper, we investigate reachable sets or some classes o open- and closed-loop control systems governed by ordinary dierential equations. We are particularly interested in convexity properties o both reachable sets and sets o trajectories. I such properties exist they can be used to estimate the reachable set under investigation. This is important in applications as, e.g., the veriication o saety properties. Convexity o the set o trajectories is also important or a constructive reormulation o some constrained dynamical optimization problems in the orm o a convex minimization problem. I. INTRODUCTION First, let us consider a nonlinear closed-loop system given by ẋ(t) = (x(t), u(x(t))), a.e. on [, t ], x() = x, where : R n R m R n is Lipschitz continuous in both components. Let U be a compact and convex subset o R m and consider measurable eedback control unctions u : R n U. Assume that or every such eedback control unction u( ) there exists a solution x u ( ) o (1) or uniqueness conditions and or some constructive existence conditions or systems (1) we reer to [6], [9]. Given an initial value x R n and such a eedback control unction u( ), the solution o (1) is an absolutely continuous unction. Let n (, t ) be the Sobolev space o all absolutely continuous R n -valued unctions y( ) such that the derivative ẏ( ) exists almost everywhere and belongs to the Lebesgue space L 1 n(, t ) o all measurable unctions y : [, t ] R n with y( ) L 1 n (,t ) = y(t)dt <. Recall that n (, t ) equipped with the norm W 1,1 n (,t ) deined by w( ) W 1,1 n (,t ) := w( ) L 1 n (,t ) + ẇ( ) L 1 n (,t ) V. Azhmyakov is with the Fachgebiet Regelungssysteme, Institut ür Energie- und Automatisierungstechnik, Technische Universität Berlin, Einsteinuer 17, D-1587 Berlin, Germany. azhmyakov@control.tu-berlin.de D. Flockerzi is with the Systems and Control Theory Group, Max Planck Institute or Dynamics o Complex Technical Systems, Sandtorstr. 1, D-3916 Magdeburg, Germany. lockerzi@mpi-magdeburg.mpg.de J. Raisch is with the Fachgebiet Regelungssysteme, Institut ür Energie- und Automatisierungstechnik, Technische Universität Berlin, Einsteinuer 17, D-1587 Berlin, Germany and with the Systems and Control Theory Group, Max Planck Institute or Dynamics o Complex Technical Systems, Sandtorstr. 1, D-3916 Magdeburg, Germany. raisch@control.tu-berlin.de (1) or w( ) n (, t ) is a Banach space. Moreover, n (, t ) is the completion o the space o all continuous dierentiable R n -valued unctions C 1 n(, t ) with respect to the norm L 1 n (,t ) (see e.g. [1], [3]). The initial value problem (1) can also be considered as a problem in the space n (, t ). To study control theoretical questions or system (1), or instance controllability, it is useul to deine the reachable set K(t, x ) at time t. K(t, x ) is the set o states o (1) which can be reached at time t, when starting at x at time t =, using all possible controls (see e.g., [19], [15]). That is, K(t, x ) := {x u (t) u( ) L 1 m (R n ), u(x) U}, where L 1 m(r n ) denotes the Lebesgue space o all measurable unctions u : R n R m. We now ormulate our standing hypothesis: (H) The reachable sets K(t, x ) are contained in an open bounded superset Ω o R n or t [, t ]. This is or example the case i Ω is a positively invariant set or the system (1). Recall that a set in the state space is said to be positively invariant or a given dynamical system i any trajectory initiated in this set remains inside the set at all urther time instants. Besides, or a dynamical system (1) with bounded right-hand side, the reachable set K(t, x ) is trivially bounded. We reer to Section V or some speciic considerations. Given l R +, we introduce the space U l o admissible eedback laws u : Ω U as the space o all Lipschitz unctions with Lipschitz constants l u l on Ω. Under the above-mentioned boundedness assumption o the reachable set, we can now consider the reachable set K(t, x ) o (1) with respect to U l given by K(t, x ) := {x u (t) u( ) U l }. For the given control system (1), we address the task o ormulating suicient conditions or the convexity o the reachable set K(t, x ) or every t [, t ]. Note that the convexity o the reachable set or the existence o convex approximations or the reachable set bear a close relation to a computational method or positively invariant sets, namely the ellipsoidal technique (see [14], [17]). Moreover, we derive conditions or the set o trajectories o (1) on [, t ], i.e. T (x ) := {x u ( ) : [, t ] Ω u( ) U l }, to be convex. The main convexity result or system (1) is based on an abstract ixed point theorem or nonexpansive ISBN:

2 WeC5.2 mappings in Hilbert spaces (see e.g. [22], [2]). For some abstract convexity results or nonlinear mappings we reer to [16], [18], or some applications to optimization and optimal control to [16], [18], [4]. For an analysis o reachable sets o dynamical systems in an abstract or hybrid setting see also [14], [11]. While the main topic o our paper is the estimation o reachable sets or closed-loop systems o type (1), we also consider open-loop control systems ẋ(t) = g(x(t), u(t)), a.e. on [, t ], x() = x, where g is a Lipschitz continuous unction (in both components) and where u(t) belongs to U or t [, t ]. Let U := {u( ) L 2 m(, t ) u(t) U} be the space o admissible control signals or system (2). Here, L 2 m(, t ) denotes the standard Lebesgue space o all square-integrable unctions u : [, t ] R m with the corresponding norm. It is assumed that or every admissible time-dependent control u( ) U system (2) has a unique solution x u ( ) n (, t ). As or the closed-loop system (1), we will obtain estimates or the reachable sets o (2) provided the right-hand sides are bounded. The paper is organized as ollows. In Section II, we provide the necessary deinitions and mathematical results. Section III contains the convexity result or reachable sets o the closed-loop control system (1). In Section IV, we discuss a possible application o our convexity criterion. Section V oers an approach to approximations o reachable sets or some classes o closed-loop and open-loop control systems with bounded right-hand sides. II. PRELIMINARIES We irst provide some relevant deinitions and acts. Let X and Y be two Banach spaces with X Y. We say that the space X is compactly embedded in Y and write X c Y, i v Y cv X or all v X and each bounded sequence in X has a convergent subsequence in Y. We recall a special case o the Sobolev Embedding Theorem (c. [1], [3]) in Proposition 1 and list some interpolation properties o Lebesgue spaces (c. [3]) in Proposition 2: Proposition 1: n (, t ) c L 2 n(, t ). Proposition 2: I 1 p q, then L q n(ω) L p n(ω) and v L p n (Ω) meas(ω) 1/p 1/q v L q n (Ω) v L q n(ω) where 1/ is understood to be. In particular one has L 2 n(, t ) L 1 n(, t ) and, or all unctions y( ) L 2 n(, t ), y( ) L 1 n (,t ) t y( ) L 2 n (,t ) (2) We now consider the concept o a nonexpansive mapping in Hilbert spaces and present a undamental ixed point theorem or such mappings in Proposition 3 (c. [21], [22], [2]. Let C be a subset o a Hilbert space H with norm H. A mapping T : C H is said to be nonexpansive i T (h 1 ) T (h 2 ) H h 1 h 2 H holds true or all h 1, h 2 C. Proposition 3: Let C be a nonempty, closed and convex subset o a Hilbert space H and let T be a nonexpansive mapping o C into itsel. Then the set F (T ) o ixed points o T is nonempty, closed and convex. Now, we return to the given control system (1) or which we introduce the system operator P : U l n (, t ) U l n (, t ) (3) deined by the ollowing ormula ( u(x(t)) P (u( ), x( ))(t) := x + t (x(τ), u(x(τ)))dτ ). (4) Using the Sobolev Embedding Theorem (as stated in Proposition 1) we extend P to the operator P : U l L 2 n(, t ) U l L 2 n(, t ) (5) with P still being given by the right hand side o ormula (4). We now consider the set o admissible eedback controls U l, which is contained in C m(ω), as a subset o the space L 2 m(ω). The ollowing result speciies properties o the set U l. Lemma 1: The set U l is a closed convex subset o the Hilbert space L 2 m(ω). Proo: The set A U o all continuous unctions u( ) with range in U and the set A l o all Lipschitz continuous unctions with Lipschitz constants l u l are both convex subsets o C m (Ω). Thereore, the intersection U l := A U Al is also convex. Because o C m (Ω) L 2 m(ω), the set A l is a closed set in the sense o the sup-norm. Hence A l is also closed in the sense o the norm o the space L m (Ω). Using Proposition 2, we deduce that this set is a closed subset o the space L 2 m(ω). Now let us consider a sequence {u k ( )} o unctions rom U l such that lim u k( ) û( ) L 2 k m (Ω) =, where û( ) L 2 m(ω). Then there exists a subsequence {u s ( )} o {u k ( )} satisying lim u s(x) û(x) R m = s or almost all x Ω (see [2]). The assumption û( ) / U l implies the existence o a set I Ω o positive measure 3276

3 WeC5.2 with û(x) / U or all x I. On the other hand, we have or a ixed x I lim s u s(x) û(x) R m =. Since U is a compact set, û(x) belongs to U or the considered x I, contradicting our assumption. Thus, we obtain û( ) U l showing that the set U l is closed. By Lemma 1, the set U l L 2 n(, t ) is a closed convex subset o the Hilbert space H := L 2 m(ω) L 2 n(, t ). Note that the scalar product and the norm in this space can be introduced as ollows (u 1 ( ), x 1 ( )), (u 2 ( ), x 2 ( )) H := u 1 ( ), u 2 ( ) L 2 m (Ω) + x 1 ( ), x 2 ( ) L 2 n (,t ), (u( ), x( )) H := u( ) 2 L 2 m (Ω) + x( )2 L 2 n (,t ). III. CONVEXITY CRITERIA FOR REACHABLE SETS OF CLOSED-LOOP SYSTEMS We next state and prove our main result concerning the operator P rom (5) and (4) under our standing hypothesis (H). It will be the basis or ormulating suicient conditions or the convexity o the set T (x ) o trajectories and o the reachable set K(t, x ). Theorem 1: Assume that satisies the Lipschitz condition (x 1, u 1 ) (x 2, u 2 ) l 1 x 1 x 2 + l 2 u 1 u 2 x 1, x 2 Ω, u 1, u 2 U. where t l2 + (l 1 + l 2 l) 2 1. Then the operator P o the corresponding system is nonexpansive, and the set F ( P ) o ixed points o P is closed and convex. Proo: Since is a Lipschitz-continuous unction, the initial value problem (1) has a solution and the set F (P ) o ixed points o the operator P is nonempty. Thereore, the set F ( P ) is also a nonempty set. We claim that P is a nonexpansive mapping. To see this, consider with P (u 1 ( ), y 1 ( )) P (u 2 ( ), y 2 ( )) 2 H = u 1 (y 1 ( )) u 2 (y 2 ( )) 2 L 2 m (Ω)+ + δ(τ) dτ 2 L 2 n (,t ) δ(τ) := (y 1 (τ), u 1 (y 1 (τ))) (y 2 (τ), u 2 (y 2 (τ))) where u 1 ( ), u 2 ( ) L 2 m(ω) and y 1 ( ), y 2 ( ) are elements o L 2 n(, t ). Consider the second term o the right-hand side o (6). Thus we obtain (6) δ(τ) dτ 2 L 2 n (,t ) = t δ(τ) dτ 2 R ndt ( t δ(τ) R ndτ ) 2 dt ( t δ(τ) R ndτ ) 2 dt = ( t t δ(τ) R ndτ ) 2. From (7) and rom the Lipschitz condition or the unction it ollows that δ(τ) dτ 2 L 2 n (,t ) t (l 1 + l 2 l) 2 ( y 1 (τ) y 2 (τ) R ndτ) 2 = t (l 1 + l 2 l) 2 y 1 ( ) y 2 ( ) 2 L 1 n (,t ). By Proposition 2, we have the ollowing estimation y 1 ( ) y 2 ( ) 2 L 1 n (,t ) t y 1 ( ) y 2 ( ) 2 L 2 n (,t ). This act and inequality (8) imply δ(τ) dτ 2 L 2 n (,t ) (7) (8) (9) t 2 (l 1 + l 2 l) 2 y 1 ( ) y 2 ( ) 2 L 2 n (,t ). Considering the irst term on the right-hand side o (6) we arrive with Proposition 2 at u 1 (y 1 ( )) u 2 (y 2 ( )) 2 L 2 m (Ω) u 1 ( ) u 2 ( ) 2 L 2 m (Ω) + l t y 1 ( ) y 2 ( ) 2 L 2 n (,t ). (1) Since t l2 + (l 1 + l 2 l) 2 1 we inally deduce rom (6), (9) and (1) P (u 1 ( ), y 1 ( )) P (u 2 ( ), y 2 ( )) H [ u1 ( ) u 2 ( ) 2 L 2 m (Ω) + y 1( ) y 2 ( ) 2 ] 1/2 L 2 n (,t ) = (u 1 ( ), y 1 ( )) (u 2 ( ), y 2 ( )) H. Thus, the introduced operator P is a nonexpansive operator in the Hilbert space H. By Lemma 1, U l L 2 n(, t ) is a nonempty closed and convex subset o H. From Proposition 3 it ollows that the set F ( P ) is closed and convex. Note that Theorem 1 establishes the convexity property o the set o ixed points or the extended system operator P on U l L 2 n(, t ) (see (5) and (4)). As a consequence o this result we also can ormulate the corresponding theorem or the operator P on U l n (, t ) (see (3) and (4)). Theorem 2: Under the assumption o Theorem 1 the set F (P ) is convex. Moreover, the set o trajectories T (x ) = {x u ( ) u( ) U l } 3277

4 WeC5.2 or the initial value problem (1) on [, t ] is also convex. Proo: Evidently, the set F (P ) is nonempty. By Proposition 1, we have Hence n (, t ) L 2 n(, t ) F (P ) = n (, t ) F ( P ). Since n (, t ) is a convex subset o L 2 n(, t ), the set F (P ) is also convex. In act, F (P ) is a subset o the product-space U l n (, t ) and the structure o the operator P deines the structure o the set o ixed points Since F (P ) and U l the set T (x ). F (P ) = U l T (x ) are convex, we obtain the convexity o We now deal with the reachable set K(t, x ) or the closedloop system (1). Our next result is an immediate consequence o the convexity criterion just presented in Theorem 2. Theorem 3: Under the assumption o Theorem 1 the reachable set K(t, x ) or the initial value problem (1) is convex or every t [, t ]. Proo: From Theorem 2 we deduce the convexity o the set T (x ). It means that or x u1 ( ), x u2 ( ) T (x ) with u 1 ( ), u 2 ( ) U l and or ixed λ (, 1) there exists an admissible control u 3 ( ) U l such that x u3 ( ) = λx u1 ( ) + (1 λ)x u2 ( ) T (x ). On the other hand, at a time-instant t [, t ] we have x u1 (t), x u2 (t) K(t, x ). Hence x u3 (t) = λx u1 (t) + (1 λ)x u2 (t) K(t, x ). This shows that the reachable set K(t, x ) is convex or every t [, t ]. We now present two illustrative examples o control systems (1) satisying the Lipschitz conditions rom the main Theorem 1. Example 1: Let us consider an with every component j being a convex unction in (u, x) U R n (not identically equal to + ). In case m j j (x, u) M j, j = 1,..., n holds or all (u, x) in the ball B(, 2 ) o radius 2 around, every j is Lipschitzian on B(, ) with j (x 1, u 1 ) j (x 2, u 2 ) (M j m j ) 3278 or all (u 1, x 1 ), (u 2, x 2 ) B(, ) (c. [12]) implying with ν := max j=1,...,n (M j m j )/ (x 1, u 1 ) (x 2, u 2 ) ν ( u 1 u 2 + x 1 x 2 ). Thereore, the condition t l2 + (l 1 + l 2 l) 2 1 rom Theorem 1 can be written as ollows t l2 + ν 2 (1 + l) 2 1 where B(, ) is taken or Ω. Note that the M j and m j may depend on too. Example 2: Consider the ollowing two-dimensional control system (communicated by W. Schmidt and V. Korobov) ẋ 1 (t) = x 2 (t), ẋ 2 (t) = sin x 1 (t) + u(x(t)), x 1 () = x 2 () =, where x := (x 1, x 2 ) T and t [,.5]. It is easy to see that l 1 = l 2 = 1. The condition t l2 + (l 1 + l 2 l) 2 1 rom Theorem 1 implies 2l 2 + 2l 3. and 7 1 < l. 2 We see that under this condition the reachable set K(t, R 2) o the presented system is convex or every t [,.5]. IV. AN APPLICATION TO CONSTRAINED OPTIMAL CONTROL PROBLEMS Let us now apply the main convexity result o Theorem 2 to the ollowing eedback optimal control problem minimize J(x( )) subject to (1), x( ) A, (11) where J is a bounded, convex and lower semi-continuous objective unctional and A a nonempty, bounded, closed and convex subset o L 2 n(, t ). The given control system (1) is supposed to satisy the conditions o Theorem 1. We consider the optimal control problem (11) on the Hilbert space L 2 n(, t ) with eedback controls rom U l. Note that the class o eedback optimal control problems o type (11) is quite general [6]. For example, the objective unctional J could be given by J(x( )) = x 2 (t)dt. and the abstract restriction x( ) A could arise by a system o inequalities h i (x(t)) or all t [, t ] or convex unctions h s : R n R, s = 1,..., S. It is clear that an optimal control problem does not always have a solution. The question o existence o an optimal eedback solution is generally a delicate one (c. [6], [15]). Let T (x ) A be nonempty. Evidently, problem (11) can be rewritten as an optimization problem over the set C := T (x ) A

5 WeC5.2 o admissible trajectories as ollows: minimize J(x( )) subject to x( ) C. (12) Hereby, the state equation (1) is included into the constraints x( ) C. We claim that (12) is a standard convex optimization problem on a bounded closed convex subset o a Hilbert space (see e.g., [8]). To see this, we note that the set o solutions T (x ) = {x u ( ) u( ) U l } is a closed subset o the space C(, t ) (see [1]). Thereore, this set is also closed in the sense o the norm o L 2 n(, t ). Moreover T (x ) is convex by Theorem 2. The intersection C o the two closed convex sets A and T (x ) is again closed and convex set in the Hilbert space L 2 n(, t ). Since A is bounded, the set C is also bounded. Using well-known existence results rom convex optimization theory (c. [8]) one can establish the ollowing existence result or optimal control problem (11). Theorem 4: Under the conditions o Theorem 2 the optimal control problem (12) with a bounded convex and lower semi-continuous objective unctional J and bounded closed convex set A has at least one optimal solution (u opt ( ), x opt ( )) U l L 2 n(, t ) provided T (x ) A is nonempty. Since n (, t ) L 2 n(, t ) and n (, t ) is convex, the intersection n (, t ) C is also a bounded closed and convex subset o L 2 n(, t ). Thereore, we also obtain the corresponding existence result or problem (11) considered on the space n (, t ). Let {x k ( )} n (, t ) be a minimizing sequence or (12) deined on the space (, t ), more precisely let n lim k J(xk ( )) = n min (,t ) T C J(x( )). It is well-known that a minimizing sequence does not always converges to an optimal solution. The question o creating a convergent minimizing sequence is a central question in the numerical analysis o optimization algorithms (see e.g. [13], [4], [5]). By Proposition 1, each bounded sequence in n (, t ) has a convergent subsequence in L 2 n(, t ). Since {x k ( )} is bounded, we have lim l xl ( ) Argmin W 1,1 n (,t ) T C J(x( )) L 2 n (,t ) = or a subsequence {x l ( )} o {x k ( )}. Thus, by Theorem 4, we deduce the existence o a L 2 n-convergent minimizing sequence {x l ( )} or the optimal control problem (12). V. ON ESTIMATING REACHABLE SETS FOR SYSTEMS WITH BOUNDED RIGHT-HAND SIDES In this section we shall discuss a special classes o closedloop and open-loop systems (1) and (2), namely, systems which satisy the ollowing condition (x, u), g(x, u) K (x, u) R n U, (13) where K is a closed convex subset o R n containing. The right-hand sides and g o control systems are assumed to be continuous in both components. Let us irst ormulate the ollowing auxiliary abstract result. Lemma 2: Let X be a separable Banach space and {S, Σ, µ} be a measurable space with a probability measure µ. Let C X be closed and convex. I q : S C is a µ-measurable unction, then q(τ)µ(dτ) C. Proo: Assume a := q(τ)µ(dτ) / C and let B(a, R) := {ξ X ξ a X < R} be the ball around a with radius R. Evidently, there is a radius R such that the sets C B(a, R) =. Using a Separating Theorem rom convex analysis (c. [8], [2]), we obtain a nontrivial L X with Lξ Lψ or all ξ B(a, R), ψ C. By X we have denoted the (topological) dual space to X. Thereby we have the inequality sup (L(a + Rξ)) = La + R L L(q(τ)), τ S. ξ X 1 and by an integration with respect to µ the corresponding inequality La + R L L(q(τ))µ(dτ). (14) Because o (14) leads to L(q(τ))µ(dτ) = L La + R L La q(τ)µ(dτ) = La, contradicting the act that L is nontrivial. Thereore a = q(τ)µ(dτ) belongs to C. Returning to the control systems o the types (1) or (2) satisying the condition (13), we introduce the ollowing Lebesgue probability measure µ = τ/t on the interval [, t]. Then we apply Lemma 2 to our control systems and compute the state o system (1) (or the state o system (2)) at time t [, t ] as x(t) = x + = x + t t t (x(τ), u(x(τ)))ds = (x(τ), u(x(τ)))µ(dτ) x + tk. 3279

6 WeC5.2 For the open-loop system (2) we have the analogous result x(t) = x + t g(x(τ), u(τ))dτ x + tk. This means that the reachable sets o systems (1) and (2) with initial value x belong to the closed convex set Ω := x + t K. Since x belongs to Ω we have the set Ω as a positively invariant set or the corresponding control system. In particular, this set Ω contains the reachable set o the considered dynamical system. VI. CONCLUDING REMARKS In this paper, we proposed a new convexity criterion or reachable sets or a class o closed-loop control systems. This suicient condition is based on a general convexity result or solution sets o the corresponding nonlinear dynamical systems. Convexity o the set o trajectories makes it also possible to study some constrained eedback optimal control problems. For some amilies o closed-loop and open-loop control systems with bounded right-hand sides we construct an over estimation o the examined reachable set, i.e. we provide a set that contains the reachable set o the dynamical system under consideration. [15] J. Macki and A. Strauss, Introduction to Optimal Control Theory, Springer, New York, NY; [16] B.T. Polyak, Convexity o nonlinear image o a small ball with applications to optimization, Set-Valued Analysis, vol. 9, pp , 21. [17] B.T. Polyak, S.A. Nazin, C. Durieu and E. Walter, Ellipsoidal parameter or state estimation under model uncertainty, Automatica J. IFAC, vol. 4, 24, pp [18] B.T. Polyak, Convexity o the reachable set o nonlinear systems under L 2 bounded control,dynamics o Continuous, Discrete and Impulsive Systems, vol. 11, 24, pp [19] E. Sontag, Mathematical Control Theory, Springer, New York, NY; [2] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan; 2. [21] E.H. Zarantonello, Projection on convex sets in Hilbert space and spectral theory, in Contributions to Nonlinear Functional Analysis, Academic Press, New York, NY; 1971, pp [22] E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems, Springer, New York, NY; 199. ACKNOWLEDGMENTS The authors would like to thank Pro. B.T. Polyak or critical comments and helpul suggestions. REFERENCES [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, NY; [2] C.D. Aliprantis and K.C. Border, Ininite Dimensional Analysis, Springer, New York, NY; [3] K. Atkinson and W. Han, Theoretical Numerical Analysis, Springer, New York, NY; 25. [4] V. Azhmyakov, A numerically stable method or convex optimal control problems, Journal o Nonlinear and Convex Analysis, vol. 5, 24, pp [5] V. Azhmyakov and W. Schmidt, Approximations o relaxed optimal control problems, Journal o Optimization Theory and Applications, vol. 13, 26, pp [6] L.D. Berkovitz, Optimal Control Theory, Springer, New York, NY; [7] F.H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, PA; 199. [8] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland; [9] H.O. Fattorini, Ininite Dimensional Optimization and Control Theory, Cambridge University Press, Cambridge, UK; [1] A.F. Filippov, Dierential Equations with Discontinuous Right-Hand Sides, Kluwer Academic Publishers, Dordrecht, Holland; [11] S. Geist, G. Reissig and J. Raisch, On the convexity o reachable sets o nonlinear dynamic systems - an important step in generating discrete abstractions o continuous systems, in Proceedings o 11th IEEE International Conerence on Methods and Models in Automation and Robotics, Miedzyzdroe, Poland, 25, pp [12] J.-B. Hiriart-Urruty and C. Lemarechal, Fundamentals o Convex Analysis, Springer, Berlin, Germany; 21. [13] A. Kaplan and R. Tichatschke, Stable Methods or Ill-Posed Problems, AkademieVerlag, Berlin, Germany, [14] A.B. Kurzhanski and P. Varaiya, Ellipsoidal techniques or reachability analysis, in Computation and Control, LNCS 179, Springer, New York, NY; 2, pp