Convexity of the Set of Fixed Points Generated by Some Control Systems

convexity results for nonlinear mappings we refer to 10 , for some applications to optimization and optimal control to 10, 11 . For an analysis of reachable sets of dynamical systems in an abstract or hybrid setting, see also 8 . While the main topic of our paper is the estimation of reachable sets for closed-loop systems of type 1.1 , we also consider open-loop control systems: ẋ t g x t , u t , a.e. on [ 0, tf ] , x 0 x0, 1.5 where g is a Lipschitz continuous function in both components andwhere u t belongs toU for t ∈ 0, tf . Let U : {u · ∈ Lm 0, tf | u t ∈ U} be the space of admissible control signals for system 1.5 . Here, Lm 0, tf denotes the Lebesgue space of all square-integrable functions u : 0, tf → R with the corresponding norm. It is assumed that for every admissible timedependent control u · ∈ U system 1.5 has a unique solution x · ∈ W n 0, tf . As for the closed-loop system 1.1 , we will obtain estimates for the reachable sets of 1.5 provided the right-hand sides are bounded. The paper is organized as follows. In Section 2, we provide the necessary definitions and mathematical results. Section 3 contains the convexity result for the sets of trajectories and for reachable sets of the closed-loop control system 1.1 . Section 4 discusses overapproximation of reachable sets for some classes of closed-loop and open-loop control systems with bounded right-hand sides. We also use some techniques from optimal control theory to obtain general approximations of convex reachable sets under consideration. In Section 5, we discuss a possible application of our convexity criterion to optimal control problems with constraints. Section 6 summarizes the paper. 2. Preliminary Results We first provide some relevant definitions and facts. LetX and Y be two Banach spaces with X ⊂ Y. We say that the space X is compactly embedded in Y and write X↪→cY, if ‖v‖Y ≤ c‖v‖X for all v ∈ X and each bounded sequence inX has a convergent subsequence in Y. We recall a special case of the Sobolev Embedding Theorem cf. 5, 6 in Proposition 2.1 and list some interpolation properties of Lebesgue spaces cf. 6 in Proposition 2.2. Proposition 2.1. It holds that W n 0, tf ↪→cLn 0, tf . Proposition 2.2. If 1 ≤ p ≤ q ≤ ∞, then Lqn 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 , 2.1 where 1/∞ is understood to be 0. In particular one has Ln 0, tf ⊂ Ln 0, tf and, for all functions y · ∈ Ln 0, tf , ∥y · ∥∥ L 1 n 0,tf ≤ √ tf ∥y · ∥∥ L 2 n 0,tf . 2.2 4 Journal of Applied Mathematics We now consider the concept of a nonexpansive mapping in Hilbert spaces and present a fundamental fixed point theorem for such mappings in Proposition 2.3 cf. 1, 2, 12 . Let C be a subset of a Hilbert space H with norm ‖ · ‖H. A mapping T : C → H is said to be nonexpansive if ‖T h1 − T h2 ‖H ≤ ‖h1 − h2‖H 2.3 holds true for all h1, h2 ∈ C. Proposition 2.3. Let C be a nonempty, closed, and convex subset of a Hilbert space H and let T be a nonexpansive mapping of C into itself. Then the set F T of fixed points of T is nonempty, closed, and convex. Now, we return to the given control system 1.1 for which we introduce the system operator P : U l ×W n ( 0, tf ) −→ U l ×W n ( 0, tf ) 2.4 defined by the following formula: P u · , x · t : ⎛ ⎜⎜ ⎝ u x t