Continuous feedback control in evasion problems with nonconvex target set

Abstract

We compare program controls and strategies continuous in the phase vector in evasion problems with disturbance. The target set is specified in the function space of trajectories. In particular, this setting covers the case of a terminal set in a finite-dimensional space at the terminal time. The target set is not required to be convex. The problem is formalized as a differential game with a nonlinear equation. The players’ controls are chosen in given sets of integrable functions. We show that if evasion can be guaranteed with the help of a continuous strategy, then it can be guaranteed with the help of a program control as well. The continuous strategies that we consider can contain a deviation of the argument. We assume that some sets constructed in the problem in the space of continuous functions are acyclic (their homology groups are degenerate). The present paper deals with the theory of positional differential games [1–5] and is motivated by the results in [6, 7], where the case of linear dynamics and a convex terminal set was comprehensively studied. It is known that the specific properties of a continuous feedback also manifest themselves in the stabilization of a nonlinear control system (see [8–10] and the bibliography therein). Our proofs are based on the Eilenberg–Montgomery fixed point theorem [11, 12]. One can assume that homology groups are obtained, for example, with the help of the Vietoris construction. In the present paper, we impose no conditions on the growth rate of the right-hand side of the differential equation. Instead, we assume that the closure in the space of continuous functions of the set of all solutions of the problem for all possible pairs of the players’ admissible controls is a compact set. This condition holds in many typical situations. In what follows, we present also some considerations as to what the set of strategies on which the strategy is defined must be so as to ensure that a theorem of this type can be proved. The generalized Tietze continuation theorem proves useful in this connection. If the independent variable can be treated as time, then one can consider a fixed initial point. Below, we consider the more general case of boundary conditions. Similar relations appear, for example, in some conflict control problems for rod heating [13]. In this case, a solution can be treated as a stationary temperature distribution along the rod. Boundary value problems with similar conditions were considered in [14, 15]. The dynamics of a control system is not necessarily linear. We also assume that there may be a nonlinear dependence on the control of the evader. However, for simplicity, we assume that the differential equations are linear in the pursuer’s control. This permits one to show (with the help of weak topology by analogy with [6]) that the graph of the corresponding multimapping is closed. However, the linearity assumption can be avoided. In this case, one should prove closedness by imposing some conditions, as in [16]. At the end of the present paper, we use the results to study a specific control system. We use the following notation: R (where n ≥ 1 is an integer) is the space of n-vectors (columns) with given norm |·|n; Rn×n is the space of n×nmatrices with real entries whose norm |·|n×n is coordinated with the vector norm (i.e., |Ab|n ≤ |A|n×n|b|n, where A ∈ Rn×n and b ∈ R are arbitrary); C is the space of continuous functions; Lr (where r ≥ 1 is a real number) is the space of Lebesgue measurable functions with r-integrable absolute value [|x(t)|n or |x(t)|n×n is integrable for functions x(t) ranging in the space of n-vectors or n× n matrices]; AC is the space of absolutely continuous functions. The symbol cl co stands for the closed convex hull, and cl stands for the closure.