Optimal Control Problem

Abstract

Consider a control system of the form $$\dot q = fu(q),q \in M,u \in U \subset {R^m}.$$ (10.1) Here M is, as usual, a smooth manifold, and U an arbitrary subset of ℝm. For the right-hand side of the control system, we suppose that: $$q \mapsto fu(q)$$ (10.2) is a smooth vector field on M for any fixed u ∈ U, $$(q,u) \mapsto {f_u}(q)$$ (10.3) is a continuous mapping for q ∈ M, u ∈ Ū, and moreover, in any local coordinates on M $$(q,u) \mapsto \frac{{\partial {f_u}}}{{\partial q}}(q)$$ (10.4) is a continuous mapping for q ∈ M, u ∈ Ū. Admissible controls are measurable locally bounded mappings \(u:t \mapsto u(t) \in U\)Substitute such a control u = u(t) for control parameter into system (10.1), then we obtain a nonautonomous ODE \(\dot q = fu(q)\). By the classical Caratheodory’s Theorem, for any point q 0 ∈ M, the Cauchy problem $$\dot q = {f_u}(q),q(0) = {q_0},$$ (10.5) has a unique solution, see Subsect. 2.4.1. We will often fix the initial point q 0 and then denote the corresponding solution to problem (10.5) as q u (t).