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 ℝ^m. 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 Carathéodory’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).