Optimal Control with State Space Constraints

Abstract

Necessary and sufficient conditions for optimality in optimal control problems with state space constraints are reviewed with emphasis on geometric aspects.

Problem Formulation and Terminology

Many practical problems in engineering or of scientific interest can be formulated in the framework of optimal control problems with state space constraints. Examples range from the space shuttle reentry problem in aeronautics (Bonnard et al. 2003) to the problem of minimizing the base transit time in bipolar transistors in electronics (Rinaldi and Schättler 2003).

An optimal control problem with state space constraints in Bolza form takes the following form: minimize a functional

$$J(u) =\displaystyle\int _{ t_{0}}^{T}L(t,x(t),u(t))dt + \Phi (T,x(T))$$

over all Lebesgue measurable functions u, u : [t ₀, T] → U that take values in a control set $U \subset\mathbb{R}^{m}$, subject to the dynamics

$$\dot{x}(t) = F(t,x(t),u(t)),\qquad x(t_{0}) = x_{0},$$

terminal constraints

$$\Psi...