Optimization Techniques for State-Constrained Control and Obstacle Problems1,2,3

The design of control laws for systems subject to complex state constraints still presents a significant challenge. This paper explores a dynamic programming approach to a specific class of such problems, that of reachability under state constraints. The problems are formulated in terms of nonstandard minmax and maxmin cost functionals, and the corresponding value functions are given in terms of Hamilton-Jacobi-Bellman (HJB) equations or variational inequalities. The solution of these relations is complicated in general; however, for linear systems, the value functions may be described also in terms of duality relations of convex analysis and minmax theory. Consequently, solution techniques specific to systems with a linear structure may be designed independently of HJB theory. These techniques are illustrated through two examples.