Exact Penalizations for Optimal Control Problems

A novel technique to solve optimal control problems with state constraints is proposed. We exploit the theory of exact penalty functions, used in mathematical programming, to construct a systematic procedure to transform two classes of problems with state constraints to equivalent penalized unconstrained problems. We focus on a special class of systems with as many states as controls and subject to a set of equality constraints, which reduces the "control authority,"both in the case of linear and nonlinear control-affine systems. Two approaches are proposed, and differences, advantages, and drawbacks of such approaches are discussed. Three simple examples illustrate the theory.