Free End-Time Optimal Control Problems: Conditions for the Absence of an Infimum Gap

This paper concerns free end-time optimal control problems, in which the dynamic constraint takes the form of a controlled differential inclusion. Such problems may fail to have a minimizer. Relaxation is a procedure for enlarging the domain of an optimization problem to guarantee existence of a minimizer. In the context of problems studied here, the standard relaxation procedure involves replacing the velocity sets in the original problem by their convex hulls. It is desirable that the original and relaxed versions of the problem have the same infimum cost. For then we can obtain a sub-optimal state trajectory, by obtaining a solution to the relaxed problem and approximating it. It is important, therefore, to investigate when the infimum costs of the two problems are the same; for otherwise the above strategy for generating sub-optimal state trajectories breaks down. We explore the relation between the existence of an infimum gap and abnormality of necessary conditions for the free-time problem. Such relations can translate into verifiable hypotheses excluding the existence of an infimum gap. Links between existence of an infimum gap and normality have previously been explored for fixed end-time problems. This paper establishes, for the first time, such links for free end-time problems.