Second order sufficient conditions for optimal control problems with non-unique minimizers

This paper concerns second order sufficient conditions of optimality, for optimal control problems whose minimizers are not locally unique, i.e. there are distinct admissible controls arbitrarily close to the nominal optimal control which also achieve the minimum cost. Problems of this nature naturally arise in periodic control, shape optimization and other areas. Here, there exist transformations of the feasible control functions, such as a time translation of a periodic control function, that preserve feasibility and the value of the cost. Standard sufficient conditions which, when they apply, give the information not only that a putative minimizer is locally optimal, but also that it is locally unique, can never be directly applied in such circumstances. We provide a framework for deriving sets of sufficient conditions which, unlike the classical ones, cover problems for which optimal controls are not locally unique. We illustrate the application of the new framework with a numerical example.