Generic Uniqueness and Conjugate Points for Optimal Control Problems

Abstract.

The paper is concerned with an optimal control problem on [Formula: see text], where the dynamics is linear w.r.t. the control functions. For a terminal cost [Formula: see text] in a dense [Formula: see text] set of [Formula: see text] (i.e., in a countable intersection of open dense subsets), two main results are proved. Namely: the set [Formula: see text] of conjugate points is closed, with locally bounded [Formula: see text]-dimensional Hausdorff measure. Moreover, the set of initial points [Formula: see text], which admit two or more globally optimal trajectories, is contained in the union of a locally finite family of embedded manifolds. In particular, the value function is continuously differentiable on an open, dense subset of [Formula: see text].