Second Order Analysis for Optimal Control Problems: Improving Results Expected From Abstract Theory

TL;DR: It is proved that, in contrast to a widespread common belief, the standard second-order conditions formulated for these control problems imply strict local optimality of the controls not only in the sense of L^\infty, but also of $L^2$.

Abstract: An abstract optimization problem of minimizing a functional on a convex subset of a Banach space is considered. We discuss natural assumptions on the functional that permit establishing sufficient second-order optimality conditions with minimal gap with respect to the associated necessary ones. Though the two-norm discrepancy is taken into account, the obtained results exhibit the same formulation as the classical ones known from finite-dimensional optimization. We demonstrate that these assumptions are fulfilled, in particular, by important optimal control problems for partial differential equations. We prove that, in contrast to a widespread common belief, the standard second-order conditions formulated for these control problems imply strict local optimality of the controls not only in the sense of $L^\infty$, but also of $L^2$.