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 suffcient 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 fnite-dimensional optimization. We demonstrate that these assumptions are fulfilled, in particular, by important optimal control problems for partial dfferential 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, but also of L2 .

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