Variants of the Kuhn–Tucker Sufficient Conditions in Cones of Nonnegative Functions

Abstract

Second-order sufficient conditions of the Kuhn–Tucker type are proved for certain constrained minimization problems on sets of nonnegative $\mathcal{L}^p $ functions, with $p \in [2,\infty ]$. The objective functions for these problems have specially structured bilinear second Gateaux differentials that are bounded with respect to the $\mathcal{L}^2 $ norm and vary continuously with respect to the $\mathcal{L}^2 $ norm on $\mathcal{L}^p $. Structure and smoothness conditions of this sort are satisfied by nontrivial classes of constrained-input Bolza optimal control problems, and in this context, the associated Kuhn–Tucker sufficient conditions yield a partial extension of the classical weak sufficiency theory in the calculus of variations.