Second-order conditions for non-uniformly convex integrands: quadratic growth in $L^1$

TL;DR

This paper develops second-order optimality conditions for a class of non-uniformly convex, non-smooth integral functionals extended to measures, with applications to non-smooth optimal control.

Contribution

It introduces no-gap second-order conditions for non-uniformly convex integrands, extending the functional to measures and involving derivatives with integrals on lower-dimensional manifolds.

Findings

Second-order conditions are established for non-uniformly convex integrals.

The integral functional is extended to the space of measures.

Applications to non-smooth optimal control problems are demonstrated.

Abstract

We study no-gap second-order optimality conditions for a non-uniformly convex and non-smooth integral functional. The integral functional is extended to the space of measures. The obtained second-order derivatives contain integrals on lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which is an integral functional on the space of continuous functions. Applications to non-smooth optimal control problems are given.