A variant of nonsmooth maximum principle for state constrained problems

TL;DR

This paper develops a new variant of the nonsmooth maximum principle tailored for state-constrained problems, providing a sufficient optimality condition for linear convex cases and extending classical results to nonsmooth contexts.

Contribution

It introduces a novel nonsmooth maximum principle for state-constrained problems that includes Weierstrass conditions and applies to linear convex problems, bridging a gap in existing optimality conditions.

Findings

The new principle coincides with classical results in smooth cases.

It provides a sufficient condition for optimality in linear convex problems.

The approach extends nonsmooth maximum principles to state-constrained scenarios.

Abstract

We derive a variant of the nonsmooth maximum principle for problems with pure state constraints. The interest of our result resides on the nonsmoothness itself since, when applied to smooth problems, it coincides with known results. Remarkably, in the normal form, our result has the special feature of being a sufficient optimality condition for linearconvex problems, a feature that the classical Pontryagin maximum principle had whereas the nonsmooth version had not. This work is distinct to previous work in the literature since, for state constrained problems, we add the Weierstrass conditions to adjoint inclusions using the joint subdifferentials with respect to the state and the control. Our proofs use old techniques developed in [16], while appealing to new results in [7].