Normality of Necessary Optimality Conditions for Calculus of Variations Problems with State Constraints

TL;DR

This paper establishes conditions under which necessary optimality conditions in calculus of variations with state constraints are normal, focusing on the interior of the Clarke tangent cone and different assumptions for local and global minimizers.

Contribution

It introduces a constraint qualification based on the Clarke tangent cone's interior and proves normality of optimality conditions for both local and global minimizers under new assumptions.

Findings

Normality of optimality conditions holds when the Clarke tangent cone's interior is non-empty.

Different assumptions are required for local versus global minimizers.

The results apply to non-autonomous problems with Lipschitz and convex Lagrangians.

Abstract

We consider non-autonomous calculus of variations problems with a state constraint represented by a given closed set. We prove that if the interior of the Clarke tangent cone of the state constraint set is non-empty (this is the constraint qualification that we suggest here), then the necessary optimality conditions apply in the normal form. We establish normality results for (weak) local minimizers and global minimizers, employing two different approaches and invoking slightly diverse assumptions. More precisely, for the local minimizers result, the Lagrangian is supposed to be Lipschitz with respect to the state variable, and just lower semicontinuous in its third variable. On the other hand, the approach for the global minimizers result (which is simpler) requires the Lagrangian to be convex with respect to its third variable, but the Lipschitz constant of the Lagrangian with respect to the state variable might now depend on time.