Explicit bounds for Lipschitz constant of solution to basic problem in calculus of variations

TL;DR

This paper provides explicit bounds for the Lipschitz constant of solutions to calculus of variations problems by leveraging control theory, enabling complexity analysis of related numerical methods.

Contribution

It introduces a novel explicit estimate for the Lipschitz constant using a control-theoretic approach, connecting calculus of variations with time-optimal control.

Findings

Derived explicit Lipschitz bounds for solutions

Applied bounds to complexity analysis of path-following methods

Connected calculus of variations with control theory techniques

Abstract

In this paper we present explicit estimate for Lipschitz constant of solution to a problem of calculus of variations. The approach we use is due to Gamkrelidze and is based on the equivalence of the problem of calculus of variations and a time-optimal control problem. The obtained estimate is used to compute complexity bounds for a path-following method applied to a convex problem of calculus of variations with polyhedral end-point constraints.