Second-order sufficient optimality conditions in the calculus of variations

TL;DR

This paper demonstrates the equivalence of classic second-order optimality conditions in the calculus of variations and introduces a new, easy-to-apply second-order condition based on integrating a linear initial value problem.

Contribution

It establishes the equivalence of existing conditions and proposes a novel, simplified second-order optimality criterion for the calculus of variations.

Findings

Classic second-order conditions are equivalent

New second-order condition is easy to verify

Condition involves integrating a linear initial value problem

Abstract

Some classic second-order sufficient optimality conditions in the calculus of variations are shown to be equivalent, while also introducing a new equivalent second-order condition which is extremely easy to apply: simply integrate a linear second-order initial value problem and check that the solution is positive over the problem domain.