Second-order sufficient optimality conditions in the calculus of variations
William W. Hager
TL;DR
The paper addresses the problem of efficiently certifying that a stationary point $y^*$ of $F(y)=\int_0^1 f(y'(x), y(x), x)\,dx$ with $y(0)=y(1)=0$ is a strict local minimizer. It unifies classical second-order conditions—Legendre-type Hessian positivity, Riccati equation, and conjugate-point criteria—by proving their equivalence under mild regularity, and introduces a new, highly practical test: if the solution of the linear Euler equation $\frac{d}{dx}(P(x)u')=Q(x)u$ with $u(0)=0$, $u'(0)=1$ remains positive on $(0,1]$, then $y^*$ is a strict local minimizer. The key contributions include an equivalence theorem among five conditions (positive Euler-solution, bounded Riccati solution, a Gamma lower bound, absence of conjugate points, and positive initial-value solution), and the demonstration that the positivity test reduces verification to a forward integration of a linear ODE. This provides a unified theoretical framework with a simple, numerically friendly criterion for certifying minimality in calculus of variations problems, with direct implications for efficient algorithmic checks.
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.