Calculus of variations and optimal control; optimization Second variation and sufficient conditions Theorem

Legendre condition

Builds on Integral functional with a Lagrangian density Weak and strong extremum Second variation Positive semidefinite matrix

Statement

Let $a < b$ , and let $L \colon [a, b] \times \R^n \times \R^n \to \R$ , written $L(x, y, p)$ , be of class $C^2$ . Let $J(y) = \int_a^b L(x, y(x), y'(x)) \, dx$ be the integral functional on $C^1([a, b]; \R^n)$ . Let $y_0 \in C^1([a, b]; \R^n)$ be a weak local minimiser of $J$ among functions with the same values at $a$ and $b$ . Then for every $x \in [a, b]$ the $n \times n$ matrix $L_{pp}(x, y_0(x), y_0'(x))$ of second derivatives in $p$ is positive semidefinite. For $n = 1$ this reads $L_{pp}(x, y_0(x), y_0'(x)) \ge 0$ .

Remarks

The second variation at a minimiser is nonnegative: $\int_a^b \big( h'^T L_{pp} h' + 2 h^T L_{yp} h' + h^T L_{yy} h \big) \, dx \ge 0$ for all $h \in C^1$ with $h(a) = h(b) = 0$ . If $L_{pp}$ had a negative direction $v$ at some interior point $x_0$ , take $h$ to be a small zigzag in direction $v$ near $x_0$ , with slopes $\pm 1$ and tiny height. Then the term with $h'^2$ dominates the others, because $h$ itself is small, and the second variation becomes negative. The endpoints follow by continuity.

Example: for $J(y) = \int_0^1 (1 - y'^2) \, dx$ one has $L_{pp} = -2 < 0$ everywhere, so $J$ has no weak local minimiser with fixed end values.

For a maximiser the matrix is negative semidefinite. The condition is only necessary: the extremal $y_0 = 0$ of $\int_0^{2\pi} (y'^2 - y^2) \, dx$ satisfies $L_{pp} = 2 > 0$ but is not a minimiser. The strengthened Legendre condition asks that $L_{pp}$ be positive definite; together with the Jacobi condition it gives a sufficient condition for a weak minimum.

Going deeper

For a student

The Legendre condition is the second necessary condition of the classical theory, after the Euler–Lagrange equation. It only involves the highest-order part of the second variation, the coefficient of $h'^2$ . The full sign of the second variation is controlled by the Jacobi equation and conjugate points. In several variables its analogue is the Legendre–Hadamard condition, and in optimal control it becomes the Legendre–Clebsch condition on the Hamiltonian.

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Integral functional with a Lagrangian density concept
Positive semidefinite matrix concept
Second variation concept
Weak and strong extremum concept
Legendre condition statement

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About	Second variation
Uses	Integral functional with a Lagrangian density, Weak and strong extremum, Second variation, Positive semidefinite matrix