Calculus of variations and optimal control; optimization Theorem

Jacobi condition

Builds on Integral functional with a Lagrangian density Weak and strong extremum Second variation Jacobi equation Euler–Lagrange equation

Statement

Let $a < b$ and let $L(x, y, p)$ be a $C^3$ function on $[a, b] \times \R \times \R$ . Let $J(y) = \int_a^b L(x, y, y') \, dx$ on $C^1([a, b])$ , with fixed values $y(a)$ and $y(b)$ . Let $y_0 \in C^2([a, b])$ solve the Euler–Lagrange equation of $J$ . Assume the strengthened Legendre condition $L_{pp}(x, y_0(x), y_0'(x)) > 0$ for all $x \in [a, b]$ . A point $c \in (a, b]$ is conjugate to $a$ if a solution $h \ne 0$ of the Jacobi equation along $y_0$ has $h(a) = h(c) = 0$ .

(Necessary condition.) If $y_0$ is a weak local minimiser of $J$ , then no point of the open interval $(a, b)$ is conjugate to $a$ .
(Sufficient condition.) If no point of $(a, b]$ is conjugate to $a$ , then $y_0$ is a strict weak local minimiser of $J$ .

Remarks

Write the second variation as $\int_a^b (P h'^2 + Q h^2) \, dx$ , as in the Jacobi equation. Without conjugate points in $(a, b]$ , there is a Jacobi solution $u > 0$ on all of $[a, b]$ . Then $w = -P u' / u$ solves $w' = w^2 / P - Q$ . Adding the zero term $\int_a^b (w h^2)' \, dx$ turns the second variation into $\int_a^b P (h' + w h / P)^2 \, dx$ , which is positive for $h \ne 0$ . For the necessary part, a Jacobi solution on $[a, c]$ extended by $0$ gives zero second variation, and smoothing its corner makes the second variation negative.

Example: for $J(y) = \int_0^b (y'^2 - y^2) \, dx$ and $y_0 = 0$ , the Jacobi equation is $h'' + h = 0$ . The solution with $h(0) = 0$ is $h = \sin x$ , so the first conjugate point is $\pi$ . For $b < \pi$ , the extremal $y_0 = 0$ is a strict weak local minimiser. For $b > \pi$ , it is not: with $\varphi(x) = \sin(\pi x / b)$ one gets $J(t\varphi) = \frac{t^2 b}{2}(\pi^2 / b^2 - 1) < 0$ for $t \ne 0$ .

The strengthened Legendre condition is needed in both parts. Conjugate points here generalise conjugate points along geodesics in Riemannian geometry.

Going deeper

For a student

The Legendre condition controls the second variation for test functions that oscillate fast. The Jacobi condition controls it for all test functions, and it reduces this question to the zeros of one solution of a linear ODE. Together with the Euler–Lagrange equation and the strengthened Legendre condition, it gives the classical sufficient condition for a weak minimum. For a strong minimum one adds the Weierstrass condition on the excess function, and the proof embeds the extremal in a field of extremals.

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Euler–Lagrange equation concept
Integral functional with a Lagrangian density concept
Jacobi equation concept
Second variation concept
Weak and strong extremum concept
Jacobi condition statement

Tag	`ZRV9T8DD05`
URI	`m2://ZRV9T8DD05`
Kind	statement, theorem
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MSC	49K05
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About	Jacobi equation, Second variation
Uses	Integral functional with a Lagrangian density, Weak and strong extremum, Second variation, Jacobi equation, Euler–Lagrange equation