Calculus of variations and optimal control; optimization Basic notions of the calculus of variations Theorem

Transversality conditions for variable endpoints

Builds on Integral functional with a Lagrangian density Euler–Lagrange equation First variation Natural boundary condition Partial derivative

Statement

Let $L \colon \R \times \R \times \R \to \R$ , written $L(x, z, p)$ , be of class $C^2$ , with partial derivative $L_p$ in $p$ . Let $\psi \colon \R \to \R$ be of class $C^1$ , and fix $a, A \in \R$ . Consider pairs $(y, b)$ with $b > a$ , $y \in C^2([a, b])$ , $y(a) = A$ and $y(b) = \psi(b)$ , and set $J(y, b) = \int_a^b L(x, y, y') \, dx$ . Suppose that $(y_0, b_0)$ minimises $J$ among all such pairs. Then:

$y_0$ solves the Euler–Lagrange equation $\frac{d}{dx} L_p(x, y_0, y_0') = L_z(x, y_0, y_0')$ on $[a, b_0]$ ;
at the free end, the transversality condition $\Big[ L(x, y_0, y_0') + \big(\psi'(x) - y_0'(x)\big) L_p(x, y_0, y_0') \Big]_{x = b_0} = 0$ holds.

Remarks

The proof computes the first variation of $J$ along a family $(y_t, b_t)$ of competitors with $y_t(b_t) = \psi(b_t)$ . Variations that fix both ends give the Euler–Lagrange equation. Moving the end along the curve $y = \psi(x)$ gives a boundary term, $L \, \delta b + L_p (\delta y - y' \delta b)$ at $b_0$ , with $\delta y = \psi' \delta b$ ; it must vanish, which is the transversality condition.

Special cases. If instead $b$ is fixed and $y(b)$ is free, the end moves on the vertical line $x = b$ , and the same computation gives the natural boundary condition $L_p = 0$ at $b$ . If $L(x, z, p) = f(x, z) \sqrt{1 + p^2}$ with $f(b_0, y_0(b_0)) \ne 0$ , the condition simplifies to $\psi'(b_0) \, y_0'(b_0) = -1$ : the extremal meets the curve $y = \psi(x)$ at a right angle.

Example: for $f = 1$ , $J$ is arc length, and the condition says that the shortest segment from the point $(a, A)$ to a curve meets the curve orthogonally.

Going deeper

For a student

The transversality condition replaces the missing boundary value. The Euler–Lagrange equation has order two and one end is fixed, so one more condition fixes the extremal; then $y(b_0) = \psi(b_0)$ fixes the unknown $b_0$ . In optimal control, the Pontryagin maximum principle has transversality conditions of the same kind on the costate at a free final state or final time.

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Euclidean space object
Euler–Lagrange equation concept
First variation concept
Function concept
Integral functional with a Lagrangian density concept
Natural boundary condition concept
Function of several variables concept
Partial derivative concept
Transversality conditions for variable endpoints statement

Tag	`DAEHVRJVSD`
URI	`m2://DAEHVRJVSD`
Kind	statement, theorem
Writing	W0 stub
MSC	49K05
Status	This copy of the site has no build data. Run `m2 status DAEHVRJVSD`.
About	Euler–Lagrange equation, Natural boundary condition
Uses	Integral functional with a Lagrangian density, Euler–Lagrange equation, First variation, Natural boundary condition, Partial derivative

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