Calculus of variations and optimal control; optimization Definition

Admissible function

Builds on Integral functional with a Lagrangian density Continuously differentiable function

Definition

Let $a < b$ and $A, B \in \R$ , and let $J(y) = \int_a^b L(x, y, y') \, dx$ be an integral functional with a continuous Lagrangian $L \colon [a, b] \times \R \times \R \to \R$ . In the fixed-endpoint problem for $J$ , a function $y \colon [a, b] \to \R$ is admissible if:

$y \in C^1([a, b])$ , that is, $y$ is continuously differentiable on $[a, b]$ ;
$y(a) = A$ ;
$y(b) = B$ .

The set $\mathcal{A}$ of admissible functions is the admissible class. More generally, the admissible class of a variational problem is the set of functions in the domain of the functional that satisfy all its constraints. Its elements are the admissible functions.

Remarks

Example: for $a = 0$ , $b = 1$ , $A = 0$ , $B = 1$ , the function $y(x) = x^2$ is admissible: it is $C^1$ , $y(0) = 0$ and $y(1) = 1$ . Non-example: for $a = 0$ , $b = 1$ , $A = B = 1$ , the function $y(x) = \abs{x - 1/2} + 1/2$ satisfies $y(0) = y(1) = 1$ but is not differentiable at $1/2$ , so it fails clause 1. Many texts use the larger class of piecewise $C^1$ functions, which admits such broken curves. If $y$ is admissible and $\eta \in C^1([a, b])$ has $\eta(a) = \eta(b) = 0$ , then $y + t\eta$ is admissible for every $t \in \R$ ; such $\eta$ are the admissible variations.

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Continuously differentiable function concept
Integral functional with a Lagrangian density concept
Admissible function definition

Tag	`993HPHG9VF`
URI	`m2://993HPHG9VF`
Kind	definition
Writing	W0 stub
MSC	49K05, 49J05
Defines	Admissible function
Uses	Integral functional with a Lagrangian density, Continuously differentiable function

Admissible function

Definition

Remarks

Reading path

Further reading

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