Calculus of variations and optimal control; optimization Definition

Field of extremals

Builds on Integral functional with a Lagrangian density Euler–Lagrange equation Continuously differentiable function

Definition

Let $L(x, y, p)$ be a $C^2$ function on $\R \times \R \times \R$ , with integral functional $J(y) = \int L(x, y, y') \, dx$ . Let $D \subseteq \R^2$ be a simply connected open set. A field of extremals of $J$ on $D$ is a family of functions $y = \varphi(x, c)$ , with $c$ in an open interval $I$ , such that:

for each $c \in I$ , the function $x \mapsto \varphi(x, c)$ solves the Euler–Lagrange equation of $J$ ;
through each point $(x, y) \in D$ passes exactly one curve $y = \varphi(x, c)$ of the family;
the parameter $c(x, y)$ of that curve is a continuously differentiable function on $D$ .

The slope function of the field is $p(x, y) = \partial_x \varphi(x, c(x, y))$ . An extremal $y_0$ is embedded in the field if $y_0 = \varphi(\cdot, c_0)$ for some $c_0 \in I$ .

Remarks

Example: for $L = \sqrt{1 + p^2}$ the extremals are straight lines. The family $\varphi(x, c) = x + c$ , $c \in \R$ , is a field on $D = \R^2$ , with $c(x, y) = y - x$ and slope function $p = 1$ .

Non-example: the family $\varphi(x, c) = c x$ , $c \in \R$ , of lines through the origin is not a field on $\R^2$ . All its curves pass through $(0, 0)$ , and no curve passes through the points $(0, y)$ with $y \ne 0$ . On the half-plane $x > 0$ it is a field, with $c = y / x$ ; it is then a central field, whose curves all start at one point on the boundary.

Let an extremal $y_0$ on $[a, b]$ satisfy $L_{pp} > 0$ along it, with no point conjugate to $a$ in $(a, b]$ . Then it can be embedded in a field on a neighbourhood of its graph.

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Continuously differentiable function concept
Euler–Lagrange equation concept
Integral functional with a Lagrangian density concept
Field of extremals definition

Tag	`PPTX47481A`
URI	`m2://PPTX47481A`
Kind	definition
Writing	W0 stub
MSC	49K05
Defines	Field of extremals
Uses	Integral functional with a Lagrangian density, Euler–Lagrange equation, Continuously differentiable function

Field of extremals

Definition

Remarks

Reading path

Further reading

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