Calculus of variations and optimal control; optimization Definition

Admissible control

Builds on Control system Measurable function Absolute continuity

Definition

Let $x' = f(x, u)$ be a control system on $\R^n$ with control set $U \subseteq \R^m$ . Fix a time horizon $T > 0$ , an initial state $x_0 \in \R^n$ and a closed target set $S \subseteq \R^n$ . An admissible control for $(x_0, T, S)$ is a function $u \colon [0, T] \to \R^m$ such that:

$u$ is measurable and essentially bounded, and $u(t) \in U$ for almost every $t \in [0, T]$ ;
there is an absolutely continuous trajectory $x \colon [0, T] \to \R^n$ with $x(0) = x_0$ and $x'(t) = f(x(t), u(t))$ for almost every $t$ ;
this trajectory satisfies $x(T) \in S$ .

Remarks

By the local Lipschitz condition on $f$ , the trajectory in clause 2 is unique when it exists. Clause 2 can fail because the solution blows up before time $T$ . Many authors call any control satisfying clause 1 admissible, as in the control system definition, and add clauses 2 and 3 as constraints of the problem. Problems with free final time, state constraints or other control classes, such as piecewise continuous controls, change the clauses accordingly.

Example: $x' = u$ on $\R$ with $U = [-1, 1]$ , $x_0 = 0$ , $T = 1$ and $S = \set{\tfrac12}$ . The constant control $u = \tfrac12$ is admissible: it is bounded with values in $U$ , and $x(t) = t/2$ gives $x(1) = \tfrac12$ .

Non-example: the constant control $u = 1$ in the same setting satisfies clauses 1 and 2, with $x(t) = t$ . It fails clause 3, since $x(1) = 1 \notin S$ .

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`8GETTS41P6`
URI	`m2://8GETTS41P6`
Kind	definition
Writing	W0 stub
MSC	49J15, 93C15
Defines	Admissible control
Uses	Control system, Measurable function, Absolute continuity

Admissible control

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here