Calculus of variations and optimal control; optimization Definition

Bang–bang control

Builds on Control system Convex set Compact set

Definition

Let $x' = f(x, u)$ be a control system whose control set $U \subseteq \R^m$ is a compact convex set, for example a box $\prod_{i=1}^m [a_i, b_i]$ with $a_i < b_i$ . An admissible control $u \colon [0, T] \to U$ is bang–bang if $u(t)$ is an extreme point of $U$ for almost every $t \in [0, T]$ . For a box, this means $u_i(t) \in \set{a_i, b_i}$ for each $i$ and almost every $t$ . The times at which $u$ jumps between extreme points are its switching times.

Remarks

Example: for $U = [-1, 1]$ , the control $u(t) = 1$ on $[0, 1)$ and $u(t) = -1$ on $[1, 2]$ is bang–bang, with one switching time $t = 1$ . For the double integrator $x_1' = x_2$ , $x_2' = u$ starting at rest at $x_1 = -1$ , it brings the state to rest at $x_1 = 0$ at time $2$ , and this is the minimum time with $\abs{u} \le 1$ . Non-example: $u(t) = \sin t$ on $[0, \pi]$ with $U = [-1, 1]$ is not bang–bang, since $\abs{u(t)} < 1$ except at $t = \pi / 2$ . Consider a linear system $x' = A x + B u$ whose control set $U$ is a compact convex polytope. The bang–bang principle says that every state reachable at time $T$ is also reachable at time $T$ by a bang–bang control.

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Compact set concept
Control system concept
Convex set concept
Bang–bang control definition

Tag	`RQBNA1ZRJW`
URI	`m2://RQBNA1ZRJW`
Kind	definition
Writing	W0 stub
MSC	49J15, 93C15
Defines	Bang–bang control
Uses	Control system, Convex set, Compact set
Changed	2 November 2026

Bang–bang control

Definition

Remarks

Reading path

Further reading

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