Controllability

Definition

Let $x' = f(x, u)$ be a control system on $\R^n$ with control set $U$ . Let $x_0, x_1 \in \R^n$ and $T > 0$ . The state $x_1$ is reachable from $x_0$ in time $T$ if some admissible control $u$ on $[0, T]$ has a trajectory with $x(0) = x_0$ and $x(T) = x_1$ . The system is controllable if for all $x_0, x_1 \in \R^n$ there is $T > 0$ such that $x_1$ is reachable from $x_0$ in time $T$ .

Remarks

For a linear system $x' = A x + B u$ with $U = \R^m$ , controllability in some time implies controllability in every time $T > 0$ . Example: the double integrator $x_1' = x_2$ , $x_2' = u$ with $U = \R$ is controllable. Given $(a_0, b_0)$ , $(a_1, b_1)$ and $T > 0$ , take the cubic polynomial $p$ with $p(0) = a_0$ , $p'(0) = b_0$ , $p(T) = a_1$ , $p'(T) = b_1$ ; then $u = p''$ gives the trajectory $(x_1, x_2) = (p, p')$ , which joins the two states. Non-example: the system $x_1' = u$ , $x_2' = 0$ is not controllable, because $x_2$ stays constant along every trajectory, so $(0, 1)$ is not reachable from $(0, 0)$ . Weaker notions are local controllability near an equilibrium and small-time local controllability.

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Control system concept
Controllability definition

Tag	`YR43719RWE`
URI	`m2://YR43719RWE`
Kind	definition
Writing	W0 stub
MSC	93B05, 34H05
Defines	Controllability
Uses	Control system
Changed	2 November 2026

Controllability

Definition

Remarks

Reading path

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