---
tag: YR43719RWE
kind: definition
name: Controllability
msc: [93B05, 34H05]
defines: A9WFM77Z4P
uses: [ZPDTVZQN5K]
sources:
  - url: https://en.wikipedia.org/wiki/Controllability
---

## Definition

Let $x' = f(x, u)$ be a [[control system|ZPDTVZQN5K]] 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.