Controllability/Reachability

Controllability and reachability are defined for discrete-time systems as:

• A system is controllable if from any state $x_0\in {\mathbb{R}}^n$ one can achieve the state $x_N=0$ using a series of inputs $u_{0:N-1}$.
• A state $x_N\in {\mathbb{R}}^n$ is reachable if it can be achieved by applying a series of inputs $u_{0:N-1}$ when starting from the initial state $x_0=0$.

For continuous-time systems they are similarly defined as:

• A system is controllable if from any state $x(0)\in {\mathbb{R}}^n$ one can achieve the state $x(t)=0$ using a control policy $u(\tau )$, $\tau \in ⟨0,t)$.
• A state $x(t)\in {\mathbb{R}}^n$ is reachable if it can be achieved by applying a control policy $u(\tau )$, $\tau \in ⟨0,t)$ when starting from the initial state $x(0)=0$.

For linear systems all states are reachable if the system is controllable.