Definition (Controllable subspace)

Consider a LTVC system $$\begin{cases} \dot{x}(t)=A(t)x(t)+B(t)u(t) \\ x(t_{0})=x_{0} \end{cases}$$with controllability gramian $W$. The subspace $\text{Image}(W(t_{0},t_{1}))$ is called the controllable subspace for the pair $(t_{0},t_{1})$.

Definition (Controllable)

A LTVC system $$\begin{cases} \dot{x}(t)=A(t)x(t)+B(t)u(t) \\ x(t_{0})=x_{0}\in\mathbb{R}^{n} \end{cases}$$is said to be controllable for the pair $(t_{0},t_{1})$ if the controllable subspace is the entire state space. i.e. $$\text{Image}(W(t_{0},t_{1}))=\mathbb{R}^{n}$$

Definition (Controllable (332))

Consider $$\frac{dx}{dt}=Ax(t)+Bu(t)$$The pair $(A,B)$ is said to be controllable if for any $x(0)=x_{0}\in\mathbb{R}^{n}$ and $x_{f}\in\mathbb{R}^{n}$, there exists $T<\infty$ and a control input $\{ u_{s},0\le s\le T \}$ so that $x_{T}=x_{f}$.