Definition (Observable)

An LTVC System is said to be observable for the pair $(t_{0},t_{1})$ if $$\text{Ker}(M(t_{0},t_{1}))=\{ \vec{0} \}(\iff \text{Image}(M(t_{0},t_{1}))=\mathbb{R}^{n})$$

Definition (Observable (332))

Consider $$\frac{dx}{dt}=Ax(t)+Bu(t),\quad y(t)=Cx(t)+Du(t)$$The pair $(A,C)$ is said to be observable if for any $x(0)=x_{0}\in\mathbb{R}^{n}$ there exists $T<\infty$ s.t. the knowledge of $\{ (y_{s},u_{s}),0\le s\le T \}$ is sufficient to uniquely determine $x(0)$.