Linear dynamical systems

Definitions
A continuous-time linear dynamical system (CT LDS) has the form

$$\frac{dx}{dt}=A(t) x(t) + B(t)u(t)$$, $$\,\,\,\,y(t) = C(t) x(t) + D(t)u(t)$$

where $$t \in \mathbb{R}$$ denotes time, $$x(t) \in \mathbb{R}^n$$ is the state (vector), $$u(t) \in \mathbb{R}^m$$ is the input or control and $$y(t) \in \mathbb{R}^p$$ is the output. Further, $$A(t) \in \mathbb{R}^{n\times n}$$ is the dynamics matrix, $$B(t) \in \mathbb{R}^{n\times m}$$ is the input matrix, $$C(t) \in \mathbb{R}^{p\times n}$$ is the output or sensor matrix and $$D(t) \in \mathbb{R}^{p\times m}$$ is the feedthrough matrix. The matrices $$A,B,C,D$$ are generally time-invariant.

For lighter appearance, the CT LDS can be written as: $$\dot{x} = Ax + Bu$$, $$y = Cx+Du$$.

A discrete-time linear dynamical system (DT LDS) has the form : $$x(t+1) = A(t)x(t) + B(t)u(t),$$ $$\,\,\,\, y(t) = C(t)x(t) + D(t)u(t)$$

where $$t \in \mathbb{Z} = \{0,\pm 1, \pm 2, \ldots\}$$ and the (vector) signals $$x, u, y$$ are sequences.