Control system

Remarks

Fix an admissible control $u$ and an initial state $x(0) = x_0$. Then the equation is an ordinary differential equation in the sense of Carathéodory, and it has a unique trajectory on some interval $[0, \tau)$. The most important case is the linear system $x' = A x + B u$ with matrices $A \in \R^{n \times n}$ and $B \in \R^{n \times m}$. Example: the double integrator $x_1' = x_2$, $x_2' = u$ with $U = [-1, 1]$ is linear, with $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$; the control $u = 1$ on $[0, 1)$ and $u = -1$ on $[1, 2]$ is admissible. Non-example: $u(t) = 1/t$ for $t \in (0, 1]$ is not an admissible control for $U = \R$ on $[0, 1]$, because it is not essentially bounded. Some authors allow locally integrable controls, or controls of a smaller class such as piecewise continuous ones.