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  "body": "\n## Definition\n\nLet $n, m \\ge 1$, and let $U \\subseteq \\R^m$ be a nonempty closed set, the control set.\nLet $f \\colon \\R^n \\times U \\to \\R^n$ be continuous and locally [[Lipschitz|0AW1E6YTJY]] in $x$, locally uniformly in $u$.\nThe *control system* given by $f$ is the equation\n$$x'(t) = f(x(t), u(t)).$$\nAn *admissible control* on an interval $[0, T]$ is a measurable, essentially bounded function $u \\colon [0, T] \\to U$.\nA *trajectory* for $u$ is an [[absolutely continuous|ZT6F4C2YHK]] function $x \\colon [0, T] \\to \\R^n$ that satisfies the equation for almost every $t \\in [0, T]$.\nOften a control system also has an output $y(t) = h(x(t))$ with a map $h \\colon \\R^n \\to \\R^p$.\n\n## Remarks\n\nFix an admissible control $u$ and an initial state $x(0) = x_0$.\nThen the equation is an [[ordinary differential equation|0NT64JV96A]] in the sense of Carathéodory, and it has a unique trajectory on some interval $[0, \\tau)$.\nThe most important case is the linear system $x' = A x + B u$ with matrices $A \\in \\R^{n \\times n}$ and $B \\in \\R^{n \\times m}$.\nExample: the double integrator $x_1' = x_2$, $x_2' = u$ with $U = [-1, 1]$ is linear, with $A = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ and $B = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}$; the control $u = 1$ on $[0, 1)$ and $u = -1$ on $[1, 2]$ is admissible.\nNon-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.\nSome authors allow locally integrable controls, or controls of a smaller class such as piecewise continuous ones.\n",
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