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  "body": "\n## Definition\n\nLet $f \\colon \\R^n \\times \\R^m \\to \\R^n$ be locally [[Lipschitz|0AW1E6YTJY]] with $f(0, 0) = 0$.\nConsider the control system $\\dot x(t) = f(x(t), u(t))$, where the inputs $u \\colon [0, \\infty) \\to \\R^m$ are measurable and essentially bounded, with norm $\\norm{u}_\\infty = \\operatorname{ess\\,sup}_{t \\ge 0} \\abs{u(t)}$.\nA function $\\gamma \\colon [0, \\infty) \\to [0, \\infty)$ is of *class $\\mathcal{K}$* if it is continuous, strictly increasing and $\\gamma(0) = 0$.\nA function $\\beta \\colon [0, \\infty) \\times [0, \\infty) \\to [0, \\infty)$ is of *class $\\mathcal{KL}$* if two conditions hold.\nFor each $t$, $\\beta(\\cdot, t)$ is of class $\\mathcal{K}$; for each $r$, $\\beta(r, \\cdot)$ is nonincreasing with limit $0$ at infinity.\nThe system is *input-to-state stable* (ISS) if there are $\\beta$ of class $\\mathcal{KL}$ and $\\gamma$ of class $\\mathcal{K}$ with this property.\nFor every $x_0 \\in \\R^n$ and every input $u$, the solution with $x(0) = x_0$ exists for all $t \\ge 0$ and satisfies\n$$\\abs{x(t)} \\le \\beta(\\abs{x_0}, t) + \\gamma(\\norm{u}_\\infty) \\quad \\text{for all } t \\ge 0.$$\n\n## Remarks\n\nExample: $\\dot x = -x + u$ on $\\R$.\nBy variation of constants, $x(t) = e^{-t} x_0 + \\int_0^t e^{-(t - s)} u(s) \\, ds$.\nHence $\\abs{x(t)} \\le e^{-t} \\abs{x_0} + \\norm{u}_\\infty$, so the system is ISS with $\\beta(r, t) = r e^{-t}$ and $\\gamma(r) = r$.\n\nNon-example: $\\dot x = -x + x u$ on $\\R$.\nFor $u = 0$ the origin is globally asymptotically stable.\nBut for the constant input $u = 2$ the equation is $\\dot x = x$, so $x(t) = e^t x_0$ is unbounded, and no estimate of the ISS form can hold.\n\nTaking $u = 0$ shows that an ISS system has a globally asymptotically stable equilibrium at $0$.\nAn equivalent characterisation uses an ISS Lyapunov function $V$: a smooth, positive definite, proper function with $\\nabla V(x) \\cdot f(x, u) \\le -\\alpha(\\abs{x}) + \\sigma(\\abs{u})$ for some class $\\mathcal{K}$ functions $\\alpha$ (unbounded) and $\\sigma$.\n\n## Explanations\n\n### L1 human\n\nA system is ISS when bounded inputs give bounded states, and when the effect of the initial state fades away with time.\nThe function $\\beta$ measures how fast the start is forgotten, and $\\gamma$ measures how strongly the input pushes the state.\n",
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