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  "body": "\n## Summary\n\nInput-to-state stability is a stability notion for systems driven by an input, $\\dot x = f(x, u)$.\nIt asks that the state decays to a size controlled by the size of the input, whatever the initial state.\nIt merges global asymptotic stability of the unforced system, a strong form of [[Lyapunov stability|0F2SYCKDT2]], with a bounded-input bounded-state property.\nIt is a basic tool in nonlinear [[control systems|ZPDTVZQN5K]], because it behaves well under cascades and feedback interconnections.\n\n## Explanations\n\n### L1 human\n\nThink of a ball in a bowl that is shaken by an outside force.\nWithout shaking, the ball rolls back to the bottom.\nWith shaking, the ball stays close to the bottom, and the distance depends on how hard the shaking is.\nFor the system $\\dot x = -x + u$, the state satisfies $\\abs{x(t)} \\le e^{-t} \\abs{x(0)} + \\sup \\abs{u}$.\nSo the effect of the start dies out, and what remains is bounded by the input.\n",
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