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  "body": "\n## Counterexample\n\nAizerman's conjecture concerns systems of the form\n$$\\dot x = P x + q\\, f(e), \\qquad e = r^{T} x,$$\nwhere $x \\in \\R^n$, $P$ is a constant real $n \\times n$ matrix, $q, r \\in \\R^n$ are constant vectors, and $f \\colon \\R \\to \\R$ satisfies $f(0) = 0$.\nAssume that $k_1 \u003c f(e)/e \u003c k_2$ for all $e \\ne 0$, and that for every $k \\in (k_1, k_2)$ the [[linear system|037ZNY7NAG]] $\\dot x = (P + k\\, q r^{T}) x$ is [[asymptotically stable|0NB689782W]].\nThe conjecture claims that then the [[equilibrium|0N8NGKDZDZ]] $x = 0$ is globally asymptotically stable.\n\nThe claim is false in general.\nLet $P$, $q$, $r$, $f$ be a [[counterexample to Kalman's conjecture|5MP7DM0V5V]].\nSo $f$ is differentiable, $f(0) = 0$, $k_1 \u003c f' \u003c k_2$, and all the linear systems above are asymptotically stable.\nYet the system has a non-constant periodic solution.\nFor $e \\ne 0$ the [[mean value theorem|0BARQY52N5]] gives $\\xi$ between $0$ and $e$ with $f(e)/e = f'(\\xi)$, so $k_1 \u003c f(e)/e \u003c k_2$.\nHence the same system satisfies the hypotheses of Aizerman's conjecture.\nIts periodic solution does not tend to $0$, so the conclusion fails.\n\n## Remarks\n\nAizerman posed the problem in 1949.\nThe hypothesis of Aizerman's conjecture bounds the secant slopes $f(e)/e$, while Kalman's conjecture bounds the slopes $f'$; the second hypothesis is stronger, so Kalman's conjecture is the weaker claim.\nThis is why a counterexample to Kalman's conjecture is also one to Aizerman's.\nIn the counterexamples constructed by Leonov, Kuznetsov and coauthors, a stable periodic orbit coexists with the stable equilibrium, a so-called hidden oscillation.\n\n## Explanations\n\n### L1 human\n\nSuppose a feedback loop is stable whenever its nonlinearity is replaced by a straight line through the origin with a slope in a given range.\nThe conjecture says that the loop is then also stable with the nonlinearity itself.\nThis sounds plausible, since the nonlinearity always lies between such lines.\nIt is false: some of these systems oscillate forever, although every linear comparison system settles down.\n",
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