Failure of Aizerman's conjecture

Counterexample

Aizerman's conjecture concerns systems of the form $$\dot x = P x + q\, f(e), \qquad e = r^{T} x,$$ where $x \in \R^n$, $P$ is a constant real $n \times n$ matrix, $q, r \in \R^n$ are constant vectors, and $f \colon \R \to \R$ satisfies $f(0) = 0$. Assume that $k_1 < f(e)/e < k_2$ for all $e \ne 0$, and that for every $k \in (k_1, k_2)$ the linear system $\dot x = (P + k\, q r^{T}) x$ is asymptotically stable. The conjecture claims that then the equilibrium $x = 0$ is globally asymptotically stable.

The claim is false in general. Let $P$, $q$, $r$, $f$ be a counterexample to Kalman's conjecture. So $f$ is differentiable, $f(0) = 0$, $k_1 < f' < k_2$, and all the linear systems above are asymptotically stable. Yet the system has a non-constant periodic solution. For $e \ne 0$ the mean value theorem gives $\xi$ between $0$ and $e$ with $f(e)/e = f'(\xi)$, so $k_1 < f(e)/e < k_2$. Hence the same system satisfies the hypotheses of Aizerman's conjecture. Its periodic solution does not tend to $0$, so the conclusion fails.