Kalman conjecture for slope-restricted Lur'e systems

Let $c6$ be a memoryless slope-restricted nonlinearity such that there exists a continuously differentiable $N:mathds{R}rightarrowmathds{R}$ and $S>0$ such that $c6(v)(t)=N(v(t))$ in continuous time, or $c6(v)(k)=N(v(k))$ in discrete time, and

$$0\leq\frac{dN(x)}{dx}\leq S,\qquad\forall x\inmathds{R}.$$

Kalman conjecture. Then the negative-feedback interconnection of the continuous-time (or discrete-time) LTI system $G\sim[A,B,C,0]$ and $c6$ is globally asymptotically stable if $A-BCk$ is Hurwitz in continuous time, or Schur in discrete time, for all $k\in[0,S]$. The conjecture concerns whether stability of every linearization implies global asymptotic stability of the corresponding Lur'e system. It is refuted in general: fourth-order continuous-time and second-order discrete-time counterexamples are known.