Kalman's convergence-rate conjecture for Lur'e systems

Let $G$ be stable and let $K_N$ denote its Nyquist value. For a class of feedback nonlinearities $\Delta\in S[0,K]$ with $K<K_N$, define the absolute convergence rate

$$\rho^*_{\{G,K\}}=\sup_{\Delta\in S[0,K]}\{\rho_{\{G,\Delta\}}\}.$$

Let

$$\underline{\rho^*_{\{G,K\}}}=\max_{\tau\in[0,1]}\left\{\left|\operatorname{eig}\left(\frac{G}{1-\tau K G}\right)\right|\right\}.$$

Kalman's convergence-rate conjecture. For any stable $G$ and $\Delta\in S[0,K]$ with $K<K_N$,

$$\rho^*_{\{G,K\}}=\underline{\rho^*_{\{G,K\}}}.$$

This is presented as a convergence-rate restatement of Kalman's stability conjecture: the worst-case nonlinear convergence rate should equal the corresponding theoretical linear rate over the slope-restricted class.