Positive boundary dissipation conjecture for BIBO stability of hBCS

Let $\widetilde{W}_B$, $R_0$, and $\Sigma$ be the matrices associated with a one-dimensional hyperbolic boundary control system (hBCS). The system is BIBO stable when

$$\widetilde{W}_B R_0^{-1} \Sigma \left(\widetilde{W}_B R_0^{-1}\right)^* > 0.$$

Positive boundary dissipation conjecture. If $\widetilde{W}_B R_0^{-1} \Sigma \left(\widetilde{W}_B R_0^{-1}\right)^* > 0$, then the hBCS is BIBO stable. In the setting where the decomposition assumption holds, this condition is equivalent to $\|M\|_{\ell^2\rightarrow\ell^2}<1$. The authors report no counterexample, but the conjecture remains open; the condition is only proposed as sufficient, since BIBO stability can also occur when the displayed positivity condition fails.