Destabilization conjecture for diffusive boundary feedback

For $\eta>0$, consider the coupled transport–diffusion system with state variables $y$ and $\hat y$ on $[0,1]$:

$$\partial_t y+\partial_x y-\eta\partial_{xx}y=0,\qquad \partial_t\hat y+\partial_x\hat y=0,$$

with boundary conditions

$$y(t,0)=y(t,1)-\hat y(t,1),\qquad \hat y(t,0)=y(t,0),\qquad \partial_x y(t,1)=0.$$

Let $\mathcal{S}_\eta=\{s\in\mathbb{C}:F_\eta(s)-1=0\}$ be its spectrum, where

$$F_\eta(s)=\frac{(\lambda_1(s)e^{\lambda_1(s)}-\lambda_2(s)e^{\lambda_2(s)})(1+e^{-s})}{(\lambda_1(s)-\lambda_2(s))e^{\lambda_1(s)+\lambda_2(s)}},$$

and

$$\lambda_1(s)=\frac{1+\sqrt{1+4\eta s}}{2\eta},\qquad \lambda_2(s)=\frac{1-\sqrt{1+4\eta s}}{2\eta}.$$

Define $\sigma_\eta=\sup\{\operatorname{Re}(s):s\in\mathcal{S}_\eta\}$. Destabilization conjecture. For all $\epsilon>0$, there exists $\eta_1>0$ such that for all $\eta\in(0,\eta_1)$, the maximal spectral abscissa satisfies

$$\sigma_\eta>-\epsilon.$$

The conjecture says that arbitrarily small diffusion destroys any uniform negative spectral gap, despite the stability of the corresponding closed-loop system without diffusion. It captures the reported destabilizing effect of diffusion and remains to be established from the characteristic equation.