Further Results on Stabilization of Shock-Like Equilibria of the Viscous Burgers PDE

In this note we show that a symmetric shock profile of the linearized viscous Burgers equation under high-gain "radiation" boundary feedback is exponentially stable, though the previously reported numerical eigenvalue calculations have reported instability. We also show limitations of the radiation feedback by deriving an analytical bound on the closed-loop decay rate for a given shock profile. We prove that the decay rate goes to zero exponentially as the shock becomes sharper. This limitation in the decay rate achievable by radiation feedback highlights the importance of backstepping designs for the Burgers equation, which achieve arbitrarily fast local convergence to arbitrarily sharp shock profiles.