A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction

Abstract

We study the isothermal Euler equations with friction and considernon-stationary solutions locally around a stationary subcriticalstate on a finite time interval. The considered control system is aquasilinear hyperbolic system with a source term. For thecorresponding initial-boundary value problem we prove the existenceof a continuously differentiable solution and present a method ofboundary feedback stabilization. We introduce a Lyapunov functionwhich is a weighted and squared $H^1$-norm of the difference betweenthe non-stationary and the stationary state. We develop boundaryfeedback conditions which guarantee that the Lyapunov function andthe $H^1$-norm of the difference between the non-stationary and thestationary state decay exponentially with time. This allows us alsoto prove exponential estimates for the $C^0$- and $C^1$-norm.