Localized Stabilization of Transport PDEs by Interior Flux Feedback

Abstract

We study stabilization of multidimensional continu- ity equations with source terms on bounded domains by means of localized interior flux feedback. The feedback is prescribed through the divergence of the flux and is chosen so that the error with respect to a reference profile satisfies a transport equation with localized damping. The main geometric condition is a finite-time characteristic damping inequality, requiring relevant characteristics to accumulate a uniform amount of damping over a time horizon. This condition is shown to yield exponential stability in L2 of the error, under a gain condition relating localized damping to compressive amplification of the transport field. Lyapunov-type entrance conditions ensure characteris- tic damping on support-restricted families of trajectories. A weighted Lyapunov functional provides a differential Lyapunov criterion and an input-to-state (ISS) estimate with respect to additive perturbations. We also discuss elliptic right-inverse realizations of the feedback flux and extend the characteristic damping argument to velocity fields depending nonlinearly on the state. A two-dimensional example finally illustrates the geometric, gain, and realization conditions.