Spectral stabilization of linear transport equations with boundary and in-domain couplings

Abstract

In this work, the problem of stabilization of general systems of linear transport equations with indomain and boundary couplings is investigated. It is proved that the unstable part of the spectrum is of finite cardinal. Then, using the pole placement theorem, a linear full state feedback controller is synthesized to stabilize the unstable finite-dimensional part of the system. Finally, by a careful study of semigroups, we prove the exponential stability of the closed-loop system. As a by product, the linear control constructed before is saturated and a fine estimate of the basin of attraction is given.

Boundary stabilization of 1D hyperbolic systems

“…C 1 norm) but that is still exponentially stable? For the linearized system this is very likely to be true, using spectral tools as shown in [12, Section 5.6] (see also [80,72])…”