Boundary controllability and stabilizability of a coupled first-order hyperbolic-elliptic system

Abstract

In this paper, we study the controllability of a coupled first-order hyperbolic-elliptic system in the interval $ (0, 1) $ by a Dirichlet boundary control acting at the left endpoint of the hyperbolic component only. Using the multiplier approach and compactness-uniqueness argument, we establish the exact controllability for the hyperbolic component of the model, at any time $ T>1 $. We explore the method of moments to conclude the exact controllability at the critical time $ T = 1 $. For the case of small time, that is for $ T<1 $, we show that the system is not null controllable. Further, using a Gramian-based approach introduced by Urquiza, we prove the exponential stabilization of the corresponding closed-loop system with arbitrary prescribed decay rate by means of boundary feedback control law.