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

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