Sharp Observability Inequalities for Hyperbolic Systems with Potentials

Abstract

This paper is devoted to a sharp internal/boundary observability inequality for a hyperbolic system with a zero order potential. For this purpose, we first establish a new Carleman estimate for hyperbolic operator in H1-norm. Based on this Carleman estimate and a modified auxiliary optimal control problem, we obtain Carleman estimate for hyperbolic operator in L2-norm. Then, by virtue of a modified energy estimate and a delicate treatment of the observation region, we obtain an internal observability estimate with the observability constant of the order exp (C‖q‖2/3L∞(Q;ℝN×N)), with q the potential involved in the system. We also address the same problem for boundary observation. Compared with the related results in the literature, the main contributions of this paper are the observability constant is sharper, the waiting time T is shorter and the internal (or boundary) observation domain is smaller.