Equivalence between the internal observability and exponential decay for the Moore-Gibson-Thompson equation

This paper is concerned with a third order in time linear Moore-Gibson-Thompson equation which describes the acoustic velocity potential in ultrasound wave program. Influenced by the work of Kaltenbacher, Lasiecka and Marchand (Control Cybernet. 2011, 40: 971–988), we establish an observability inequality of the conservative problem, and then discuss the equivalence between the exponential stabilization of a dissipative system and the internal observational inequality of the corresponding conservative system.