Modified Gronwall's inequality and its application to stabilization problem for semilinear parabolic systems

Abstract This paper deals with the stabilization of semilinear parabolic distributed systems by means of boundary controls. The partial differential equation describing the system is formulated as a semilinear evolution equation in a Hilbert space by using a variable transformation. Here it is shown that the solution globally exists for the closed-loop system which installs a finite-dimensional dynamic compensator by using the properties of an analytic semigroup. Moreover, a sufficient condition for stabilization of the system is derived by making use of the modified Gronwall's inequality developed in this paper.