Stabilization Of The Linear Parabolic Systems Via Internal Feedback Control

In this paper, we consider the stabilization of the linear parabolic systemwhere Ω is a bounded domain of Rn, n ∈ N with a sufficiently smooth boundary Ω, Q = Ω ×(0,+∞),S = Ω × (0,+∞). is the nonempty open subset of Ω which satisfies u C Ω, and v(z), , is the normal outward versor to at the point x. a(x),b(x),c(x),d(x) ∈ L∞(Ω), y0(x),z0(x) ∈L∞(Ω). m(-) is the characteristic function of , i.e.We study the stabilization of system (I) through two steps as follows: 1. Comparision principle. First we prove the solutions of system (I) and system (II) as follows satisfy the comparision principle.where M1 = ||y0(x)||L (Ω), M2 = ||z0(x)|| L∞(Ω).The Comparision principle means the solutions of system (I) and system (II) satisfy :2. The asymptotic stabilization of system (II) is proved through Carleman estimates. We also give some decay estimates about system (II).The paper is made up of four parts.Part one introduces the relevant research progress. Furthermore, we state our main results following from some retropection of the results obtained by previous mathematicians.Part two proves the comparision principle.Part three states the Carleman estimates about system (II).Part four proves the asympotic stability of system (I), i.e., Theorem 4.2.