The Exact Null Controllability And Approximate Controllability Of The Coupled Linear Parabolic System In Unbounded Domains

In this paper, we shall study the exact null controllability and approximate controllability of the coupled linear parabolic system as follows when the control force acts on a single equation :where Ω is an open and unbounded set of Rn with boundary (?)Ω of C2 uniformly, ω is an open and nonempty subset of Ω such that Ω\ ω is bounded. Xω is the characteristic function of ω.System (1.1) is said to be approximate controllable in time T > 0, if, for any final state (y1,z1) ∈ L2(Ω) × L2(Ω) and ε > 0, there exists fε × L2(ω × (0,T)) such that the solution of (1.1) satisfiesSystem (1.1) is said to be exact null controllable in time T > 0, if, for any , there exists f ∈ L2(ω× (0, T)) such that-the solution of (1.1) satisfies(y,z)(·,T) = 0 a.e in Ω . (1.4)The main difficulty of this work is to establish observability inequality. First, we get the Carleman estimate. Then based on it, the observability inequality is obtained. Finally we get the controllability of the system. The above problems will be discussed as the following four parts: Part 1. Introduce the background of the controllability problem and the relavant research progress, what's more, we state our main results.Part 2. Get Carleman estimate of the adjoint system . Part 3. Obtain observability inequality.Part 4. Prove the exact null controllability and approximate controllability of the system.