A Class Of Semi-linear Parabolic System Stabilization Control Of The Relationship, And Zero Energy

In this paper, we first discuss the relations between the exponential stabilization and null controllability of a kind of semilinear parabolic systems. By using a generalized implicit function theorem, we obtain the equivalance between exponential stabilization and null controllability. Moreover, We disscuss the object reachability with coefficient as control quantity, and obtain the corresponding results.At first, we linearized the semilinear parabolic system by variational approach, and proved the compactness of the solution mapping of the linearized system, we obtain the local null controllability of the semilinear system, by applying a gener-' alized implicit function theorem, and combining the "good" property of solution mapping of the semilinear system and null controllability results of linearized system. Thereout, we obtain the equivalance between exponential stabilization and null controllability.Then, we adopt the same method appearing in the first part, obtain and improve some results in the literature in [11]. At the same time, we also discuss the object reachability, by using a simple transformation, our problem is a generalized problem in the literature in [11]. we obtain the corresponding results by combining the results in reference and the above method.