Controllability Of Parabolic Type Equations With Memory

This dissertation investigate mainly some controllability problems on control-lability of parabolic type equations with memory. We discuss the approximatecontrollability of a linear system with constant coefcients by interior or boundarycontrol, the null controllability of a sublinear system, and the controllability andnoncontrollability of superlinear systems.This dissertation consists of three parts.In the first part, i.e. Chapter2and Chapter3of this dissertation, we study theapproximate controllability of a linear system with constant coefcients. In Chapter2, we turn the problem to proving the approximate controllability of a third ordersystem. By the operators semigroup and the extension property of real analyticfunctions, we obtain the approximate controllability of the linear system by interiorcontrol. In Chapter3, using the duality method and Hahn-Banach theorem, weget the the approximate controllability of the linear system by boundary control.In the second part, i.e. Chapter4of this dissertation, we are concerned withthe approximate controllability of a linear system with variable coefcients. Weuse a sublinear system as a bridge to show the main result. By a Carleman es-timate and a fixed point theorem, we prove the null controllability of the systemunder the condition of sublinear growth. Then, applying the controllability to thetrajectories which is equivalent to the null controllability and the theory of func-tion approximation, we get the approximate controllability of a linear system withvariable coefcients.In the third part, i.e. Chapter5of this dissertation, we discuss the control-lability of two kind of superlinear system. For the first system, we prove that nomatter what control function is chosen, making use of a localized estimate, theblow-up phenomena will still happen. So, the system fail to be controllable. At the same time, we show that for some special initial data and target, the systemis approximately controllable and null controllable. For the second system, weprove that the system with a superlinear growth is null controllable. In addition,the nonlinear growth of the system with memory which is null controllable can behigher than that of classical parabolic equation.