Controllability Of Second Order Parabolic Type And Cahn-Hilliard Type Equations

This dissertation investigate mainly some controllability problems on bilinear control systems governed by second order parabolic type equations, control systems governed by four order semilinear Cahn- Hilliard type equations with traditionally additive distributed controls and semilinear second order parabolic control systems with bounded constrained controls. This dissertation consists of three parts.In Part 1, we are concerned with the controllability of bilinear control systems. In Chapter 2, we dicuss the control problems on bilinear control systems with the reaction-diffusion term satisfying Newton's Law governed by second order linear parabolic equations. Moreover, by this result we further prove the exactly null controllability in long time of a semilinear parabolic equation y_t—Δy + b| (?)y|~2 = x_ωu with traditionally additive locally distributed controls. Then, we give a result of the reachability of the system for some special targets. In Chapter 3, we discuss the controllability of a bilinear control system governed by a nonlinear degenerate parabolic equation not in divergence form and give the result that neither is the system globally approximate controllable, nor globally exact null controllable, but is globally approximate null controllable. We also prove the reachability of the system for some special targets.In Part 2, i.e. Chapter 4 of this dissertation, we discuss control systems governed by two types of four order semilinear Cahn-Hilliard type equations and prove that both of them are null controllable.In Part 3, i.e. Chapter 5 of this dissertation, we discuss locally null controllability problems of a semilinear second order parabolic control system with bounded constrained controls and prove that if the nonlinear term satisfies globally Lipschitz conditions, then the system is locally null controllable.