Controllability Theory Of Degenerate And Singular Parabolic Equations And Systems

In the thesis, we study the controllability of degenerate and singular parabolicequations and systems, including null controllability and approximate controllability.In the first chapter, we investigate a class of degenerate parabolic equations withconvection terms and prove the null controllability. In the previous work about thenull controllability of the degenerate parabolic equations with convection terms, theconvection terms are depending on the difusion terms, while we study the case that theconvection term cannot be controlled by the difusion term. As we know, to prove thenull controllability of the equation, the Carleman estimate for its conjugate equationis needed. For the equation with boundary degeneracy, the convection term cannotcause the essential difculty for the Carleman estimate when the convection term onthe boundary has certain degeneracy. However, the general convection term will bringthe essential difculty. By the method of undetermined exponent to take the auxiliaryfunction, we establish the Carleman estimate.In the second chapter, we investigate the approximate controllability of the semi-linear system governed by the degenerate equation with convection term. We firstconsider the linear case of the semilinear equation, do the estimates for the solutionof the linearized problem and prove the approximate controllability of the linearizedequation. Then, we prove the approximate controllability of the semilinear system by the Kakutani fixed point theorem. It is well known that the key to prove the approx-imate controllability is to do suitable compact estimates for solution to the linearizedproblem and its conjugate problem, which are more precise and complicated than theones needed for the well-posedness. Since the equation may be degenerate on a portionof the lateral boundary, weak solutions with poor regularity should be considered andsome compactness estimates for solutions to nondegenerate equations are missing. Toget the estimates needed for the approximate controllability of the system, we mustovercome the technical difculties caused by the degeneracy and the convection term.Further, we can not get the L2estimate of utfor the linearized problem owing to theconvection term. In order to compensate the lack of this estimate, we establish theequicontinuity of u(·, t) with respect to t∈(0, T).In the third chapter, we prove the null controllability of a class of semilinear sin-gular parabolic equation of the initial boundary value problem. By coordinate trans-formation, the singular equation that we consider in this chapter can be transformedinto a weakly degenerate singular equation. Since the singularity of the equation maycause new difcult for establishing the Carleman estimate, we need some restrictionson the coefcient of the reaction term. In the proof, we choose the new auxiliary func-tion, establishing the new Carleman estimate and observability. Then, we get the nullcontrollability of the problem by the Schauder fixed point theorem.In the fourth chapter, we consider two class of coupled degenerate systems andprove the null controllability of the problems. First, we investigate the null controllabil-ity of a “single” nonlinear coupled degenerate system. Since the difusion terms of thecoupled equations we investigate can be diferent, the auxiliary functions we choosefor diferent equations are also diferent when we establish the Carleman estimates.By choosing the suitable auxiliary functions and applying the Carleman estimate forthe single equation and some precise estimates, we establish the Carleman estimatefor the conjugate problem. Then by constructing the control and the Schauder fixed point theorem, we prove the null controllability. We also study the null controllabilityof a “double” linear coupled degenerate system. In the proof, we first prove the nullcontrollability of the system with two controls, which is also based on the Carlemanestimate for its conjugate problem. Then, the control function of the system with onecontrol can be constructed by the solution of the system with two controls.