Theoretical Analysis And Numerical Methods For Some Control Problems With Partial Differential Equations

The control theory of partial differential equations has been widely used in various areas of engineering.Its theoretical analysis and algorithm design have significant value in scientific research.In this dissertation,we study the optimal controls of two kinds of steady-state partial differential equations and the controllability of a class of unsteady equations(phase-field systems).The main research contents are as follows:First,for a class of fractional-order equations with uncertain inputs,this dissertation proves the existence and uniqueness of the solution under the framework of stochastic optimal control theory.The first-order necessary conditions that the optimal solution satisfies are also derived.Then,the numerical algorithm for such optimal control problems is given by using the stochastic Galerkin spectral method.Aiming at solving large-scaled linear equations caused by high-dimensional uncertain parameters,this dissertation combines the matrix decomposition method to construct preconditioners for MINRES and PPCG iterative algorithms and provides the eigenvalue distribution of the preconditioned matrix.In addition,in order to reduce the computational cost,this dissertation presents a matrix iterative acceleration algorithm based on low-rank technique.Numerical experiments verify that the preconditioners proposed in this dissertation can effectively reduce the number of iteration counts and improve computational efficiency compared with existing methods.Then,for the elliptic equation with uncertain inputs,this dissertation proves the uniqueness of the solution under the framework of the deterministic optimal control theory and derives the first-order necessary conditions that the optimal solution satisfies.The numerical algorithm of the optimal control problem is developed under the stochastic Galerkin method.In the study of optimal control problems with uncertain inputs,the deterministic assumption of the control function is more reasonable,but it increases the difficulty of constructing the preconditioner.To this end,this dissertation proposes SMS preconditioner and SMS-RI preconditioner for the discrete optimal control systems,and gives the eigenvalue distribution of the preconditioned matrix.In addition,the coefficient matrix of the state equation results from the discretization by the stochastic Galerkin method is often complicated,and the eigenvalue research of the existing preconditioner cannot guarantee its effectiveness in the square approximation.For this reason,this dissertation also develops square preconditioners which are more suitable for the optimal control system.Using the hierarchical structure of coefficient matrix,the hierarchical Gauss Seidel square preconditioner and the semi-hierarchical Gauss Seidel square preconditioner are proposed.The fast solver of the preconditioners is presented and the relevant eigenvalue distribution estimation is given.The results of numerical experiments are consistent with the conclusions of theoretical analysis,which proves the effectiveness of the preconditioners constructed in this dissertation.In addition,the above research problems are all steady-state problems,while in practical applications,time-varying non-steady-state control problems are more common,and their research also has greater challenges.Therefore,the local null-controllability of a class of quasi-linear phase field systems is also studied in this dissertation.Firstly,the corresponding Carleman inequality and the observability estimation of the adjoint linear system is given.Then,the local null-controllability of the linear system is proved by establishing a series of optimal control problems combined with iterative algorithm.Finally,combined with Kakutani’s fixed point theorem,the proof of local null-controllability of such quasi-linear equations is given.Controllability is an important part of the study of unsteady partial differential equation control theory,and it is closely related to the optimal control problem.Although the research in this part only considers the case of deterministic inputs,its theoretical results can be extended to the case with uncertain inputs by using the boundedness of parameters.This part of the research will provide a theoretical basis for further research on numerical algorithms for phase field systems with uncertain inputs.