Spectral-Galerkin Method For Optimal Control Problem Governed By Space Fractional Differential Equation

Optimal control problem governed by fractional differential equations(FDEs)have received more and more attention not only in model problems,but also in numerical methods,since its wide applications in groundwater pollution problem and other practical problems.This dissertation mainly studies the Spectral-Galerkin method for two types of optimal control problem governed by space fractional differential equation with control integral constraint.Firstly,we study the following optimal control problem governed by Riesz frac-tional differential equation:#12 subject to#12Here y is the state variable,u is the control variable,Uad is the constraint set of u,?>0 is the regularization parameter,D? is the Riesz fractional derivative.By using Lagrange functional we derive the first order optimality condition and further discuss the regularity of the optimality system.Based on first discretize then optimize approach a Spectral-Galerkin approximation of the control problem is developed.By introducing some auxiliary problems,? priori error analysis for state variable,adjoint state variable and control variable is presented.Numerical experiments are presented to support our theoretical results.Secondly,we study the following optimal control problem governed by two side fractional differential equation with a reaction term:#12 subject to#12For the above control problem,we derive the continuous first order optimality condition,and on this basis,the regularity of the solution of the optimal control prob-lem in symmetric case(?=0.5)is analyzed.The generalized Jacobi polynomials are used to approximate the state variable and the adjoint state variable.For the control variable variational discretization approach was adopted.For symmetric(?=0.5)and asymmetric(??0.5)cases,the Spectral-Galerkin approximation and the Spec-tral Petrov-Galerkin approximation of the control problem are developed,respective-ly.Then we present a priori error analysis for state variable,adjoint state variable and control variable by introducing some auxiliary problems.Due to the presence of reaction term in the problem,the coefficient matrixes of the discrete state and adjoint state equation are dense.A direct solver will require a lot of storage and complexity.Therefore we present two projection gradient algorithms based on fast polynomial transformation.Finally,numerical experiments are presented to demon-strate our theoretical results.