Spectral Numerical Simulation For Optimal Control Problem Governed By Two Kinds Of Fractional Partial Differential Equations

With the development of theory and application about fractional differential equation,the study of fractional optimal control has attracted wide attention.In this thesis,spectral Galerkin of optimal control problems governed by two types of fractional differential equations is investigated.Firstly,the optimal control problem governed by distributed order fractional differential equations is considered:#12 s.t.#12 Here-1Dt? denotes the left Riemann-Liouville fractional derivative.y and u denote state variable and control variable,respectively.yd is desired state.? denotes the regularization parameter.?(?)?L1([?min,?max])(0??min??max?1)is a non-negative function.For this problem we firstly derive the first order optimality condition by La-grangian functional.A spectral Petrov-Galerkin scheme is constructed based on first optimize then discretize approach,where the Jacobi polyfractonomials of first kind and second kind are used as the basis functions to approximate the state and ad-joint state variable,respectively.A prior error estimate of spectral Petrov-Galerkin scheme is proved.Numerical implementations are discussed and numerical example is given to illustrate the effectiveness of the discrete scheme.Secondly,the optimal control problem governed by two-dimension fractional diffusion-reaction equation is discussed:#12 s.t.#12 Here ?? 0.The function f(x)is given and pd is the desired state,?={x=(x1,x2)|x12+x22<1} and ? is complement of the domain in R2.For this problem the first order optimality condition is derived by using La-grangian functional and the regularity of the solution is analyzed based on boot-strapping approach.Using the orthogonal polynomial basis function over the circu-lar domain to approximate the state variable,a spectral Galerkin discrete scheme is constructed.The optimal error estimate of spectral Galerkin discrete scheme is es-tablished in the weighted Sobolev space.Based on the discrete first order optimality condition,the projection gradient algorithm is constructed.Finally the theoretical results are verified by a numerical example.