Numerical Simulation Of Fractional Optimal Control Problem By Using Spectral Collocation Method

In this paper,the spectral collocation methods for two classes of fractional optimal control problem with control constrained are studied.We first consider the optimal control problem governed by time fractional dif-fusion equation:Here 0Dt?u denotes the left Caputo fractional derivative of order ?(0<?<1)of the state u.Based on 'first discretize,then optimize' approach,we present a numerical scheme for above time-fractional optimal control problem.For spatial discretization,we employ the Legendre spectral collocation method.The semi-discrete scheme of the optimal control problem is established,and semi-discrete first order optimality condition is derived.By using the L1 scheme which based on the finite difference method for time discretization,we construct the fully discrete scheme of the optimal control problem.The fully discrete first order optimality condition is deduced.Furthermore,the projected gradient algorithm is designed based on the fully discrete optimality conditions.Finally,numerical experiments are given to illustrate the effectiveness of the proposed scheme and algorithm.A comparison of this method with the finite element method[18]is presented.Next,we investigate the optimal control problem governed by the space frac-tional diffusion equation:Here-1RLDx1+? denotes the left Riemann-Liouville fractional derivative of order 1+?(0<?<1)of state u.Based on 'first optimize,then discretize' approach,we proposed two fractional spectral collocation numerical schemes for the above space-fractional optimal control problem.The continuous first order optimality condition is derived by using the Pontryagin's principle.We adopt the eigenfunctions of two classes of fractional Strum-Liouville problem as basis function to approximate state variable and adjoint state variable.Then the fractional spectral collocation scheme of optimal control problem is constructed.Note that the solutions of state equation and adjoint state equation have singularity at endpoints,we employ the generalized fractional spectral collocation method[39]to discrete the state and adjoint state e-quation.And the generalized fractional spectral collocation scheme is constructed.We design the projected gradient algorithm based on the discrete optimality con-ditions.Finally,numerical experiment is carried to verify the effectiveness of the proposed numerical schemes and algorithm.This two methods are compared with standard collocation method.