| Due to the capability of efficiently modeling long-term memory,long-range spatial interactions and anomalous diffusion in physical systems,fractional differential equations find important applications in many scientific and engineering fields.The inherent non-local properties and singularities of fractional calculus operators affect regularity of the solution,which brings great challenges to the construction of high-accuracy approximations for related problems.This thesis focuses on the spectral Petrov-Galerkin method for solving optimal control problems governed by three types of fractional differential equations.Without any regularity assumption on the analytical solution,regularity of the optimal control problems is firstly established in weighted Sobolev space,then a framework of convergence analysis and error estimate on spectral Petrov-Galer kin method is provided for the optimal control problems.The obtained results of the thesis may enrich the theory and numerical methods of optimal control problems with non-local differential equation constraints,and to some extent provide theoretical support in making control decision in practical optimization problems.Firstly,we investigate a spectral Petrov-Galerkin method for a fractional differential equation with initial value.To compensate the weak singularity of the solution at the origin,we analyze the regularity of the problem in weighted Sobolev space by the regularity lifting technique.Using weighted Jacobi polynomials as basis functions,a spectral Petrov-Galerkin method is established,and its optimal error estimate is given based on the regularity results.To increase the efficiency of computation,by selecting the appropriate preconditioner,an iterative algorithm with a quasi-linear complexity is proposed to solve the produced dense linear system.Numerical results show the validity of the theoretical analysis and availability of the numerical method,where the iteration numbers are independent of the fractional orders and the degree of polynomials.According to the above results,we study a spectral Petrov-Galerkin method for an optimal control problem governed by a fractional ordinary differential equation.Using the regularity results of the fractional differential equation with initial value,the regularity of the optimal control problem is established by analyzing the regularity connection between the functions t-α u and u in the weighted Sobolev space,and an optimal error estimate of the spectral PetrovGalerkin method is given.Based on the fast iterative algorithm,a fast projected gradient algorithm is presented to solve the resulting discrete optimal control problem.The theoretical findings are verified by numerical experiments.Secondly,we consider a spectral Petrov-Galerkin method for an optimal control problem governed by a two-sided fractional diffusion-advection-reaction equation.To overcome the weak singularity of the solution near boundaries caused by the fractional differential operator,we analyze the regularity of the optimal control problem in the weighted Sobolev space by the lifting technique.With the regularity results,we prove the stability and convergence of the spectral Petrov-Galerkin method.To save the computational cost,based on the fast polynomial transform a fast projected gradient algorithm is presented to solve the resulting discrete optimal control problem.Numerical experiments show the validity of the numerical method and the proposed fast algorithm.Finally,we develop a spectral Petrov-Galerkin method for an optimal control problem governed by a two-sided fractional diffusion-reaction equation with fractional noise.For smooth or rough data,we construct a framework of regularity for the corresponding deterministic optimal control problem,and analyze regularity of the fractional noise by the connection between two Jacobi polynomials in weighted Sobolev space.Then taking the fractional noise as the rough input,we obtain the regularity results for the stochastic optimal control problem in weighted Sobolev space.By employing truncated spectral expansion of the fractional noise,a spectral Petrov-Galerkin method is established.Based on the regularity results,error estimates are given for the optimal control problem with fractional noise as well as its deterministic counterpart with rough source term.The observed numerical results of experiments are well consistent with the theoretical findings. |
