Spectral Galerkin Method For Optimal Control Problem Governed By Fractional Convection Diffusion Equation

Fractional convection diffusion reaction equations are widely used in many phys-ical phenomena,such as the transport of pollutants in groundwater,chaotic dynam-ics and so on.In recent years,the research on numerical methods or algorithms for optimal control problems of fractional differential equations has become a hot topic.In this thesis,two kinds of optimal control problem governed by fractional convection diffusion equation were studied.Spectral-Galerkin discrete schemes and corresponding numerical analysis theory were established.Firstly,we study the following optimal control problem governed by fractional convection diffusion reaction equation with control constraint:#12 subject to#12 and the control constraint set#12 where ?=(-1,1),?c=R\? and D is the first-order derivative with respect to x.Here ?1?0 and ?2?0 are constants.The function f(x)is given and yd is the desired state.(-?)?/2(??(1,2))is an integral fractional Laplace operator.In order to solve the above problem,the first order optimality condition was derived,and then the regularity of the solutions was analyzed.Using the weighted Jacobi polynomials as the basis function,a Spectral-Galerkin discrete scheme was constructed.The error estimates of state,adjoint state and control variable in H?/2 and the weighted L2 norms were derived.Finally,in order to improve the computational efficiency,a fast projection gradient algorithm was constructed based on fast polynomial transformation and numerical examples were given to verify the theoretical results.Secondly,we study the following optimal control problem governed by fractional convection diffusion reaction equation with state constraint:#12 subject to#12 and the state constraint set Gad={??L2(?):||?(x)||?d},where ||·|| denotes the L2 norm.For the above control problem,the first order optimality condition was derived.Then using the weighted Jacobi polynomials as the basis function,the Spectral-Galerkin discrete scheme was constructed.The error estimates of state,adjoint state,control variable and Lagrange multiplier in H?/2 and the weighted L2 norms were derived.Finally,a numerical example was given to verify the above theoretical results.