Finite Element Method For State Constrained Fractional Optimal Control Problems

Fractional differential equations are widely used in fluid,continuum mechanics,solute transport in groundwater and other problems.In recent years,the research on numerical algorithm for optimal control problem of fractional differential equations has become a hot issue,which has been widely concerned by researchers.In this paper,the finite element method for two kinds of state constrained frac-tional order optimal control problems is studied.Firstly,we consider the following time fractional optimal control problem.#12 subject to#12 and#12 Here ? is a bounded domain of Rd(d=1,2,3),?T=?×(0,T),?T=?×(0,T),? is a fixed constant,ud is the observed value,u is the state variable and q is the control variable.?>0 is the regularization parameter.Uad=L2(?T),f?L2(?T)is the given function.0Dt?(0<?<1)is the Riemann Lioville fractional derivative.For above control problem,the continuous first order optimality condition is derived and the corresponding regularity of the solutions is discussed.A space-time finite element discrete scheme is built up based on piecewise constant discontinuous Galerkin discretization for temporal discretization and conforming linear finite ele-ments for spatial discretization.A priori error estimate of state,adjoint state,control variable and multiplier is derived.A projection gradient algorithm is designed to solve the optimal control problem with state constraints,and numerical examples are given to verify the correctness of the theoretical analysis and the effectiveness of the algorithm.Secondly,we consider the following space fractional optimal control problem:#12 subject to#12 and#12 Here ?(?)R2 is a bounded domain,ud? L2(?)is the observed value,? is the regularization parameter,? is a given constant,Uad=L2(?)and(-?)s(0<s<1)is a fractional Laplacian operator in integral form.For the above control problem,the continuous first-order optimality condition and the regularity of the solution are derived.Piecewise linear finite element method is used to approximate the equation of state.Then we build up the finite element scheme for the control problem.The a priori error estimates of states,adjoint s-tates,control variables and multipliers are derived.Finally,the theoretical results are verified by numerical examples.