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In this paper a priori error analysis for time-stepping discontinuous Galerkin finite element approximation of optimal control problem governed by time fractional diffusion equation is presented. A time-stepping discontinuous Galerkin finite element method and variational discretization approach are used to approximate the state and control variables respectively. Regularity of the optimal control problem is discussed. Since the time fractional derivative is nonlocal, in order to reduce the computational cost a fast gradient projection algorithm is designed for the control problem based on the block triangular Toeplitz structure of the discretized state equation and adjoint state equation. Numerical examples are carried out to illustrate the theoretical findings and fast algorithm.

In this paper a posteriori error analysis of virtual element method (VEM) for the optimal control problem governed by general elliptic equation is presented. The virtual element discrete scheme is constructed with virtual element approximation of the state equation and variational discretization of the control variable. Based on the a posteriori error estimates of virtual element method for general elliptic equation and approximated error equivalence of the solution of the optimal control problem to solutions of the state and adjoint problems we build up upper and lower a posteriori error estimates of the optimal control problem. Under the Dorfler’s marking strategy, the traditional projected gradient algorithm and adaptive VEM algorithm drived by the state and adjoint error estimators are used to solve the optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

In this paper we investigate finite element approximation of optimal control problem governed by space fractional diffusion equation with control constraints. The control variable is approximated by piecewise constant. Regularity estimate for the control problem is proved based on the first order optimality system and a priori error estimates for the state, the adjoint state and the control variables are derived. Due to the nonlocal property of fractional derivative, which will leads to a full stiff matrix, we develop a fast primal dual active set algorithm for the control problem. Numerical examples are given to illustrate the theoretical findings and the efficiency of the fast algorithm.

In this paper, a piecewise constant time-stepping discontinuous Galerkin method combined with a piecewise linear finite element method is applied to solve control constrained optimal control problem governed by time fractional diffusion equation. The control variable is approximated by variational discretization approach. The discrete first-order optimality condition is derived based on the first discretize then optimize approach. We demonstrate the commutativity of discretization and optimization for the time-stepping discontinuous Galerkin discretization. Since the state variable and the adjoint state variable in general have weak singularity near t = 0and t = T, a time adaptive algorithm is developed based on step doubling technique, which can be used to guide the time mesh refinement. Numerical examples are given to illustrate the theoretical findings.

Finite element approximations of Dirichlet boundary control problems governed by parabolic PDEs on convex polygonal domains are studied in this paper. The existence of a unique solution to optimal control problems is guaranteed based on very weak solution of the state equation and $$L^2(0,T;L^2(\varGamma ))$$L2(0,T?L2(Γ)) as control space. For the numerical discretization of the state equation we use standard piecewise linear and continuous finite elements for the space discretization of the state, while a dG(0) scheme is used for time discretization. The Dirichlet boundary control is realized through a space---time $$L^2$$L2-projection. We consider both piecewise linear, continuous finite element approximation and variational discretization for the controls and derive a priori $$L^2$$L2-error bounds for controls and states. We finally present numerical examples to support our theoretical findings.

We propose a characteristic finite element discretization of evolutionary type convection-diffusion optimal control problems. Nondivergence-free velocity fields and bilateral inequality control constraints are handled. Then some residual type a posteriori error estimates are analyzed for the approximations of the control, the state, and the adjoint state. Based on the derived error estimators, we use them as error indicators in developing efficient multi-set adaptive meshes characteristic finite element algorithm for such optimal control problems. Finally, one numerical example is given to check the feasibility and validity of multi-set adaptive meshes refinements.

In this paper, we provide a numerical scheme--RT mixed FEM/DG scheme for the constrained optimal control problem governed by convection dominated diffusion equations. A priori and a posteriori error estimates are obtained for both the state, the co-state and the control. The adaptive mesh refinement can be applied indicated by a posteriori error estimator provided in this paper. Numerical examples are presented to illustrate the theoretical analysis.

In this paper, we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations. The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L2-norm and for state y and costate p measured in energy norm. Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.

In this paper a posteriori error estimate for continuous interior penalty Galerkin approximation of transient convection dominated diffusion optimal control problems with control constraints is presented. The state equation is discretized by the continuous interior penalty Galerkin method with continuous piecewise linear polynomial space and the control variable is approximated by implicit discretization concept. By use of the elliptic reconstruction technique proposed for parabolic equations, a posteriori error estimates for state variable, adjoint state variable and control variable are proved, which can be used to guide the mesh refinement in the adaptive algorithm.