Recovery Type a Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. We derive the superconvergence properties of finite element solutions. By using the superconvergence results, we obtain recovery type a posteriori error estimates. Some numerical examples are presented to verify the theoretical results.     

A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

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.     

Robust error estimates for the finite element approximation of elliptic optimal control problems

In this paper, we consider the finite element approximation of an elliptic optimal control problem. Based on an assumption on the adjoint state of the continuous problem with a small parameter, which represents a regularization of the bang-bang type control problem, we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented. Numerical experiments confirm our theoretical results.     

Finite Element Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

In this paper finite element approximation of space fractional optimal control problem with integral state constraint is investigated. First order optimal condition and regularity of the control problem are discussed. A priori error estimates for control, state, adjoint state and lagrange multiplier are derived. The nonlocal property of the fractional derivative results in a dense coefficient matrix of the discrete state and adjoint state equation. To reduce the computational cost a fast projection gradient algorithm is developed based on the Toeplitz structure of the coefficient matrix. Numerical experiments are carried out to illustrate the theoretical findings.     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state

We analyze a finite element approximation of an elliptic optimal control problem with pointwise bounds on the gradient of the state variable. We derive convergence rates if the control space is discretized implicitly by the state equation. In contrast to prior work we obtain these results directly from classical results for the W ^1,∞-error of the finite element projection, without using adjoint information. If the control space is discretized directly, we first prove a regularity result for the optimal control to control the approximation error, based on which we then obtain analogous convergence rates.     

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.     

Adaptive characteristic finite element approximation of convection–diffusion optimal control problems

In this article, a characteristic finite element approximation of quadratic optimal control problems governed by linear convection–diffusion equations is given. We derive some a posteriori error estimates for both the control and the state approximations, where the control variable is constrained by pointwise inequality. The derived error estimators are then used as an error indicator to guide the mesh refinement. In this sense, they are very important in developing adaptive finite element algorithm for the optimal control problems. Finally, a numerical example is given to validate the efficiency and reliability of the theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints

In this article, we investigate local discontinuous Galerkin approximation of stationary convection‐dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first‐order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection‐dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L² norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014     

State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations

In this paper, an optimal control problem for the stationary Navier-Stokes equations in the presence of state constraints is investigated. Existence of optimal solutions is proved and first order necessary conditions are derived. The regularity of the adjoint state and the state constraint multiplier is also studied. Lipschitz stability of the optimal control, state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated.     

Residual Based a Posteriori Error Estimates for Convex Optimal Control Problems Governed by Stokes-Darcy Equations

In this paper, we derive a posteriori error estimates for finite element approximations of the optimal control problems governed by the Stokes-Darcy system. We obtain a posteriori error estimators for both the state and the control based on the residual of the finite element approximation. It is proved that the a posteriori error estimate provided in this paper is both reliable and efficient.     

A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems

In this paper, a constrained distributed optimal control problem governed by a first-order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient (PCG) and algebraic multi-grid (AMG) can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in $L^2(Ω)$-norm, for the original state and adjoint state in $H^1(Ω)$-norm, and for the flux state and adjoint flux state in $H$(div; $Ω$)-norm. Finally, we use one numerical example to validate the theoretical findings.     

Linear and Discontinuous Approximations for Optimal Control Problems

Abstract

An optimal control problem for 2D and 3D elliptic equations is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear but discontinuous functions. The state and the adjoint state are discretized by linear finite elements. The paper is focused on similarities and differences to piecewise constant and piecewise linear (continuous) approximation of the controls. Approximation of order h in the L ^∞-norm is proved in the main result.     

A Priori Error Estimates for Obstacle Optimal Control Problem,Where the Obstacle Is the Control Itself

In this paper we deal with the convergence analysis of the finite element method for an elliptic penalized unilateral obstacle optimal control problem where the control and the obstacle coincide. Error estimates are established for both state and control variables. We apply a fixed point type iteration method to solve the discretized problem.To corroborate our error estimations and the efficiency of our algorithms, the convergence results and numerical experiments are illustrated by concrete examples.     

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h ² + τ ^k ) in the energy norm and O((h ² + τ ^k )(h ² + τ)) in the L ²(Q)-norm.     

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.     

A priori error estimates for elliptic optimal control problems with a bilinear state equation

In this paper a priori error analysis for the finite element discretization of an optimal control problem governed by an elliptic state equation is considered. The control variable enters the state equation as a coefficient and is subject to pointwise inequality constraints. We derive a priori error estimates for the discretization error in the control variable and confirm our theoretical results by numerical examples.     

Error estimates for the approximation of multibang control problems

In the present work we apply an augmented Lagrange method to solve pointwise state constrained elliptic optimal control problems. We prove strong convergence of the primal variables as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. In addition, we show that the sequence of generated penalty parameters is bounded only in exceptional situations, which is different from classical results in finite-dimensional optimization. In addition, numerical results are presented.     

Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state

In this paper we are concerned with optimal control problems governed by an elliptic semilinear equation, the control being distributed in . The existence of constraints on the control as well as pointwise constraints on the gradient of the state is assumed. A convenient choice of the control space permits us to derive the optimality conditions and study the adjoint state equation, which has derivatives of measures as data. In order to carry out this study, we prove a trace theorem and state Green's formula by using the transposition method.This research was partially supported by Dirección General de Investigation Cientifica y Técnica (Madrid).     

Some results on pointwise second-order necessary conditions for stochastic optimal controls

The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms. When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that, quite different from the first-order necessary conditions, the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex.