Parabolic Optimal Control Problems Research Articles

Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part II: Outer Approximation Algorithm

Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part II: Outer Approximation Algorithm

Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part I: Convex Relaxations

Parabolic Optimal Control Problems with Combinatorial Switching Constraints, Part I: Convex Relaxations

一种基于对角化的抛物型最优控制问题的预处理子

一种基于对角化的抛物型最优控制问题的预处理子

A duality‐based approach for linear parabolic optimal control problems

SummaryThis paper is concerned with the optimal control problem governed by linear parabolic equation with box constraints on control variables. We employ the Fenchel duality scheme to derive an unconstrained dual problem. Compared with the primal problem, the objective functional of the dual problem includes a projection onto the box constraints. We prove the existence and uniqueness of solutions to the dual problem and derive the first‐order optimality conditions. Furthermore, we investigate the saddle point property between the solutions of the primal problem and the solutions of the dual problem. To solve the dual problem, we design two implementable methods: the conjugate gradient method and the semi‐smooth Newton method. The solutions of the primal problem can be easily obtained through the solutions of the dual problem. We demonstrate the effectiveness and accuracy of the proposed methods by solving three example problems.

An iterative proper orthogonal decomposition method for a parabolic optimal control problem

An iterative proper orthogonal decomposition method for a parabolic optimal control problem

Solution stability of parabolic optimal control problems with fixed state-distribution of the controls

The paper presents results about strong metric subregularity of the optimality mapping associated with the system of first order necessary optimality conditions for a problem of optimal control of a semilinear parabolic equation. The control has a predefined spacial distribution and only the magnitude at any time is a subject of choice. The obtained conditions for subregularity imply, in particular, sufficient optimality conditions that extend the known ones.
 The paper is complementary to a companion one by the same authors, in which a distributed control is considered.

Error Analysis for Parabolic Optimal Control Problems with Measure Data in a Nonconvex Polygonal Domain

Error Analysis for Parabolic Optimal Control Problems with Measure Data in a Nonconvex Polygonal Domain

Convergence analysis and optimization of a Robin Schwarz waveform relaxation method for time-periodic parabolic optimal control problems

This paper is concerned with a novel convergence analysis of the optimized Schwarz waveform relaxation method (OSWRM) for the solution of optimal control problems governed by periodic parabolic partial differential equations (PDEs). The new analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition. This leads to a precise characterization of the convergence factor of the method at the semidiscrete level. Using this characterization, the optimal transmission condition parameter is obtained at the semidiscrete level and its asymptotic behavior as the time discretization converges to zero is analyzed in detail.

Reduced-order finite element approximation based on POD for the parabolic optimal control problem

Reduced-order finite element approximation based on POD for the parabolic optimal control problem

Space−time spectral approximations of a parabolic optimal control problem with an L2‐norm control constraint

AbstractThis article deals with the spectral approximation of an optimal control problem governed by a parabolic partial differential equation (PDE) with an ‐norm control constraint. The investigations employ the space−time spectral method, which is, more precisely, a dual Petrov‐Galerkin spectral method in time and a spectral method in space to discrete the continuous system. As a global method, it uses the global polynomials as the trial functions for discretization of PDEs. After obtaining the optimality condition of the continuous system and that of its spectral discrete surrogate, we establish a priori and a posteriori error estimates for the spectral approximation in detail. Three numerical examples in different spatial dimensions then confirm the theoretical results and also show the efficiency as well as a good precision of the adopted space−time spectral method.

On the solution stability of parabolic optimal control problems

The paper investigates stability properties of solutions of optimal control problems constrained by semilinear parabolic partial differential equations. Hölder or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may nonlinearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper–the metric subregularity of the mapping associated with the system of first order necessary optimality conditions. This property also enables error estimates for approximation methods. A Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product.

Variational discretization combined with fully discrete splitting positive definite mixed finite elements for parabolic optimal control problems

Variational discretization combined with fully discrete splitting positive definite mixed finite elements for parabolic optimal control problems

Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems

<abstract><p>In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order $ m = 1 $ Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results.</p></abstract>

A priori error estimates of finite volume element method for bilinear parabolic optimal control problem

<abstract><p>In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is $ O(h^{3/2}+\triangle t) $ in the sense of $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. Finally, a numerical experiment is presented to test these theoretical results.</p></abstract>

Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems

<abstract><p>In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.</p></abstract>

Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases

<p style='text-indent:20px;'>In this paper, we are concerned with the space-time spectral discretization of an optimal control problem governed by a fourth-order parabolic partial differential equations (PDEs) in three control constraint cases. The dual Petrov-Galerkin spectral method in time and the spectral method in space are adopted to discrete the continuous system. By means of the obtained optimality condition for the continuous system and that of its spectral discrete system, we establish a priori error estimate for the spectral approximation in details. Four numerical examples are, subsequently, executed to confirm the theoretical results. The experiment results show the high efficiency and a good precision of the space-time spectral method for this kind of problems.</p>

Adaptive space–time finite element methods for parabolic optimal control problems

Abstract We present, analyze, and test locally stabilized space–time finite element methods on fully unstructured simplicial space–time meshes for the numerical solution of space–time tracking parabolic optimal control problems with the standard L 2-regularization.We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space– time cylinder.

Stability for Semilinear Parabolic Optimal Control Problems with Respect to Initial Data

A distributed optimal control problem for a semilinear parabolic partial differential equation is investigated. The stability of locally optimal solutions with respect to perturbations of the initial data is studied. Based on different types of sufficient optimality conditions for a local solution of the unperturbed problem, Lipschitz or Hölder stability with respect to perturbations are proved. Moreover, a particular example with semilinear equation, constant initial data, and standard quadratic tracking type objective functional is constructed that has at least two different locally optimal solutions. By the perturbation analysis, the existence of a problem with non-constant initial data is shown that also has at least two different locally optimal solutions.

Application of the Alternating Direction Method of Multipliers to Control Constrained Parabolic Optimal Control Problems and Beyond

Application of the Alternating Direction Method of Multipliers to Control Constrained Parabolic Optimal Control Problems and Beyond

Crank–Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problems

In this paper, we adopt the optimize‐then‐discretize approach to solve parabolic optimal Dirichlet boundary control problems. First, we derive the first‐order necessary optimality system, which includes the state, co‐state equations, and the optimality condition. Then, we propose Crank–Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second‐order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible, and second‐order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present several numerical examples to validate our theoretical analysis.