Parabolic Optimal Control Problems Research Articles

Parabolic optimal control problems with combinatorial switching constraints, part III: branch-and-bound algorithm

We present a branch-and-bound algorithm for globally solving parabolic optimal control problems with binary switches that have bounded variation and possibly need to satisfy further combinatorial constraints. More precisely, for a given tolerance ε>00$$\\end{document}]]>, we show how to compute in finite time an ε-optimal solution in function space, independently of any prior discretization. The main ingredients in our approach are an appropriate branching strategy in infinite dimension, an a posteriori error estimation in order to obtain safe dual bounds, and an adaptive refinement strategy in order to allow arbitrary switching points in the limit. The performance of our approach is demonstrated by extensive experimental results.

Regularity of Multipliers and Second-Order Optimality Conditions for Semilinear Parabolic Optimal Control Problems with Mixed Pointwise Constraints

A class of optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is considered. We give some criteria under which the first and second-order optimality conditions are of KKT-type. We then prove that the Lagrange multipliers belong to L p -spaces. Moreover, we show that if the initial value is good enough and boundary ∂ Ω has a property of positive geometric density, then multipliers and optimal solutions are Hölder continuous.

Analysis and approximation to parabolic optimal control problems with measure-valued controls in time

In this paper, we investigate an optimal control problem governed by parabolic equations with measure-valued controls over time. We establish the well-posedness of the optimal control problem and derive the first-order optimality condition using Clarke's subgradients, revealing a sparsity structure in time for the optimal control. Consequently, these optimal control problems represent a generalization of impulse control for evolution equations. To discretize the optimal control problem, we employ the space-time nite element method. Here, the state equation is approximated using piecewise linear and continuous nite elements in space, alongside a Petrov-Galerkin method utilizing piecewise constant trial functions and piecewise linear and continuous test functions in time. The control variable is discretized using the variational discretization concept. For error estimation, we initially derive a priori error estimates and stabilities for the nite element discretizations of the state and adjoint equations. Subsequently, we establish weak-* convergence for the control under the norm M(\bar I_c;L2(\omega)), with a convergence order of O(h^{1/2} + k^{1/4}) for the state.

Locally Lipschitz Stability of Solutions to a Parametric Parabolic Optimal Control Problem with Mixed Pointwise Constraints

Locally Lipschitz Stability of Solutions to a Parametric Parabolic Optimal Control Problem with Mixed Pointwise Constraints

New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms

New Time Domain Decomposition Methods for Parabolic Optimal Control Problems I: Dirichlet–Neumann and Neumann–Dirichlet Algorithms

An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems

An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems

A parallel-in-time preconditioner for Crank–Nicolson discretization of a parabolic optimal control problem

A parallel-in-time preconditioner for Crank–Nicolson discretization of a parabolic optimal control problem

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

This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain. Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain. The low regularity of the solution allows the finite element approximations to converge at lower orders. We prove the existence, uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition. For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables, whereas piecewise constant functions are employed to approximate the control variable. The temporal discretization is based on the implicit Euler scheme. We derive both a priori and a posteriori error bounds for the state, control and co-state variables. Numerical experiments are performed to validate the theoretical rates of convergence.

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

We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. While such combinatorial constraints are often seen as an additional complication that is treated in a heuristic postprocessing, the core of our approach is to investigate the convex hull of all feasible switching patterns in order to define a tight convex relaxation of the control problem. The convex relaxation is built by cutting planes derived from finite-dimensional projections, which can be studied by means of polyhedral combinatorics. A numerical example for the case of a bounded number of switchings shows that our approach can significantly improve the dual bounds given by the straightforward continuous relaxation, which is obtained by relaxing binarity constraints.

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.

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

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

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>