Elliptic Optimal Control Problems Research Articles

Conjugate gradient-boundary element solution for distributed elliptic optimal control problems

An optimality system of equations for the optimal control problem governed by Helmholtz-type equations is derived. By the associated first-order necessary optimality condition, we obtain the conjugate gradient method (CGM) in the continuous case. Introducing the sequence of higher-order fundamental solutions, we propose an iterative algorithm based on the conjugate gradient-boundary element method using the multiple reciprocity method (CGM+MRBEM) for solving the discrete control input. This algorithm has an advantage over that of the existing literatures because the main attribute (the reduced dimensionality) of the boundary element method is fully utilized. Finally, the local error estimates for this scheme are obtained, and a test problem is given to illustrate the efficiency of the proposed method.

Analysis of block matrix preconditioners for elliptic optimal control problems

AbstractIn this paper, we describe and analyse several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear‐quadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term with a parameter α is included. The first algorithm reduces the saddle point system to a symmetric positive definite Schur complement system for the control variable and employs conjugate gradient (CG) acceleration, however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence independent of the mesh size h is described for Ω ⊂ R2 or R3, and a preconditioner independent of h and α when Ω ⊂ R2. Next, two algorithms avoiding double iteration are described using an augmented Lagrangian formulation. One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of h. In addition to the above algorithms, two heuristic algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms. Copyright © 2007 John Wiley & Sons, Ltd.

Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem

We consider an elliptic optimal control problem with pointwise state constraints. The cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation. The corresponding minima are shown to converge in $L^2$ to the exact control as the discretization parameter tends to zero. Furthermore, error bounds for the control and the state are obtained in both two and three space dimensions. Finally, we present numerical examples which confirm our analytical findings.

Lipschitz Stability of Solutions to Some State-Constrained Elliptic Optimal Control Problems

In this paper, optimal control problems with pointwise state constraints for linear and semilinear elliptic partial differential equations are studied. The problems are subject to perturbations in the problem data. Lipschitz stability with respect to perturbations of the optimal control and the state and adjoint variables is established initially for linear-quadratic problems. Both the distributed and Neumann boundary control cases are treated. Based on these results, and using an implicit function theorem for generalized equations, Lipschitz stability is also shown for an optimal control problem involving a semilinear elliptic equation.

A globalization strategy for the multigrid solution of elliptic optimal control problems

A globalization strategy for multigrid schemes solving optimal control problems is presented. This approach searches for possible negative eigenvalues of the reduced Hessian considered at the coarsest grid of the multigrid process. If negative eigenvalues are detected, a globalization step in the direction of negative curvature is performed to escape undesired maxima or saddle points. It is shown that the multigrid solution step provides a descent update. Examples are given to illustrate and validate the present approach.

Domain decomposition methods for advection dominated linear-quadratic elliptic optimal control problems

We present an optimization-level domain decomposition (DD) preconditioner for the solution of advection dominated elliptic linear-quadratic optimal control problems, which arise in many science and engineering applications. The DD preconditioner is based on a decomposition of the optimality conditions for the elliptic linear-quadratic optimal control problem into smaller subdomain optimality conditions with Dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces. These subdomain optimality conditions are coupled through Robin transmission conditions for the states and the adjoints. The parameters in the Robin transmission condition depend on the advection. This decomposition leads to a Schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces. The Schur complement operator is the sum of subdomain Schur complement operators, the application of which is shown to correspond to the solution of subdomain optimal control problems, which are essentially smaller copies of the original optimal control problem. We show that, under suitable conditions, the application of the inverse of the subdomain Schur complement operators requires the solution of a subdomain elliptic linear-quadratic optimal control problem with Robin boundary conditions for the state. Numerical tests for problems with distributed and with boundary control show that the dependence of the preconditioners on mesh size and subdomain size is comparable to its counterpart applied to a single advection dominated equation. These tests also show that the preconditioners are insensitive to the size of the control regularization parameter.

Existence of Regular Lagrange Multipliers for a Nonlinear Elliptic Optimal Control Problem with Pointwise Control-State Constraints

A class of optimal control problems for semilinear elliptic equations with mixed control-state constraints is considered. The existence of bounded and measurable Lagrange multipliers is proven. As a particular application, the Lavrentiev-type regularization of pointwise state constraints is discussed. Here, the existence of associated regular multipliers is shown, too.

Sufficient Second‐Order Optimality Conditions for an Elliptic Optimal Control Problem with Pointwise Control‐State Constraints

An optimal control problem for a semilinear elliptic equation is investigated, where pointwise constraints are given on control and state. The state constraints are of mixed (bottleneck) type, where associated Lagrange multipliers can be chosen as bounded and measurable functions. Based on this property, a second-order sufficient optimality condition is established that takes into account strongly active constraints.

An inexact Newton method combined with Hestenes multipliers’ scheme for the solution of Karush–Kuhn–Tucker systems

In this work a Newton interior-point method for the solution of Karush–Kuhn–Tucker systems is presented.A crucial feature of this iterative method is the solution, at each iteration, of the inner subproblem. This subproblem is a linear-quadratic programming problem, that can solved approximately by an inner iterative method such as the Hestenes multipliers’ method.A deep analysis on the choices of the parameters of the method (perturbation and damping parameters) has been done.The global convergence of the Newton interior-point method is proved when it is viewed as an inexact Newton method for the solution of nonlinear systems with restriction on the sign of some variables.The Newton interior-point method is numerically evaluated on large scale test problems arising from elliptic optimal control problems which show the effectiveness of the approach.

Optimal control of radiating panels via sequential quadratic programming

The optimal control of the steady‐state temperature distribution in radiating panels using control heat sources is considered. The problem has important applications in the thermal control of space structures. A mathematical model leads to an elliptic nonlinear optimal control problem. A numerical optimal control method, based on finite element (FE) discretization and sequential quadratic programming (SQP), is employed. Results are presented for some specific examples.

Optimal Controls of 3-Dimensional Navier--Stokes Equations with State Constraints

This work is concerned with the maximum principles for optimal control problems governed by 3-dimensional Navier--Stokes equations. Some types of state constraints (time variables) are considered.

Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems

In this paper, sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems. These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach. Our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work.

Stability of elliptic optimal control problems

We consider optimal problems governed by elliptical equations on parameters and give sufficient conditions for the continuous dependence of the solutions on the parameters. The techniques are based on variational methods.

Solution of static optimal control problems in non‐linear elasticity via quadratic programming

A class of static (elliptic) optimal control problems in non-linear elasticity is considered. A new direct finite element formulation is used which results in a sequence of quadratic programming (QP) problems. The latter are solved using a standard QP algorithm. The method is applied to the problems of minimizing the lateral deflection of a membrane on a nonlinear elastic foundation and minimizing the in-plane large deformation of a hyperelastic plate.

A direct approach to the finite element solution of elliptic optimal control problems

A direct approach to the finite element solution of elliptic optimal control problems

Discretization of nonlinear elliptic optimal control problems

We consider an optimal control problem for systems defined by nonlinear elliptic partial differential equations, with state constraints. Using relaxation theory, we prove the existence of optimal controls and derive necessary conditions for optimality for the continuous problem. We then discretize the problem by using a finite element method and a control approximation and give the analogue theorems for the discrete problems. The main result is that sequences of optimal (or extremal) discrete controls contain subsequences which converge to optimal (or extremal) controls for the continuous relaxed problem.

On some elliptic optimal control problems with state constraints

In this paper optimality conditions will be derived for elliptic optimal control problems with a restriction on the state or on the gradient of the state. Essential tools are the method of transposition and generalized trace theorems and green's formulas from the theory of elliptic differential equations.