Optimal Control Problems for Elliptic–Parabolic Variational Inequalities with Time-Dependent Constraints

In this paper, we study optimal control problems for quasi-linear elliptic–parabolic variational inequalities with time-dependent constraints. We prove the existence of an optimal control that minimizes the nonlinear cost functional. Moreover, we apply our general results to some model problems. In particular, we show the necessary condition of optimal pair for a problem of partial differential equation (PDE) with a non-homogeneous Dirichlet boundary condition.     

Optimization methods for Dirichlet control problems

We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitly using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analysed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau–Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results.     

A Domain Embedding Method Using the Optimal Distributed Control and a Fast Algorithm

We propose a domain embedding method to solve second order elliptic problems in arbitrary two-dimensional domains. The method is based on formulating the problem as an optimal distributed control problem inside a disc in which the arbitrary domain is embedded. The optimal distributed control problem inside the disc is solved rapidly using a fast algorithm developed by Daripa et al. [3,7,10–12]. The arbitrary domains can be simply or multiply connected and the proposed method can be applied, in principle, to a large number of elliptic problems. Numerical results obtained for Dirichlet problems associated with the Poisson equation in simply and multiply connected domains are presented. The computed solutions are found to be in good agreement with the exact solutions with moderate number of grid points in the domain.     

A priori estimates for optimal Dirichlet boundary control problems

We consider variational discretization of control constrained elliptic Dirichlet boundary control problems on smooth twoand three-dimensional domains, where we take into account the domain approximation. The state is discretized by linear finite elements, while the control variable is not discretized. We obtain optimal error bounds for the optimal control in two and three space dimensions. Furthermore we prove a superconvergence result in two space dimensions under the assumption that the underlying finite element meshes satisfy certain regularity requirements. We confirm our findings by a numerical experiment. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A STUDY OF THE PROFITABILITY FOR AN OPTIMAL CONTROL PROBLEM WHEN THE SIZE OF THE DOMAIN CHANGES

ABSTRACT. In this work we consider the increase in benefit for a control problem when the size of domain increases. Our control problem involves the study of the profitability of a biological growing species whose growth is confined to a bounded domain Ω? R^N and is modeled by a logistic elliptic equation with different boundary conditions (Dirichlet or Neumann). The payoff-cost functional considered, J, is of quadratic type. We prove that, under Dirichlet boundary conditions, the optimal benefit (sup J) increases when the domain ? increases. This is not true under Neumann boundary conditions.     

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.     

Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEs

In this paper, two meshless schemes are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The first scheme uses radial basis function collocation method (RBF-CM) for both state equation and adjoint state equation, while the second scheme employs the method of fundamental solution (MFS) for the state equation when it has a zero source term, and RBF-CM for the adjoint state equation. Numerical examples are provided to validate the efficiency of the proposed schemes.     

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

The main objective of this paper is optimization of second‐order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi‐infinite strip is considered. The boundary impedance (Neumann‐to‐Dirichlet map) is computed as the square root of an operator using the standard three‐point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx‐imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.     

Sparse Dirichlet optimal control problems

Computational Optimization and Applications - In this paper, we analyze optimal control problems governed by an elliptic partial differential equation, in which the control acts as the Dirichlet...     

On a Fourier Method of Embedding Domains Using an Optimal Distributed Control

We propose a domain embedding method to solve second order elliptic problems in arbitrary two-dimensional domains. This method can be easily extended to three-dimensional problems. The method is based on formulating the problem as an optimal distributed control problem inside a rectangle in which the arbitrary domain is embedded. A periodic solution of the equation under consideration is constructed easily by making use of Fourier series. Numerical results obtained for Dirichlet problems are presented. The numerical tests show a high accuracy of the proposed algorithm and the computed solutions are in very good agreement with the exact solutions.     

On Neumann boundary control problem for ill-posed strongly nonlinear elliptic equation with p-Laplace operator and $$L^1$$L1-type of nonlinearity

Ricerche di Matematica - In this paper we study an optimal control problem for the mixed Dirichlet–Neumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace...     

Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints: Part 1. Boundary Control

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve these problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.     

FINITE ELEMENT APPROXIMATION OF A NONLINEAR STEADY-STATE HEAT CONDUCTION PROBLEM

1. Introduction Anciellt Chinese mathematicians have done many fund~al discoveries, even thoughmad of them are now usually called by western names, e.g., the Pascal triangle, the HornerIcheme, the Gaussian elimination, see [231. In the modern era3 Chinese mathematiciare have.jalso got many priorities. For instance, the fast Proof Of convergence of the finite element methodfor a linear elliptic boundary vajue problem was done in the pioneering work [61 by K. Feng in1965 (for the English tra…     

A multigrid method for weakly nonlinear elliptic equations of the second order

We consider the Dirichlet problem for a weakly nonlinear elliptic equation of the second order in the divergence form. We prove the convergence of a multigrid method for solving this problem. The method under consideration is based on the application of conformal finite elements and the Jacobi smoothing procedure.     

在边界附近蜕化的椭圆型Monge-Ampère方程解的正则性(英文)

我们证明了在边界附近蜕化的椭圆型Ｍｏｎｇｅ－Ａｍｐèｒｅ方程Ｄｉｒｉｃｈｌｅｔ问题解的整体Ｃ１,１正则性,并举例说明了我们的结果是最佳的。     

OPTIMIZATION APPROACH FOR THE MONGE-AMPèRE EQUATION

In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampère equation with Dirichlet boundary conditions. We formulate the Monge-Ampère equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.     

Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions

We study an optimal design problem for the domain of an elliptic equation with Dirichlet boundary conditions. We introduce a relaxed formulation of the problem which always admits a solution, and we prove some necessary conditions for optimality both for the relaxed and for the original problem.     

On the exterior two-dimensional Dirichlet problem for elliptic equations

A necessary and sufficient condition is given on the boundary datum in order to the Dirichlet problem for an elliptic equation in a two-dimensional exterior Lipschitz domain has a unique solution with a finite Dirichlet integral which converges uniformly at infinity to an assigned constant value.     

半线性椭圆方程支配系统的最优性条件

本文讨论了可能具有多值解的椭圆型偏微分方程支配系统的最优控制问题,我们通过构造一个抛物方程控制问题的逼近序列,并利用抛物方程控制问题的结果,得到了椭圆系统最优控制的必要条件．     

Solving elliptic control problems with interior point and SQP methods: control and state constraints

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. Both boundary control and distributed control problems are considered with boundary conditions of Dirichlet or Neumann type. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. Necessary conditions of optimality are discussed both for the continuous and the discretized control problem. It is shown that the recently developed interior point method LOQO of [35] is capable of solving these problems even for high discretizations. Four numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang–bang controls.