椭圆型变分不等式的障碍优化控制问题

讨论了椭圆型变分不等式的障碍优化控制问题,获得了优化控制问题的解的存在性、唯一性和相关问题的正则性等,并研究了优化控制问题的逼近等．     

变双障碍问题解的抽象稳定性

本文考虑了主部为非线性变双障碍问题解的抽象稳定性 (连续依赖性 ) .由于采用了弱收敛原理和文 [2 ]中取检验函数的技巧 ,我们的证明无需像 [1 ]那样应用Minty引理 .     

拟线性椭圆变分不等式的障碍最优控制问题

对拟线性椭圆变分不等式的障碍最优控制问题(即以障碍为控制变量)进行了研究．指标泛函为Lagrange型,其中含有控制变量二阶导数的p次幂,这使得最优性条件的推导颇为不易．对所考虑的问题给出了最优控制的存在性定理以及必要条件．     

A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.     

Optimal Control of the Obstacle for an Elliptic Variational Inequality

An optimal control problem for an elliptic obstacle variational inequality is considered. The obstacle is taken to be the control and the solution to the obstacle problem is taken to be the state. The goal is to find the optimal obstacle from H ¹ ₀ (Ω) so that the state is close to the desired profile while the H ¹ (Ω) norm of the obstacle is not too large. Existence, uniqueness, and regularity as well as some characterizations of the optimal pairs are established. Accepted 11 September 1996     

Numerical Approximation of a Unilateral Obstacle Problem

We consider a reformulation of the unilateral obstacle problem presented by the authors (Addou and Mermri in Math-Rech. Appl. 2:59–69, 2000). This reformulation introduces a continuous function, whose subdifferential characterizes the noncontact domain. Our goal in this paper is to give a numerical approximation of the solution of the reformulated problem. We consider discretization of the problem based on finite element method. Then we prove the convergence of the approximate solution to the exact one. Some numerical tests on one-dimensional obstacle problem are provided.     

Optimal Control of the Obstacle for a Parabolic Variational Inequality

An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L²(0, T; H²(Ω) ∩ H¹₀(Ω)) with ψ_t ∈ L²(Q) is taken as the control, and the solution to the obstacle problem is taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Indirect Obstacle Minimax Control for Elliptic Variational Inequalities

A minimax control problem for a coupled system of a semilinear elliptic equation and an obstacle variational inequality is considered. The major novelty of such problem lies in the simultaneous presence of a nonsmooth state equation (variational inequality) and a nonsmooth cost function (sup norm). In this paper, the existence of optimal controls and the optimality conditions are established.     

Bilateral Obstacle Optimal Control for a Quasilinear Elliptic Variational Inequality

ABSTRACT

In this paper, we consider an obstacle control problem where the state satisfies a quasilinear elliptic bilateral variational inequality and the control functions are the upper and the lower obstacles. Existence and necessary conditions for the optimal control are established.     

An Obstacle Control Problem with a Source Term

Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control, and the goal is to keep the solution of the variational inequality close to the desired profile while the H ¹ norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for the existence of a minimizer.     

A Dynamical Method for Solving the Obstacle Problem

In this paper, we consider the unilateral obstacle problem, trying to find the numerical solution and coincidence set. We construct an equivalent format of the original problem and propose a method with a second-order in time dissipative system for solving the equivalent format. Several numerical examples are given to illustrate the effectiveness and stability of the proposed algorithm. Convergence speed comparisons with existent numerical algorithm are also provided and our algorithm is fast.     

An Obstacle Control Problem with a Source Term

   Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control, and the goal is to keep the solution of the variational inequality close to the desired profile while the H ¹ norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for the existence of a minimizer.     

Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems

In this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.     

Optimal Control of the Obstacle Problem in a Perforated Domain

We study the problem of optimally controlling the solution of the obstacle problem in a domain perforated by small periodically distributed holes. The solution is controlled by the choice of a perforated obstacle which is to be chosen in such a fashion that the solution is close to a given profile and the obstacle is not too irregular. We prove existence, uniqueness and stability of an optimal obstacle and derive necessary and sufficient conditions for optimality. When the number of holes increase indefinitely we determine the limit of the sequence of optimal obstacles and solutions. This limit depends strongly on the rate at which the size of the holes shrink.     

Use of Augmented Lagrangian Methods for the Optimal Control of Obstacle Problems

We investigate optimal control problems governed by variational inequalities involving constraints on the control, and more precisely the example of the obstacle problem. In this paper, we discuss some augmented Lagrangian algorithms to compute the solution.     

Augmented Lagrangian Active Set Methods for Obstacle Problems

Active set strategies for two-dimensional and three-dimensional, unilateral and bilateral obstacle problems are described. Emphasis is given to algorithms resulting from the augmented Lagrangian (i.e., primal-dual formulation of the discretized obstacle problems), for which convergence and rate of convergence are considered. For the bilateral case, modifications of the basic primal-dual algorithm are also introduced and analyzed. Finally, efficient computer realizations that are based on multigrid and multilevel methods are suggested and different aspects of the proposed techniques are investigated through numerical experiments.     

Generalized Derivatives for the Solution Operator of the Obstacle Problem

We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed Dirichlet problems. The generalized derivatives can be used to obtain a novel necessary optimality condition for the optimal control of the obstacle problem with control constraints. A comparison shows that this system is stronger than the known system of C-stationarity.

     

Mathematical Justification of an Obstacle Problem in the Case of a Plate

In this paper the modeling of a thin plate in unilateral contact with a rigid plane is properly justified.Starting from the three-dimensional nonlinear Signorini problem,by an asymptotic approach the convergence of the displacement field as the thickness of the plate goes to zero is studied.It is shown that the transverse mechanical displacement field decouples from the in-plane components and solves an obstacle problem.     

Obstacle problem for nonlinear parabolic equations

We show the existence of a continuous solution to a nonlinear parabolic obstacle problem with a continuous time-dependent obstacle. The solution is constructed by an adaptation of the Schwarz alternating method. Moreover, if the obstacle is Hölder continuous, we prove that the solution inherits the same property.     

Furtivity and Masking Problems in Time-Dependent Electromagnetic Obstacle Scattering

In this paper, we consider furtivity and masking problems in time-dependent three-dimensional electromagnetic obstacle scattering. That is, we propose a criterion based on a merit function to minimize or to mask the electromagnetic field scattered by a bounded obstacle when hit by an incoming electromagnetic field and, with respect to this criterion, we drive the optimal strategy. These problems are natural generalizations to the context of electromagnetic scattering of the furtivity problem in time-dependent acoustic obstacle scattering presented in Ref. 1. We propose mathematical models of the furtivity and masking time-dependent three-dimensional electromagnetic scattering problems that consist in optimal control problems for systems of partial differential equations derived from the Maxwell equations. These control problems are approached using the Pontryagin maximum principle. We formulate the first-order optimality conditions for the control problems considered as exterior problems defined outside the obstacle for systems of partial differential equations. Moreover, the first-order optimality conditions derived are solved numerically with a highly parallelizable numerical method based on a perturbative series of the type considered in Refs. 2–3. Finally, we assess and validate the mathematical models and the numerical method proposed analyzing the numerical results obtained with a parallel implementation of the numerical method in several experiments on test problems. Impressive speedup factors are obtained executing the algorithms on a parallel machine when the number of processors used in the computation ranges between 1 and 100. Some virtual reality applications and some animations relative to the numerical experiments can be found in the website http://www.econ.unian.it/recchioni/w10/.