Optimal Control Of Variational Inequalities Research Articles

Solvability and stationarity for the optimal control of variational inequalities with point evaluations in the objective functional

AbstractMotivated by applications in economics and engineering, we consider the optimal control of a variational inequality with point evaluations of the state variable in the objective. This problem class constitutes a specific mathematical program with complementarity constraints (MPCC). In our context, the problem is posed in an adequate function space and the variational inequality involves second order linear elliptic partial differential operators. The necessary functional analytic framework complicates the derivation of stationarity conditions whereas the non‐convex and non‐differentiable nature of the problem challenges the design of an efficient solution algorithm. In this paper, we present a penalization and smoothing technique to derive first order type conditions related to C‐stationarity in the associated Sobolev space setting. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Path-following for optimal control of stationary variational inequalities

Moreau-Yosida based approximation techniques for optimal control of variational inequalities are investigated. Properties of the path generated by solutions to the regularized equations are analyzed. Combined with a semi-smooth Newton method for the regularized problems these lead to an efficient numerical technique.

Optimal Control of Variational Inequalities

We consider control problems for the variational inequality describing a single degree of freedom elasto-plastic oscillator. We are particularly interested in finding the critical excitation, i.e., the lowest energy input excitation that drives the system between the prescribed initial and final states within a given time span. This is a control problem for a state evolution described by a variational inequality. We obtain Pontryagin’s necessary condition of optimality. An essential difficulty lies with the non continuity of adjoint variables.

Mini-Workshop: Control of Free Boundaries

The field of the mathematical and numerical analysis of systems of nonlinear pdes involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in ma- terial science and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multiphase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimiza- tion problems with pde constraints including free boundaries. It is now timely to consider such control problems because of the maturity of the field of com- putational free boundary problems. The aim of the mini-workshop was to bring together leading experts and young researchers from the separate fields of numerical free boundary problems and optimal control in order to estab- lish links and to identify suitable model problems to serve as paradigms for progressing knowledge of optimal control of free boundaries.

Optimal Control of Obstacle Problems by H1-Obstacles

Optimal control of variational inequalities with the controls given by the obstacles is considered. Existence optimal solutions are proved for obstacles with H1 regularity and first-order optimality conditions are derived which, under additional assumptions, are also sufficient. A numerical algorithm is proposed and its practical feasibility is investigated.

Optimal Control of Variational Inequalities with Delays in the Highest Order Spatial Derivatives

In this paper, an optimal control problem for parabolic variational inequalities with delays in the highest order spatial derivatives is investigated. The well–posedness of such kinds of variational inequalities is established. The existence of optimal controls under a Cesari–type condition is proved, and the necessary conditions of Pontryagin type for optimal controls is derived.

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.

Optimization of Structures in Unilateral Contact

This article gives a review of optimization of structures in mechanical contact. Emphasis is put on linear elastic structures in frictionless contact. In particular, for optimization problems where an energy objective is used, a unified framework is given in parallel with the review. Papers related to optimal control of variational inequalities or dealing with pure sensitivity analysis are treated in less detail. Problems involving friction are also reviewed at a less detailed level. It is explained why structural optimization problems involving contact cannot be treated within classical smooth optimization theory and how they relate to modern fields such as nonsmooth optimization and mathematical programs with equilibrium constraints (MPECs). Throughout the article, discrete and continuous problems are treated in parallel. This review article includes 106 references.

Pontryagin Maximum Principle for Optimal Control of Variational Inequalities

In this paper we investigate optimal control problems governed by variational inequalities. We present a method for deriving optimality conditions in the form of Pontryagin's principle. The main to...

Pontryagin Maximum Principle for Optimal Control of Variational Inequalities

In this paper we investigate optimal control problems governed by variational inequalities. We present a method for deriving optimality conditions in the form of Pontryagin's principle. The main tools used are the Ekeland's variational principle combined with penalization and spike variation techniques.

Interior point techniques for optimal control of variational inequalities

This paper is devoted to a new application of an interior point algorithm to solve optimal control problems of variational inequalities. We propose a Lagrangian technique to obtain a necessary optimality system. After the discretization of the optimality system we prove its equivalence to Karush-Kuhn-Tucker conditions of a nonlinear regular minimization problem. This problem can be efficiently solved by using a modification of Herskovits' interior point algorithm for nonlinear optimization. We describe the numerical scheme for solving this problem and give some numerical examples of test problems in 1-D and 2-D.

Optimal control of variational inequality with applications to axisymmetric shells

The optimal control problem of variational inequality with applications to axisymmetric shells is discussed. First an existence result for the solution of the optimal control problem is given. Next is presented the formulation of first order necessary conditionas of optimality for the control problem governed by a variational inequality with its coefficients as control variables.

Optimal control of variational inequalities. Approximation theory and numerical realization

An optimal-control problem of a variational inequality of the elliptic type is investigated. The problem is approximated by a family of finite-dimensional problems and the convergence of the approximated optimal controls is shown. The finite-dimensional problems, being nonsmooth, are to be optimized by a bundle algorithm, which requires an element of Clarke's generalized gradient of the minimized function. A simple algorithm which yields this element is proposed. Some numerical experiments with a simple model problem have also been carried out.

On the optimal control for variational inequalities

We prove an existence theorem for the optimal control of variational inequalities governed by a pseudomonotone operator: the cost is assumed to be quadratic. Then, we give an extension of the theorem to more general costs (assuming the operator to be monotone); we also give a result on a perturbation problem.