Optimal control of the obstacle in semilinear variational inequalities

We consider an optimal control problem where the state satisfies an obstacle type semilinear variational inequality and the control function is the obstacle. The state is chosen to be close to a desired profile while the obstacle is not too large in H ₀ ¹ (), and H ²-bounded. We prove that an optimal control exists and give necessary optimality conditions, using approximation techniques.     

Criteria of optimality in the infinite-time optimal control problem

The present paper gives a systematic presentation of different definitions of optimality in the infinite-time optimal control problem. Some of these definitions are new, while others have been used throughout the literature, sometimes with different names. The logical implications between them are clearly stated, corresponding comparison criteria for solutions are defined, and other relations as well as two types of equivalence relations are established.This work was supported in part by Simmons College Fund for Research, Grant No. 201.     

Optimal shape design for systems governed by variational inequalities,part 3: Necessary conditions in the elliptic case

General optimality conditions are obtained for optimal shape design for systems governed by a class of elliptic variational inequalities. The conditions are established by making use of the necessary conditions for optimal control of systems governed by strongly monotone variational inequalities. These conditions are then applied to an electrochemical machining problem.     

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.     

Optimal obstacle control problem for semilinear evolutionary bilateral variational inequalities

This paper is concerned with an optimal control problem for semilinear evolutionary bilateral variational inequalities. The pair of the upper and lower obstacles is taken as the control and the corresponding state is chosen close to a desired target profile with the norms of the obstacles not too large. Existence and optimality conditions for the problem are derived.     

Optimal control of an obstacle problem

We investigate optimal control problems governed by variational inequalities, and more precisely the obstacle problem. Since we adopt a numerical point of view, we first relax the feasible domain of the problem; then using both mathematical programming methods and penalization methods we get optimality conditions with smooth lagrange multipliers.     

Necessary optimality conditions in an abnormal optimization problem with equality constraints

An abnormal minimization problem with equality constraints and a finite-dimensional image is examined. Second-order necessary conditions for this problem are given that strengthen previously known results.     

双值约束非凸三次规化问题的全局最优性条件

考虑一类带有双值约束的非凸三次优化问题, 给出了该问题的一个全局最优充分必要条件. 结果改进并推广了一些文献中所给出的全局最优性条件, 同时还通过数值例子来说明所给出的全局最优充要条件是易验证的.     

Necessary optimality conditions for singular controls

A definition of singular controls with respect to components is given that includes, in particular, the conventional definition. On the basis of this definition, new necessary optimiality conditions for singular controls with respect to components are derived for the processes governed by systems of ordinary differential equations.     

Second order optimality conditions for a class of control problems governed by non-linear integral equations with application to parabolic boundary control

In the paper necessary and sufficient second order optimality conditions for optimal control problems governed by weakly singular non linear Hammerstein integral equations are derived. They are applied to a semilinear parabolic boundary control problem for the one dimensional heat equation.     

Optimal control of unilateral obstacle problem with a source term

We consider an optimal control problem for the obstacle problem with an elliptic variational inequality. The obstacle function which is the control function is assumed in H². We use an approximate technique to introduce a family of problems governed by variational equations. We prove optimal solutions existence and give necessary optimality conditions. The author is grateful to Prof. M. Bergounioux for her instructive suggestions.     

Fast solution of the obstacle problem

The purpose of this paper is to present a fast algorithm for the numerical solution of the one-dimensional obstacle problem. It is proven that the algorithm converges in a finite number of steps; application examples showing its efficiency are presented.This work was supported by the National Research Council of Argentina, Grant No. PID-3091000.     

Generalized Bellman-Hamilton-Jacobi optimality conditions for a control problem with a boundary condition

In this paper we study conditions for optimality of a deterministic control problem where the state of the system is required to stop at the boundary. Using the Clarke generalized gradient, we refine the classical verification theorem and show that it is not only sufficient but also necessary for optimality. It is also shown that the solution to the generalized Bellman-Jacobi-Hamilton equation involving the Clarke generalized gradient is unique among the class of regular functions.     

Optimal control of a two-dimensional contact problem

We consider a frictionless contact problem with unilateral constraints for a 2D bar. We describe the problem, then we derive its weak formulation, which is in the form of an elliptic variational inequality of the first kind. Next, we establish the existence of a unique weak solution to the problem and prove its continuous dependence with respect to the applied tractions and constraints. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.     

Second-order optimality conditions in a discrete optimal control problem

Discrete optimal control problems with variable endpoints and with inequality type constraints on control are considered. We derive first- and second-order necessary optimality conditions that are meaningful without a priori normality assumptions.     

Sufficient optimality conditions and duality for a quasiconvex programming problem

Under differentiability assumptions, Fritz John Sufficient optimality conditions are proved for a nonlinear programming problem in which the objective function is assumed to be quasiconvex and the constraint functions are assumed to quasiconcave/strictly pseudoconcave. Duality theorems are proved for Mond-Weir type duality under the above generalized convexity assumptions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are thankful to Professor B. Mond for suggestions that improved the original draft of the paper.     

On sufficient optimality conditions for a quasiconvex programming problem

Some remarks are made on a paper by Bector, Chandra, and Bector (see Ref. 1) concerning the Fritz John and Kuhn-Tucker sufficient optmality conditions as well as duality theorems for a nonlinear programming problem with a quasiconvex objective function.This research was supported by the Italian Ministry of University Scientific and Technological Research.