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.     

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     

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.     

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.     

Optimal control of the obstacle in a quasilinear elliptic variational inequality

In this paper we consider an obstacle control problem where the state satisfies a quasilinear elliptic variational inequality and the control function is the obstacle. The state is chosen to be close to the desire profile while the H² norms of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

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.     

Components identification based method for box constrained variational inequality problems with almost linear functions

We present a method for solving a class of box constrained variational inequality problems. The method makes use of a procedure for identifying some components of the solution by bounding it with an interval vector. It is shown that the method computes an approximate solution of the variational inequality problem by solving at most n reduced systems of equations, where n is the dimension of the problem. Among those systems, only the one of the smallest dimension has to be solved with high accuracy. The others are solved merely to identify some components of the solution, and so the computation can be done under a very mild requirement of accuracy. Numerical results are presented for the obstacle problem, to illustrate the efficiency of the method. AMS subject classification (2000)  90C33, 65G30, 65K10     

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.     

Regularization in Sobolev Spaces with Fractional Order

We study the minimization of a quadratic functional where the Tichonov regularization term is an H ^s -norm with a fractional s > 0. Moreover, pointwise bounds for the unknown solution are given. A multilevel approach as an equivalent norm concept is introduced. We show higher regularity of the solution of the variational inequality. This regularity is used to show the existence of regular Lagrange multipliers in function space. The theory is illustrated by two applications: a Dirichlet boundary control problem and a parameter identification problem.     

Solving a class of variational inequalities with inexact oracle operators

Consider a class of variational inequality problems of finding ${x^*\in S}Consider a class of variational inequality problems of finding x^* ? S{x^*\in S}, such that

f(x*)T (z-x*) 3 0,    "z ? S,f(x^*)^\top (z-x^*)\geq 0,\quad \forall z\in S,      11. Energy–based localization and multiplicity of radially symmetric states for the stationary p–Laplace diffusion      Radu Precup  Csaba Varga 《复变函数与椭圆型方程》2020,65(7):1198-1209 ABSTRACT It is investigated the role of the state–dependent source–term for the localization by means of the kinetic energy of radially symmetric states for the stationary p–Laplace diffusion. It is shown that the oscillatory behavior of the source–term, with respect to the state amplitude, yields multiple possible states, located in disjoint energy bands. The mathematical analysis makes use of critical point theory in conical shells and of a version of Pucci–Serrin three–critical point theorem for the intersection of a cone with a ball. A key ingredient is a Harnack type inequality in terms of the energetic norm.      12. Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques      Michele Matzeu 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(11):4347-4359 In this paper we consider a semilinear variational inequality with a gradient-dependent nonlinear term. Obviously the nature of this problem is non-variational. Nevertheless we study that problem associating a suitable semilinear variational inequality, variational in nature, with it, and performing an iterative technique used in De Figueiredo et al. (2004) [6] in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity. We prove the existence of a non-trivial non-negative weak solution u for our problem using essentially variational methods, a penalization technique and an iterative scheme. Via Lewy-Stampacchia’s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is α-H?lder continuous on for any α∈(0,1).      13. Variational problem with an obstacle in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> for a class of quadratic functionals      A. A. Arkhipova 《Journal of Mathematical Sciences》2009,159(4):391-410   A variational problem with an obstacle for a certain class of quadratic functionals is considered. Admissible vector-valued functions are assumed to satisfy the Dirichlet boundary condition, and the obstacle is a given smooth (N − 1)-dimensional surface S in ℝ N . The surface S is not necessarily bounded. It is proved that any minimizer u of such an obstacle problem is a partially smooth function up to the boundary of a prescribed domain. It is shown that the (n − 2)-Hausdorff measure of the set of singular points is zero. Moreover, u is a weak solution of a quasilinear system with two kinds of quadratic nonlinearities in the gradient. This is proved by a local penalty method. Bibliography: 25 titles. Dedicated to V. A. Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh. Seminarov POMI, Vol. 362, 2008, pp. 15–47.      14. Characterizations of K -Functionals Built from Fractional Powers of Infinitesimal Generators of Semigroups      Trebels  Westphal 《Constructive Approximation》2008,19(3):355-371    Abstract. On a Banach space X consider an equibounded (C_0)-semigroup of linear operators { T(t): t ≥ 0} with infinitesimal generator A . We introduce fractional powers (-A) α , α >0 , of A with domain D((-A) α )) and characterize the K -functionals with respect to (X,D((-A) α )) via fractional differences [I-T(t)] α , via appropriate truncated hypersingular integrals and via some type of fractional integral over the resolvent of A . Immediate consequences are an abstract Marchaud-type inequality for moduli of smoothness arising from (semi-) groups of operators as well as optimal and nonoptimal approximation results.      15. The Neumann Problem on Unbounded Domains of ℝ d and Stochastic Variational Inequalities      Viorel Barbu  Giuseppe Da Prato 《偏微分方程通讯》2013,38(8):1217-1248 ABSTRACT An elliptic equation with Neumann boundary conditions and unbounded drift coefficients is studied in a space L 2(? d , ν) where ν is an invariant measure. The corresponding semigroup generated by the elliptic operator is identified with the transition semigroup associated with a stochastic variational inequality.      16. On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability      M.A. Tawhid 《Journal of Optimization Theory and Applications》2002,113(1):149-164 In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C 1 and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C 2 and C 1 nonlinear complementarity problems.      17. On a variational inequality associated with a stopping game combined with a control      Kenji Kamizono  Hiroaki Morimoto 《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(1-2):99-123 We study a non-linear elliptic variational inequality which corresponds to a zero-sum stopping game (Dynkin game) combined with a control. Our result is a generalization of the existing works by Bensoussan [ Stochastic Control by Functional Analysis Methods (North-Holland, Amsterdam), 1982], Bensoussan and Lions [ Applications des Inéquations Variationnelles en Contrôle Stochastique (Dunod, Paris), 1978] and Friedman [ Stochastic Differential Equations and Applications (Academic Press, New York), 1976] in the sense that a non-linear term appears in the variational inequality, or equivalently, that the underlying process for the corresponding stopping game is subject to a control. By using the dynamic programming principle and the method of penalization, we show the existence and uniqueness of a viscosity solution of the variational inequality and describe it as the value function of the corresponding combined-stochastic game problem.      18. SOLVING SYMMETRIC MONOTONE LINEAR VARIATIONAL INEQUALITIES BY SOME MODIFIED LEVITIN-POLYAK PROJECTION METHODS      GAOYI 《高校应用数学学报(英文版)》1998,13(2):150-158 Some modified Levitin-Polyak projection methods are proposed in this paper for solving monotone linear variational inequality x∈Ω，（x′-x)^T(Hx c)≤0，for any x′∈Ω.It is pointed out that there are similar methods for solving a general linear variational inequality.      19. Optimal control of the obstacle problem: optimality conditions      Hadi  Khalid Ait 《IMA Journal of Mathematical Control and Information》2006,23(3):325-334 ** E-mail: k.aithadi{at}ucam.ac.ma In this paper, we investigate optimal control problem governedby variational inequality of the obstacle type. Existence ofsolution for the problem is proved and we also show how to obtainoptimality conditions for a penalized problem issued from theoriginal one.      20. Walrasian Equilibrium Problem with Memory Term      Maria Bernadette Donato  Monica Milasi  Laura Scrimali 《Journal of Optimization Theory and Applications》2011,151(1):64-80 The aim of this paper is to study the Walrasian equilibrium problem when the data are time-dependent. In order to have a more realistic model, the excess demand function depends on the current price and on previous events of the market. Hence, a memory term is introduced; it describes the precedent states of the equilibrium. This model is reformulated as an evolutionary variational inequality in the Lebesgue space L 2([0,T],ℝ), and, thanks to this characterization, existence and qualitative results on equilibrium solution are given.

Energy–based localization and multiplicity of radially symmetric states for the stationary p–Laplace diffusion

ABSTRACT

It is investigated the role of the state–dependent source–term for the localization by means of the kinetic energy of radially symmetric states for the stationary p–Laplace diffusion. It is shown that the oscillatory behavior of the source–term, with respect to the state amplitude, yields multiple possible states, located in disjoint energy bands. The mathematical analysis makes use of critical point theory in conical shells and of a version of Pucci–Serrin three–critical point theorem for the intersection of a cone with a ball. A key ingredient is a Harnack type inequality in terms of the energetic norm.     

Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques

In this paper we consider a semilinear variational inequality with a gradient-dependent nonlinear term. Obviously the nature of this problem is non-variational. Nevertheless we study that problem associating a suitable semilinear variational inequality, variational in nature, with it, and performing an iterative technique used in De Figueiredo et al. (2004) [6] in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity. We prove the existence of a non-trivial non-negative weak solution u for our problem using essentially variational methods, a penalization technique and an iterative scheme. Via Lewy-Stampacchia’s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is α-H?lder continuous on for any α∈(0,1).     

Variational problem with an obstacle in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> for a class of quadratic functionals

  A variational problem with an obstacle for a certain class of quadratic functionals is considered. Admissible vector-valued functions are assumed to satisfy the Dirichlet boundary condition, and the obstacle is a given smooth (N − 1)-dimensional surface S in ℝ ^N . The surface S is not necessarily bounded. It is proved that any minimizer u of such an obstacle problem is a partially smooth function up to the boundary of a prescribed domain. It is shown that the (n − 2)-Hausdorff measure of the set of singular points is zero. Moreover, u is a weak solution of a quasilinear system with two kinds of quadratic nonlinearities in the gradient. This is proved by a local penalty method. Bibliography: 25 titles. Dedicated to V. A. Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh. Seminarov POMI, Vol. 362, 2008, pp. 15–47.     

Characterizations of K -Functionals Built from Fractional Powers of Infinitesimal Generators of Semigroups

   Abstract. On a Banach space X consider an equibounded (C_0)-semigroup of linear operators { T(t): t ≥ 0} with infinitesimal generator A . We introduce fractional powers (-A) ^α , α >0 , of A with domain D((-A) ^α )) and characterize the K -functionals with respect to (X,D((-A) ^α )) via fractional differences [I-T(t)] ^α , via appropriate truncated hypersingular integrals and via some type of fractional integral over the resolvent of A . Immediate consequences are an abstract Marchaud-type inequality for moduli of smoothness arising from (semi-) groups of operators as well as optimal and nonoptimal approximation results.     

The Neumann Problem on Unbounded Domains of ℝ d and Stochastic Variational Inequalities

ABSTRACT

An elliptic equation with Neumann boundary conditions and unbounded drift coefficients is studied in a space L ²(? ^d , ν) where ν is an invariant measure. The corresponding semigroup generated by the elliptic operator is identified with the transition semigroup associated with a stochastic variational inequality.     

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

On a variational inequality associated with a stopping game combined with a control

We study a non-linear elliptic variational inequality which corresponds to a zero-sum stopping game (Dynkin game) combined with a control. Our result is a generalization of the existing works by Bensoussan [ Stochastic Control by Functional Analysis Methods (North-Holland, Amsterdam), 1982], Bensoussan and Lions [ Applications des Inéquations Variationnelles en Contrôle Stochastique (Dunod, Paris), 1978] and Friedman [ Stochastic Differential Equations and Applications (Academic Press, New York), 1976] in the sense that a non-linear term appears in the variational inequality, or equivalently, that the underlying process for the corresponding stopping game is subject to a control. By using the dynamic programming principle and the method of penalization, we show the existence and uniqueness of a viscosity solution of the variational inequality and describe it as the value function of the corresponding combined-stochastic game problem.     

SOLVING SYMMETRIC MONOTONE LINEAR VARIATIONAL INEQUALITIES BY SOME MODIFIED LEVITIN-POLYAK PROJECTION METHODS

Some modified Levitin-Polyak projection methods are proposed in this paper for solving monotone linear variational inequality x∈Ω，（x′-x)^T(Hx c)≤0，for any x′∈Ω.It is pointed out that there are similar methods for solving a general linear variational inequality.     

Optimal control of the obstacle problem: optimality conditions

^** E-mail: k.aithadi{at}ucam.ac.ma In this paper, we investigate optimal control problem governedby variational inequality of the obstacle type. Existence ofsolution for the problem is proved and we also show how to obtainoptimality conditions for a penalized problem issued from theoriginal one.     

Walrasian Equilibrium Problem with Memory Term

The aim of this paper is to study the Walrasian equilibrium problem when the data are time-dependent. In order to have a more realistic model, the excess demand function depends on the current price and on previous events of the market. Hence, a memory term is introduced; it describes the precedent states of the equilibrium. This model is reformulated as an evolutionary variational inequality in the Lebesgue space L ²([0,T],ℝ), and, thanks to this characterization, existence and qualitative results on equilibrium solution are given.