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A minimax inequality with applications to existence of equilibrium point and fixed point theorems

Xie Ding, Kok-Keong Tan (1992)

Colloquium Mathematicae

Ky Fan’s minimax inequality [8, Theorem 1] has become a versatile tool in nonlinear and convex analysis. In this paper, we shall first obtain a minimax inequality which generalizes those generalizations of Ky Fan’s minimax inequality due to Allen [1], Yen [18], Tan [16], Bae Kim Tan [3] and Fan himself [9]. Several equivalent forms are then formulated and one of them, the maximal element version, is used to obtain a fixed point theorem which in turn is applied to obtain an existence theorem of an...

A minimum effort optimal control problem for elliptic PDEs

Christian Clason, Kazufumi Ito, Karl Kunisch (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation...

A minimum effort optimal control problem for elliptic PDEs

Christian Clason, Kazufumi Ito, Karl Kunisch (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation...

A min-max theorem for multiple integrals of the Calculus of Variations and applications

David Arcoya, Lucio Boccardo (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper we deal with the existence of critical points for functionals defined on the Sobolev space W 0 1 , 2 ⁢ Ω by J ⁢ v = ∫ Ω I ⁢ x , v , D ⁢ v ⁢ d ⁢ x , v ∈ W 0 1 , 2 ⁢ Ω , where Ω is a bounded, open subset of R N . Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of Calculus of Variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions. Then we...

A mixed finite element method for plate bending with a unilateral inner obstacle

Ivan Hlaváček (1994)

Applications of Mathematics

A unilateral problem of an elastic plate above a rigid interior obstacle is solved on the basis of a mixed variational inequality formulation. Using the saddle point theory and the Herrmann-Johnson scheme for a simultaneous computation of deflections and moments, an iterative procedure is proposed, each step of which consists in a linear plate problem. The existence, uniqueness and some convergence analysis is presented.

A Mixed Formulation of the Monge-Kantorovich Equations

John W. Barrett, Leonid Prigozhin (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce and analyse a mixed formulation of the Monge-Kantorovich equations, which express optimality conditions for the mass transportation problem with cost proportional to distance. Furthermore, we introduce and analyse the finite element approximation of this formulation using the lowest order Raviart-Thomas element. Finally, we present some numerical experiments, where both the optimal transport density and the associated Kantorovich potential are computed for a coupling problem and problems...

A moving mesh fictitious domain approach for shape optimization problems

Raino A.E. Mäkinen, Tuomo Rossi, Jari Toivanen (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A new numerical method based on fictitious domain methods for shape optimization problems governed by the Poisson equation is proposed. The basic idea is to combine the boundary variation technique, in which the mesh is moving during the optimization, and efficient fictitious domain preconditioning in the solution of the (adjoint) state equations. Neumann boundary value problems are solved using an algebraic fictitious domain method. A mixed formulation based on boundary Lagrange multipliers is...

A multidimensional singular stochastic control problem on a finite time horizon

Marcin Boryc, Łukasz Kruk (2015)

Annales UMCS, Mathematica

A singular stochastic control problem in n dimensions with timedependent coefficients on a finite time horizon is considered. We show that the value function for this problem is a generalized solution of the corresponding HJB equation with locally bounded second derivatives with respect to the space variables and the first derivative with respect to time. Moreover, we prove that an optimal control exists and is unique

A new approach to the constrained controllability problem

Ali Boutoulout, Layla Ezzahri, Hamid Bourray (2014)

Applicationes Mathematicae

We consider the problem of internal regional controllability with output constraints. It consists in steering a hyperbolic system to a final state between two prescribed functions only on a subregion of the evolution system domain. This problem is solved by characterizing the optimal control in terms of a subdifferential associated with the minimized functional.

Currently displaying 81 – 100 of 389