Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.     

Quasistatic evolution of damage in an elastic body with random inputs

Abstract

A model for the quasistatic evolution of the motion of an elastic body which is subject to material damage is presented and analyzed. The model takes the form of an elliptic system for the displacements coupled with a parabolic inclusion for the damage field. In both the system and the inclusion the coefficient and input functions that are present are assumed to be stochastic processes dependent on a random variable. The existence of a weak solution to the model is established using a sequence of approximate problems and passage to a limit. Moreover, the weak solution is shown to be product measurable.     

A frictional contact problem with wear and damage for electro-viscoelastic materials

We consider a quasistatic contact problem for an electro-viscoelastic body. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformation is taken into account, its evolution is described by an inclusion of parabolic type. We present a weak formulation for the model and establish existence and uniqueness results. The proofs are based on classical results for elliptic variational inequalities, parabolic inequalities and fixed point arguments.     

An Interface Problem in Solid Mechanics with a Linear Elastic and a Hyperelastic Material

The three dimensional interface problem is considered with the homogeneous Lamé system in an unbounded exterior domain and some quasistatic nonlinear elastic material behavior in a bounded interior Lipschitz domain. The nonlinear material is of the Mooney-Rivlin type of polyconvex materials. We give a weak formulation of the interface problem based on minimizing the energy, and rewrite it in terms of boundary integral operators. Then, we prove existence of solutions.     

Beam evolution equation with variable coefficients in non‐cylindrical domains

In this article, we present results concerning with the existence of global solutions and a rate decay estimate for energy associated with an initial and boundary value problem for a beam evolution equation with variable coefficients in non‐cylindrical domains. Copyright © 2007 John Wiley & Sons, Ltd.     

带损伤弹性反问题的数值分析

考虑一类由椭圆性方程和热传导方程共同来刻画的准静态弹性模型,通过给定观测值来反演边界的牵引力.首先构造一个凸目标泛函,并引入Tikhonov正则化方法,使之极小化得到一个稳定的近似解.再用有限元离散求解,导出误差估计.最后,用数值例子说明算法的可行性和有效性.     

Numerical analysis of an elastic contact problem with damage

In this work, we consider a mathematical model for the quasistatic contact problem between an elastic body and a deformable obstacle, including the effect of the damage of the material, within the framework of the small deformation theory. The numerical analysis of the variational problem is provided using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Finally, a two-dimensional numerical problem is presented to show the performance of the method. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance,friction and damage

We consider a model for quasistatic frictional contact between a viscoelastic body and a foundation. The material constitutive relation is assumed to be nonlinear. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, the evolution of which is determined by a parabolic inclusion. The contact is modeled with the normal compliance condition and the associated version of Coulomb's law of dry friction. We derive a variational formulation for the problem and prove the existence of its unique weak solution. We then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions.     

Variational approach and optimal control of a PEM fuel cell

The purpose of this paper is to propose and study a mathematical model and a boundary control problem associated to the miscible displacement of hydrogen through the porous anode of a PEM fuel cell. Throughout the paper, we study certain variational problems with a priori regularity properties of the weak solutions. We obtain the existence of less regular solutions and then we prove the desired regularity of these solutions. We consider a control problem that permits to determine the boundary distribution of the pressure which provides an optimal configuration for the temperature and for the concentration, as well. Since the solution of the problem is not unique, the control variable does not appear explicitly in the definition of our cost functional. To overcome this difficulty, we introduce a family of penalized control problems which approximates our boundary control problem. The necessary conditions of optimality are derived by passing to the limit in the penalized optimality conditions.     

Quasistatic Frictional Problems for Elastic and Viscoelastic Materials

We consider two quasistatic problems which describe the frictional contact between a deformable body and an obstacle, the so-called foundation. In the first problem the body is assumed to have a viscoelastic behavior, while in the other it is assumed to be elastic. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of evolution variational inequalities and fixed-point arguments. We also prove that the solution of the viscoelastic problem converges to the solution of the corresponding elastic problem, as the viscosity tensor converges to zero. Finally, we describe a number of concrete contact and friction conditions to which our results apply.     

A class of optimal control problems for piezoelectric frictional contact models

We consider control problems for a mathematical model describing the frictional bilateral contact between a piezoelectric body and a foundation. The material’s behavior is modeled with a linear electro–elastic constitutive law, the process is static and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity conditions on the contact surface are described with the Clarke subdifferential boundary conditions. The weak formulation of the problem consists of a system of two hemivariational inequalities. We provide the results on existence and uniqueness of a weak solution to the model and, under additional assumptions, the continuous dependence of a solution on the data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.     

Dynamical stability of compressible drops and stars

Interest is directed to linearized free boundary motion of a compressible liquid subject to surface tension and self‐gravitation respectively. Linearization relative to an a‐priori given solution to the non‐linear equations leads to a non‐local second order evolution problem to be posed in a space‐time cylinder with variable cross section subject to Fréchet boundary conditions along the lateral boundary part. Well‐posedness of the corresponding initial value problem in a natural weak formulation is proved. Copyright © 2005 John Wiley & Sons, Ltd.     

Beam evolution equation with variable coefficients

We investigate a initial‐boundary value problem for the nonlinear beam equation with variable coefficients on the action of a linear internal damping. We show the existence of a unique global weak solution and that the energy associated with this solution has a rate decay estimate. Besides, we prove the existence and uniqueness of non‐local strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Modeling via the internal energy balance and analysis of adhesive contact with friction in thermoviscoelasticity

In this paper we introduce and investigate a model for adhesive contact with friction between a thermoviscoelastic body and a rigid support.A PDE system, consisting of the evolution equations for the temperatures in the bulk domain and on the contact surface, of the momentum balance, and of the equation for the internal variable describing the state of the adhesion, is derived on the basis of a surface damage theory by M. Frémond.The existence of global-in-time solutions to the associated initial–boundary value problem is proved by passing to the limit in a carefully tailored time-discretization scheme.     

Esistenza ed unicita’ delle soluzioni di un problema misto per un sistema iperbolico simmetrico con coefficienti discontinui

In this paper we give existence and uniqueness theorems for weak solutions of a mixed initial and boundary value problem with conservative boundary conditions relating to a particular symmetric hyperbolic linear system with discontinuous coefficients. The unicity is proved also for a mixed non linear problem.     

Optimal <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Control in Coefficients for Dirichlet Elliptic Problems: <Emphasis Type="Italic">W</Emphasis>-Optimal Solutions

In this paper, we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The coefficients may degenerate and, therefore, the problems may exhibit the so-called Lavrentieff phenomenon and non-uniqueness of weak solutions. We consider the solvability of this problem in the class of W-variational solutions. Using a concept of variational convergence of constrained minimization problems in variable spaces, we prove the existence of W-solutions to the optimal control problem and provide the way for their approximation. We emphasize that control problems of this type are important in material and topology optimization as well as in damage or life-cycle optimization.     

Elastic antiplane contact problem with adhesion

We study a mechanical problem modeling the antiplane shear deformation of a linearly elastic body in adhesive contact with a foundation. The material is assumed to be homogeneous and isotropic and the process is quasistatic. The adhesion process on the contact surface is modeled by a surface internal variable, the bonding field, and the tangential shear due to the bonding is included. We establish the existence of a unique weak solution for the problem, by construction of an appropriate mapping which is shown to be a contraction on a Banach space.     

A bending–stretching model in adhesive contact for elastic rods obtained by using asymptotic methods

This paper is devoted to the derivation and mathematical justification of models for the bending–stretching of an elastic rod in adhesive contact with a deformable foundation. The process is assumed to be quasistatic, and therefore the effects of inertia are neglected. Contact is modeled with normal compliance and the adhesion is modeled by introducing a surface internal variable, the bonding function, the evolution of which is described by an ordinary differential equation. To derive the models we consider the three-dimensional contact problem of an elastic body in adhesive contact with a foundation, introduce a change of variable together with the scaling of the unknowns and parameters of the problem, and we obtain a limit model under the assumption of suitable asymptotic expansions for the scaled unknowns. After that, we obtain error estimates and convergence results which legitimate the limit model. Finally we show that our limit model contains as particular cases models previously considered by other authors. To our knowledge it is for the first time that a rigorous justification and a generalization of those models is provided.     

Differential variational–hemivariational inequalities with application to contact mechanics

In this paper we study a dynamical system which consists of the Cauchy problem for a nonlinear evolution equation of first order coupled with a nonlinear time-dependent variational–hemivariational inequality with constraint in Banach spaces. The evolution equation is considered in the framework of evolution triple of spaces, and the inequality which involves both the convex and nonconvex potentials. We prove existence of solution by the Kakutani–Ky Fan fixed point theorem combined with the Minty formulation and the theory of hemivariational inequalities. We illustrate our findings by examining a nonlinear quasistatic elastic frictional contact problem for which we provide a result on existence of weak solution.