Continuous Dependence Results Research Articles

Optimal control of an evolutionary variational–hemivariational inequality model with new results in numerical analysis and control of the dynamic frictional contact problem

In this paper, we are concerned with numerical analysis on optimal control of a dynamic contact model. Our work builds on some existing results of this model. We derive a weak formulation, and present an approximate problem for the weak formulation by discretization in space. We prove that the approximate problem has a unique solution, which converges to the unique solution of the original problem. The existence of optimal solutions to the control problem is established based on a known continuous dependence result. We consider the numerical solutions of the optimal control problem and prove a convergence result of numerical solutions.

Continuous dependence of renormalized solution for convection-diffusion problems involving a nonlocal operator

In Ouédraogo A. et al (cf. [30]), it is provided existence and uniqueness results of L1-renormalized entropy solution for the Cauchy problem associated to the following vast class of nonlinear anisotropic degenerate parabolic-hyperbolic equations involving a nonlocal diffusion term: \begin{equation*}\label{problem_(CP)} \partial_{t}u+\nabla.F(u)-\displaystyle\sum_{i,j=1}^{N}\partial^{2}_{x_{i}x_{j}}A_{ij}(u) -{\cal L}_{\mu}[u]= f(u) \hskip0.3 cm\hbox{in }\ \ Q=(0,T)\times \RR^N \ \text{with}\ T>0\ \text{and}\ N\geq 1. \end{equation*} Our goal is to complement this previous work with a continuous dependence result of the L1-solution with respect to the data set (F,a,μ,f, u0). The strategy is to follow the approach developed by Karlsen and Ulusoy in [28]. However, we must manage the difficulties due to the fact that we are working in the whole space RN with an only integrable initial datum u0 and the term source f depends on the unknown function u.

On a Cahn–Hilliard system with source term and thermal memory

A nonisothermal phase field system of Cahn–Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn–Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem.

Inverse problems for constrained parabolic variational-hemivariational inequalities *

In this paper we study a novel class of inverse problems for parabolic variational–hemivariational inequalities with a unilateral constraint. A theorem on the well-posedness for weak solution is established. Based on a new continuous dependence result, we prove the nonemptiness and stability of the set of optimal solutions to the corresponding inverse problems. We illustrate the results by a quasistatic nonsmooth frictional viscoelastic contact problem with a unilateral constraint for which we derive results on the unique weak solvability, and existence and stability of solutions to the associated inverse problems.

A stabilized finite element method for the Stokes–Temperature coupled problem

In this work, we introduce and analyze a new stabilized finite element scheme for the Stokes–Temperature coupled problem. This new scheme allows equal order of interpolation to approximate the quantities of interest, i.e. velocity, pressure, temperature, and stress. We analyze an equivalent variational formulation of the coupled problem inspired by the ideas proposed in [3]. The existence of the discrete solution is proved, decoupling the proposed stabilized scheme and using the help of continuous dependence results and Brouwer's theorem under the standard assumption of sufficiently small data. Optimal convergence is proved under classic regularity assumptions of the solution. Finally, we present some numerical examples to show the quality of our scheme, in particular, we compare our results with those coming from a standard reference in geosciences described in [38].

A new class of fully history-dependent variational-hemivariational inequalities with application to contact mechanics

In this paper, we consider the behaviour of solutions to a class of fully history-dependent variational-hemivariational inequalities with respect to the perturbation of the data. First, the existence and uniqueness of the solution to a class of fully history-dependent variational-hemivariational inequalities is obtained by using a fixed point theorem. Second, we obtain continuous dependence result of solutions with respect to all the data of variational-hemivariational inequalities. Meanwhile, the convergence results of the solutions for the special case of abstract variational inequalities are also given. Finally, to illustrate our main results, we consider a class of viscoelastic contact problem with a long memory. By using our abstract result, we get the continuous dependence of the solutions to frictional contact problem with respect to all the data.

Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications

In this paper, we consider an abstract system which consists of a nonlinear differential inclusion and a parabolic hemivariational inequality (DPHVI) in Banach spaces. The objective of this paper is four fold. The first target is to deal with the existence of solutions and the properties which involve the boundedness and continuous dependence results of the solution set to parabolic hemivariational inequality. The second aim is to investigate the existence of mild solutions to DPHVI by means of a fixed point technique. The third one is to study the existence of a pullback attractor for the multivalued processes governed by DPHVI. Finally, the fourth goal is to demonstrate a concrete application of our main results arising from the dynamic thermoviscoelasticity problems.

Convergence Theorems for Varying Measures Under Convexity Conditions and Applications

In this paper, convergence theorems involving convex inequalities of Copson’s type (less restrictive than monotonicity assumptions) are given for varying measures, when imposing convexity conditions on the integrable functions or on the measures. Consequently, a continuous dependence result for a wide class of differential equations with many interesting applications, namely measure differential equations (including Stieltjes differential equations, generalized differential problems, impulsive differential equations with finitely or countably many impulses and also dynamic equations on time scales) is provided.

Existence and continuous dependence results for fractional evolution integrodifferential equations of order [formula omitted

The article analyzes the existence of Caputo fractional evolution integrodifferential equations of order 1<r<2 in Hilbert space with delay. A new set of adequate requirements for the existence outcomes of fractional delay evolution integrodifferential equations have been developed and are shown using the fractional derivative, Krasnoselskii’s fixed point theorem, and Henry-Gronwall inequalities. In addition, for the provided system, we developed continuous dependence results. Afterward, we apply our findings to the concept of nonlocal conditions. Then, to demonstrate our primary outcomes, two examples are given.

Continuous Dependence for the Boussinesq Equations under Reaction Boundary Conditions in R2

In this paper, we studied the continuous dependence result for the Boussinesq equations. We considered the case where Ω was a bounded domain in R2. Temperatures T and C satisfied reaction boundary conditions. A first-order inequality for the differences of energy could be derived. An integration of this inequality produced a continuous dependence result. The result told us that the continuous dependence type stability was also valid for the Boussinesq coefficient λ of the Boussinesq equations with reaction boundary conditions.

Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term

&lt;p style='text-indent:20px;'&gt;The paper treats the problem of optimal distributed control of a Cahn–Hilliard–Oono system in &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ {{\mathbb{R}}}^d $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ 1\leq d\leq 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ d = 2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain.&lt;/p&gt;

Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory

A nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. Two equations, resulting from phase dynamics and the universal balance law for internal energy, are written in terms of the phase variable (representing a non-conserved order parameter) and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. Existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem. Then, an optimal control problem is investigated for a suitable cost functional, in which two data act as controls, namely, the distributed heat source and the initial temperature. Fr\'echet differentiability between suitable Banach spaces is shown for the control-to-state operator, and meaningful first-order necessary optimality conditions are derived in terms of variational inequalities involving the adjoint variables. Eventually, characterizations of the optimal controls are given.

A new elliptic mixed boundary value problem with (p,q)-Laplacian and Clarke subdifferential: Existence, comparison and convergence results

The goal of this paper is to investigate a new class of elliptic mixed boundary value problems involving a nonlinear and nonhomogeneous partial differential operator [Formula: see text]-Laplacian, and a multivalued term represented by Clarke’s generalized gradient. First, we apply a surjectivity result for multivalued pseudomonotone operators to examine the existence of weak solutions under mild hypotheses. Then, a comparison theorem is delivered, and a convergence result, which reveals the asymptotic behavior of solution when the parameter (heat transfer coefficient) tends to infinity, is obtained. Finally, we establish a continuous dependence result of solution to the boundary value problem on the data.

Well-posedness for thermo-electro-viscoelasticity of Green–Naghdi type

We study the linear theory of thermo-electro-viscoelasticity of Green–Naghdi type for the case of a one-dimensional body. For the corresponding mathematical model, we prove a uniqueness theorem of the solution to the mixed boundary-initial-value problem by means of the Laplace transform after rewriting the constitutive equations in an appropriate form. Moreover, we derive a result of continuous dependence upon the supply terms.

A semigroup of contractions in elasticity of porous bodies

The mixed initial-boundary value problem in the context of elasticity of porous bodies having a dipolar structure is considered. By means of a semigroup of contractions, we can obtain some results regarding the existence and uniqueness of solutions for this mixed problem, after proving the equivalence between this problem and a Cauchy problem attached to an abstract equation of evolution. Also, by means of this Cauchy problem, we deduce two continuous dependence results, regarding the supply terms of the original mixed problem and upon initial data.

Initial value problem for a fractional neutral differential equation with infinite delay

We consider the initial value problem for a class of nonlinear fractional neutral functional differential equations with infinite delay involving the standard fractional derivative in the sense of Caputo. By using a variety of tools of fractional calculus including the Banach contraction principle and the Schaefer fixed point theorem, the existence, uniqueness and continuous dependence results are obtained in the space of continuous functions.

Hemivariational inverse problem for contact problem with locking materials

The aim of this work is to study an inverse problem for a frictional contact model for locking material. The deformable body consists of electro-elastic-locking materials. Here, the locking character makes the solution belong to a convex set, the contact is presented in the form of multivalued normal compliance, and frictions are described with a sub-gradient of a locally Lipschitz mapping. We develop the variational formulation of the model by combining two hemivariational inequalities in a linked system. The existence and uniqueness of the solution are demonstrated utilizing recent conclusions from hemivariational inequalities theory and a fixed point argument. Finally, we provided a continuous dependence result and then we established the existence of a solution to an inverse problem for piezoelectric-locking material frictional contact problem.

Analysis results for dynamic contact problem with friction in thermo-viscoelasticity

We present a mathematical model which describes the dynamic frictional contact between a thermo-viscoelastc body and a conductive foundation. The contact is modeled using the normal compliance condition, the quasistatic version of Coulomb's law of fry friction. We derive the weak formulation and we prove the existence and uniqueness result. The proofs are based on the theory of first-order and second-order evolution inequalities and Banach fixed point theorem. We introduce a new problem on perturbation of the contact boundary condition and we establish its continuous dependence result. <br><br>Представлено математичну модель, що описує динамічний контакт тертя між термов’язкопружним тілом і основою-провідником. Контакт моделюється з використанням умови нормальної еластичності, квазістатичної версії закону фрикційного тертя Кулона. Пропонується означення слабкого розв’язку, доведена теорема існування та єдиності. Доведення базуються на теорії еволюційних нерівностей першого та другого порядків і теоремі Банаха про нерухому точку. Ставиться нова задача про збурення контактної граничної умови; отримана теорема про неперервну залежність.

Optimal Control of History-Dependent Evolution Inclusions with Applications to Frictional Contact

In this paper, we study a class of subdifferential evolution inclusions involving history-dependent operators. First, we improve an existence and uniqueness theorem and prove the continuous dependence result in the weak topologies. Next, we establish the existence of optimal solution to an optimal control problem for the evolution inclusion. Finally, we illustrate the results by an example of an optimal control of a dynamic frictional contact problem in mechanics, whose weak formulation is the evolution variational inequality.

A Continuous Dependence Result for a Dynamic Debonding Model in Dimension One

In this paper we address the problem of continuous dependence on initial and boundary data for a one-dimensional debonding model describing a thin film peeled away from a substrate. The system underlying the process couples the weakly damped wave equation with a Griffith's criterion which rules the evolution of the debonded region. We show that under general convergence assumptions on the data the corresponding solutions converge to the limit one with respect to different natural topologies.