# American Institute of Mathematical Sciences

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2163-2480

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## Evolution Equations & Control Theory

December 2020 , Volume 9 , Issue 4

Special issue on modeling, analysis and control of contact problems

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2020, 9(4): i-ii doi: 10.3934/eect.2020090 +[Abstract](556) +[HTML](303) +[PDF](13635.75KB)
Abstract:
2020, 9(4): 915-934 doi: 10.3934/eect.2020042 +[Abstract](1050) +[HTML](396) +[PDF](1434.04KB)
Abstract:

In this paper, we show how the approach of nonsmooth dynamical systems can be used to develop a suitable method for the modelling of a rotary oil drilling system with friction. We study different kinds of frictions and analyse the mathematical properties of the involved dynamical systems. We show that using a general Stribeck model for the frictional contact, we can formulate the rotary drilling system as a well-posed evolution variational inequality. Several numerical simulations are also given to illustrate both the model and the theoretical results.

2020, 9(4): 935-960 doi: 10.3934/eect.2020055 +[Abstract](784) +[HTML](300) +[PDF](396.77KB)
Abstract:

This work establishes the existence of measurable solutions to evolution inclusions involving set-valued pseudomonotone operators that depend on a random variable \begin{document}$\omega\in \Omega$\end{document} that is an element of a measurable space \begin{document}$(\Omega, \mathcal{F})$\end{document}. This result considerably extends the current existence results for such evolution inclusions since there are no assumptions made on the uniqueness of the solution, even in the cases where the parameter \begin{document}$\omega$\end{document} is held constant, which leads to the usual evolution inclusion. Moreover, when one assumes the uniqueness of the solution, then the existence of progressively measurable solutions under reasonable and mild assumptions on the set-valued operators, initial data and forcing functions is established. The theory developed here allows for the inclusion of memory or history dependent terms and degenerate equations of mixed type. The proof is based on a new result for measurable solutions to a parameter dependent family of elliptic equations. Finally, when the choice \begin{document}$\omega = t$\end{document} is made, where \begin{document}$t$\end{document} is the time and \begin{document}$\Omega = [0, T]$\end{document}, the results apply to a wide range of quasistatic inclusions, many of which arise naturally in contact mechanics, among many other applications.

2020, 9(4): 961-980 doi: 10.3934/eect.2020059 +[Abstract](853) +[HTML](329) +[PDF](353.79KB)
Abstract:

We consider a contact process between a body and a foundation. The body is assumed to be viscoelastic and piezoelectric and the contact is dynamic. Unlike many related papers, the body is assumed to be non-clamped. The contact conditions has a form of inclusions involving the Clarke subdifferential of locally Lipschitz functionals and they have nonmonotone character. The problem in its weak formulation has a form of two coupled Clarke subdifferential inclusions, from which the first one is dynamic and the second one is stationary. The main goal of the paper is numerical analysis of the studied problem. The corresponding numerical scheme is based on the spatial and temporal discretization. Furthermore, the spatial discretization is based on the first order finite element method, while the temporal discretization is based on the backward Euler scheme. We show that under suitable regularity conditions the error between the exact solution and the approximate one is estimated in an optimal way, namely it depends linearly upon the parameters of discretization.

2020, 9(4): 981-993 doi: 10.3934/eect.2020060 +[Abstract](816) +[HTML](306) +[PDF](361.18KB)
Abstract:

The aim of this work is to study a dynamic problem that constitutes a unified approach to describe some rate-depending interactions between the boundaries of two viscoelastic bodies, including relaxed unilateral contact, pointwise friction or adhesion conditions. The classical formulation of the problem is presented and two variational formulations are given as three and four-field evolution implicit equations. Based on some approximation results and an equivalent fixed point problem for a multivalued function, we prove the existence of solutions to these variational evolution problems.

2020, 9(4): 995-1008 doi: 10.3934/eect.2020046 +[Abstract](959) +[HTML](404) +[PDF](337.71KB)
Abstract:

In this paper, we provide stability analysis for a stationary Stokes hemivariational inequality where along the tangential direction of the slip boundary, an inclusion relation involving the generalized subdifferential of a superpotential is specified. With viscous incompressible fluid flows as application background, stability is analyzed for solutions with respect to perturbations in the superpotential and the density of external forces. We also present a result on the existence of a solution to an optimal control problem for the stationary Stokes hemivariational inequality.

2020, 9(4): 1009-1026 doi: 10.3934/eect.2020049 +[Abstract](939) +[HTML](440) +[PDF](556.87KB)
Abstract:

We formulate a dynamic problem which governs the displacement of a viscoelastic body which, on one hand, can come into frictional contact with a penetrable foundation, and, on the other hand, may undergo material damage. We formulate and prove the theorem on the existence and uniqueness of the weak solution to the formulated problem.

2020, 9(4): 1027-1040 doi: 10.3934/eect.2020050 +[Abstract](941) +[HTML](408) +[PDF](343.7KB)
Abstract:

We consider a nonlinear optimal control problem with dynamics described by a nonlinear evolution equation defined on an evolution triple of spaces. Both the dynamics and the cost functional are not convex and so an optimal pair need not exist. For this reason using tools from multivalued analysis and from convex analysis, we introduce a relaxed version of the problem. No Young measures are involved in our relaxation method. We show that the relaxed problem is admissible.

2020, 9(4): 1041-1055 doi: 10.3934/eect.2020041 +[Abstract](800) +[HTML](386) +[PDF](322.63KB)
Abstract:

This paper obtains existence of random variable solutions to elliptic and evolution inclusions. As a special case, surprising theorems are obtained for the quasistatic problems. A new existence theorem is also presented for evolution inclusions with set valued operators dependent on elements of a measurable space.

2020, 9(4): 1057-1071 doi: 10.3934/eect.2020056 +[Abstract](766) +[HTML](283) +[PDF](335.8KB)
Abstract:

The paper investigates a new type of differential inclusion problem driven by a weighted (p, q)-Laplacian and subject to Dirichlet boundary condition. The problem fully depends on the solution and its gradient. The main novelty is that the problem exhibits simultaneously a nonlocal term involving convolution with the solution and a multivalued term describing discontinuous nonlinearities for the solution. Results stating existence, uniqueness and dependence on parameters are established.

2020, 9(4): 1073-1087 doi: 10.3934/eect.2020044 +[Abstract](1060) +[HTML](475) +[PDF](341.36KB)
Abstract:

The primary objective of this paper is to explore a complicated differential variational-hemivariational inequality involving a history-dependent operator in Banach spaces. A well-posedness result for the inequality, including the existence, uniqueness, and continuous dependence on the initial data of the solution is established by using a fixed point principle for history-dependent operators. Moreover, to illustrate the applicability of the theoretical results, an elastic contact problem with wear and long time dependent effort is explored.

2020, 9(4): 1089-1114 doi: 10.3934/eect.2020047 +[Abstract](851) +[HTML](397) +[PDF](409.27KB)
Abstract:

In this paper we study a new class of abstract evolution first order hemivariational inequalities which involves constraints and history-dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected bifunctions combined with a fixed-point principle for history-dependent operators. Next, we deduce existence, uniqueness and regularity results for some special subclasses of problems which include a constrained history-dependent variational–hemivariational inequality, an evolution quasi-variational inequality with constraints, and an evolution second order hemivariational inequality with constraints. Then, we provide an application of the results to a dynamic unilateral viscoelastic frictional contact problem and show its unique weak solvability.

2020, 9(4): 1115-1131 doi: 10.3934/eect.2020043 +[Abstract](866) +[HTML](402) +[PDF](3475.97KB)
Abstract:

We investigate a fixed domain approach in shape optimization, using a regularization of the Heaviside function both in the cost functional and in the state system. We consider the compliance minimization problem in linear elasticity, a well known application in this area of research. The optimal design problem is approached by an optimal control problem defined in a prescribed domain including all the admissible unknown domains. This approximating optimization problem has good differentiability properties and a gradient algorithm can be applied. Moreover, the paper also includes several numerical experiments that demonstrate the descent of the obtained cost values and show the topological and the boundary variations of the computed domains. The proposed approximation technique is new and can be applied to state systems given by various boundary value problems.

2020, 9(4): 1133-1151 doi: 10.3934/eect.2020057 +[Abstract](639) +[HTML](285) +[PDF](354.99KB)
Abstract:

We consider the model problem of an elastic beam vibrating between two stops. More precisely the beam is clamped at its left end while its right end may undergo contact and collision events with two stops. We model the interaction between the beam and the stops either with Signorini complementarity conditions when the stops are perfectly rigid or with a normal compliance contact law allowing some penetration within the stops and given by a linear relationship between the shear stress and the penetration at some positive power \begin{document}$\beta$\end{document} when contact occurs.

Motivated by computational issues we study the evolution of the energy functional defined as the sum of the kinetic energy and the potential energy of elastic deformation of the beam. When contact is modelled with a normal compliance law we prove an energy conservation property. Then we interpret the relationship between the shear stress and the penetration in case of contact as a penalization of the non-penetration condition. We show that the solutions of the penalized problems converge to a strong solution of the problem with Signorini conditions as defined in [26] and we prove that the limit satisfies an energy conservation property through instantaneaous collision events.

2020, 9(4): 1153-1165 doi: 10.3934/eect.2020058 +[Abstract](828) +[HTML](315) +[PDF](333.66KB)
Abstract:

This paper deals with an evolution inclusion which is an equivalent form of a variational-hemivariational inequality arising in quasistatic contact problems for viscoelastic materials. Existence of a weak solution is proved in a framework of evolution triple of spaces via the Rothe method and the theory of monotone operators. Comments on applications of the abstract result to frictional contact problems are made. The work extends the known existence result of a quasistatic hemivariational inequality by S. Migórski and A. Ochal [SIAM J. Math. Anal., 41 (2009) 1415-1435]. One of the linear and bounded operators in the inclusion is generalized to be a nonlinear and unbounded subdifferential operator of a convex functional, and a smallness condition of the coefficients is removed. Moreover, the existence of a hemivariational inequality is extended to a variational-hemivariational inequality which has wider applications.

2020, 9(4): 1167-1185 doi: 10.3934/eect.2020048 +[Abstract](930) +[HTML](420) +[PDF](374.83KB)
Abstract:

We consider an initial and boundary value problem \begin{document}${{\mathcal{P}}}$\end{document} which describes the frictionless contact of a viscoplastic body with an obstacle made of a rigid body covered by a layer of elastic material. The process is quasistatic and the time of interest is \begin{document}$\mathbb{R}_+ = [0,+\infty)$\end{document}. We list the assumptions on the data and derive a variational formulation \begin{document}${{\mathcal{P}}}_V$\end{document} of the problem, in a form of a system coupling an implicit differential equation with a time-dependent variational-hemivariational inequality, which has a unique solution. We introduce the concept of Tykhonov triple \begin{document}${{\mathcal{T}}} = (I,\Omega, {{\mathcal{C}}})$\end{document} where \begin{document}$I$\end{document} is set of parameters, \begin{document}$\Omega$\end{document} represents a family of approximating sets and \begin{document}${{\mathcal{C} }}$\end{document} is a set of sequences, then we define the well-posedness of Problem \begin{document}${{\mathcal{P}}}_V$\end{document} with respect to \begin{document}${{\mathcal{T}}}$\end{document}. Our main result is Theorem 3.4, which provides sufficient conditions guaranteeing the well-posedness of \begin{document}${{\mathcal{P} }}_V$\end{document} with respect to a specific Tykhonov triple. We use this theorem in order to provide the continuous dependence of the solution with respect to the data. Finally, we state and prove additional convergence results which show that the weak solution to problem \begin{document}${{\mathcal{P}}}$\end{document} can be approached by the weak solutions of different contact problems. Moreover, we provide the mechanical interpretation of these convergence results.

2020, 9(4): 1187-1201 doi: 10.3934/eect.2020045 +[Abstract](890) +[HTML](412) +[PDF](385.71KB)
Abstract:

We consider a heat conduction problem \begin{document}$S$\end{document} with mixed boundary conditions in a n-dimensional domain \begin{document}$\Omega$\end{document} with regular boundary \begin{document}$\Gamma$\end{document} and a family of problems \begin{document}$S_{\alpha}$\end{document}, where the parameter \begin{document}$\alpha>0$\end{document} is the heat transfer coefficient on the portion of the boundary \begin{document}$\Gamma_{1}$\end{document}. In relation to these state systems, we formulate simultaneous distributed-boundary optimal control problems on the internal energy \begin{document}$g$\end{document} and the heat flux \begin{document}$q$\end{document} on the complementary portion of the boundary \begin{document}$\Gamma_{2}$\end{document}. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient \begin{document}$\alpha$\end{document} goes to infinity. Finally, we prove estimations between the simultaneous distributed-boundary optimal control and the distributed optimal control problem studied in a previous paper of the first author.

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