Subdifferential Inclusions Research Articles

Feedback control systems governed by evolution equations

ABSTRACTThe goal of this paper is to provide systematic approaches to study the feedback control systems governed by evolution equations in separable reflexive Banach spaces. We firstly give some existence results of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray–Schauder alternative fixed point theorem with Lipschitz conditions and some types of boundedness conditions. Next, by using the Filippove theorem and the Cesari property, a new set of sufficient assumptions are formulated to guarantee the existence of feasible pairs for the feedback control systems. Some existence results for an optimal control problem are given. Finally, we apply our main result to obtain a controllability result for semilinear evolution equations and existence results for a class of differential variational inequalities and Clarke's subdifferential inclusions.

Existence results for impulsive feedback control systems

The goal of this paper is to provide systematic approaches to study the feedback control systems governed by impulsive evolution equations in separable reflexive Banach spaces. We firstly give some existence results of mild solutions for the equations by applying the Banach’s fixed point theorem and the Leray–Schauder alternative fixed point theorem with Lipschitz conditions and different types of boundedness conditions. Then, by using the Filippove theorem and the Cesari property, a new set of sufficient assumptions are formulated to guarantee the existence results of feasible pairs for the feedback control systems. Finally, we apply our main result to obtain a controllability result for impulsive evolution equations and two existence results for a class of impulsive differential variational inequalities and impulsive Clarke’s subdifferential inclusions.

Variational–hemivariational inequality for a class of dynamic nonsmooth frictional contact problems

In this paper, a dynamic frictional contact problem for viscoelastic materials with long memory is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality for the velocity. A result on the unique weak solvability to this inequality is proved through a recent contribution on evolutionary subdifferential inclusions and a fixed point approach.

Elastic contact problem with Coulomb friction and normal compliance in Orlicz spaces

A static contact problem for inhomogeneous elastic materials is studied with a non-polynomial growth of the elasticity under the Coulomb’s law of dry friction and the normal compliance condition. We demonstrate the results on existence and uniqueness of a solution to an abstract subdifferential inclusion and a variational–hemivariational inequality in the reflexive Orlicz–Sobolev space which are applied to the static elastic frictional problem.

Filippov--Ważewski Theorem for Subdifferential Inclusions with an Unbounded Perturbation

An evolution inclusion of subdifferential type with time dependent subdifferentials and a multivalued perturbation is considered in a separable Hilbert space. The perturbation has closed unbounded values. We also consider the inclusion with the values of the perturbation being convexified (convexified inclusion). The notion of a regular solution for the unbounded convexified inclusion is introduced. Any solution of the convexified inclusion with a bounded perturbation is such a solution. The theorems on existence and relaxation for regular solutions of the unbounded convexified inclusion are proved. In contrast to the known results of this kind, we do not suppose that the convex function whose subdifferential appears in the inclusion has the compactness property. Additionally, instead of making the traditional assumption for such problems of Lipschitz continuity of the perturbation in the phase variable in the Hausdorff metric, we use a more natural notion of $(\rho-H)$ Lipschitzness for mappings with unbounded values. An example of unbounded convexified inclusion, every solution of which is regular, is given.

Dynamic thermoviscoelastic thermistor problem with contact and nonmonotone friction

The paper studies the evolution of the thermomechanical and electric state of a thermoviscoelastic thermistor that is in frictional contact with a reactive foundation. The mechanical process is dynamic, while the electric process is quasistatic. Friction is modeled with a nonmonotone relation between the tangential traction and tangential velocity. Frictional heat generation is taken into account and so is the strong dependence of the electric conductivity on the temperature. The mathematical model for the process is in the form of a system that consists of dynamic hyperbolic subdifferential inclusion for the mechanical state coupled with a nonlinear parabolic equation for the temperature and an elliptic equation for the electric potential. The paper establishes the existence of a weak solution to the problem by using time delays, a priori estimates and a convergence method.

Subdifferential inclusions for stress formulations of unilateral contact problems

We consider two classes of inclusions involving subdifferential operators, both in the sense of Clarke and in the sense of convex analysis. An inclusion that belongs to the first class is stationary while an inclusion that belongs to the second class is history-dependent. For each class, we prove existence and uniqueness of the solution. The proofs are based on arguments of pseudomonotonicity and fixed points in reflexive Banach spaces. Then we consider two mathematical models that describe the frictionless unilateral contact of a deformable body with a foundation. The constitutive law of the material is expressed in terms of a subdifferential of a nonconvex potential function and, in the second model, involves a memory term. For each model, we list assumptions on the data and derive a variational formulation, expressed in terms of a multivalued variational inequality for the stress tensor. Then we use our abstract existence and uniqueness results on the subdifferential inclusions and prove the unique weak solvability of each contact model. We end this paper with some examples of one-dimensional constitutive laws for which our results can be applied.

Numerical analysis of a dynamic bilateral thermoviscoelastic contact problem with nonmonotone friction law

We study a fully dynamic thermoviscoelastic contact problem. The contact is assumed to be bilateral and frictional, where the friction law is described by a nonmonotone relation between the tangential stress and the tangential velocity. Weak formulation of the problem leads to a system of two evolutionary, possibly nonmonotone subdifferential inclusions of parabolic and hyperbolic type, respectively. We study both semidiscrete and fully discrete approximation schemes, and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the linear element solution. This theoretical result is illustrated numerically.

Existence and relaxation for subdifferential inclusions with unbounded perturbation

We consider a differential inclusion of subdifferential type with a nonconvex and unbounded valued perturbation. Existence and relaxation results are obtained for this inclusion. By relaxation we mean approximation of a solution of the differential inclusion with convexified perturbation by solutions of the given inclusion. The traditional condition of Lipschitz continuity for such kind of problems is weakened and a somehow more appropriate in the context of unbounded valued multifunctions “truncated” version of it is considered instead.

Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation

An evolution inclusion is considered in a separable Hilbert space. The right-hand side of the inclusion contains the subdifferential of a time-dependent proper convex lower semicontinuous function and a multivalued perturbation with nonempty closed, not necessarily, bounded values. Along with this inclusion we consider the inclusion with the perturbation term being convexified. We prove the existence of solutions and a density of the solution set of the original inclusion in a closure of the solution set of the inclusion with the convexified perturbation. In contrast to the known results of this kind we do not suppose that the convex function has the compactness property and that the values of the perturbation are bounded sets. An example of a perturbation with closed unbounded values satisfying the conditions of the main theorems is given.

Subdifferential inclusions with unbounded perturbation: Existence and relaxation theorems

The paper studies an evolution inclusion in a separable Hilbert space whose right-hand side contains the subdifferential of a proper convex lower semicontinuous function of time and a set-valued perturbation. Together with this inclusion, an inclusion with convexified perturbation values is considered. The existence and density of the solution set of the initial inclusion in the closure of the solution set of the inclusion with convexified perturbation are proved. This property is usually called relaxation. Traditional assumptions for relaxation theorems are the compactness property of the convex function and the boundedness of the perturbation. In the present paper, such assumptions are not made. Assumptions for subdifferential inclusions described by polyhedral sweeping processes and variational inequalities with time-dependent obstacles and constraints are specified.

Balanced-Viscosity solutions for multi-rate systems

Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients εα and ε, where 0 < ε « 1 and α > 0 is a fixed parameter. Therefore for α ≠ 1 u and z have different relaxation rates.We address the vanishing-viscosity analysis as ε ↓ 0 of the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system, and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α > 1, α =1, and α є (0,1).

A Class of Subdifferential Inclusions for Elastic Unilateral Contact Problems

We consider a class of stationary subdifferential inclusions in a reflexive Banach space. We reformulate the problem in terms of a variational inequality with multivalued term and prove an existence result using the Kakutani-Fan-Glicksberg fixed point theorem. This approach allows to consider, in a natural way, a dual variational formulation of the problem. Next, we study the link between the primal and dual formulations and provide an equivalence result. Then, we consider a new mathematical model which describes the contact of an elastic body with a foundation. We apply the abstract formalism to derive the primal and the dual variational formulations of the problem, in terms of displacement and stress, respectively. Finally, we present existence and equivalence results in the study of this contact model.

History-Dependent Problems with Applications to Contact Models for Elastic Beams

We prove an existence and uniqueness result for a class of subdifferential inclusions which involve a history-dependent operator. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Problems of such kind arise in a large number of mathematical models which describe quasistatic processes of contact. To provide an example we consider an elastic beam in contact with a reactive obstacle. The contact is modeled with a new and nonstandard condition which involves both the subdifferential of a nonconvex and nonsmooth function and a Volterra-type integral term. We derive a variational formulation of the problem which is in the form of a history-dependent hemivariational inequality for the displacement field. Then, we use our abstract result to prove its unique weak solvability. Finally, we consider a numerical approximation of the model, solve effectively the approximate problems and provide numerical simulations.

On a time-dependent subdifferential evolution inclusion with Carathéodory perturbation

On a time-dependent subdifferential evolution inclusion with Carathéodory perturbation

Numerical Analysis of a Hyperbolic Hemivariational Inequality Arising in Dynamic Contact

In this paper a fully dynamic viscoelastic contact problem is studied. The contact is assumed to be bilateral and frictional, where the friction law is described by a nonmonotone relation between the tangential stress and the tangential velocity. A weak formulation of the problem leads to a second order nonmonotone subdifferential inclusion, also known as a second order hyperbolic hemivariational inequality. We study both semidiscrete and fully discrete approximation schemes and bound the errors of the approximate solutions. Under some regularity assumptions imposed on the true solution, optimal order error estimates are derived for the linear element solution. This theoretical result is illustrated numerically.

Sensitivity of Optimal Solutions to Control Problems for Second Order Evolution Subdifferential Inclusions.

In this paper the sensitivity of optimal solutions to control problems described by second order evolution subdifferential inclusions under perturbations of state relations and of cost functionals is investigated. First we establish a new existence result for a class of such inclusions. Then, based on the theory of sequential Gamma -convergence we recall the abstract scheme concerning convergence of minimal values and minimizers. The abstract scheme works provided we can establish two properties: the Kuratowski convergence of solution sets for the state relations and some complementary Gamma -convergence of the cost functionals. Then these two properties are implemented in the considered case.

Inclusion of Subdifferentials, Linear Well-Conditioning, and Steepest Descent Equation

Given a normed space $X$, we study the following problem: characterize the functions $f:X\to\mathbb{R}\cup\{+\infty\}$ such that for some $\overline{x} \in {\rm dom}\kern 0.12em f$ and some neighborhood $U$ of $\overline{x}$, $\partial f(x) \subset Q \text{for all }x\in U.$ The operator $\partial$ denotes an abstract subdifferential, and $Q$ is a fixed subset of $X^*$ (the topological dual space of $X$). When $X$ is a Banach space and $f$ is lower semicontinuous, we provide a characterization via the support function of $Q$, based on the Zagrodny mean value theorem. If $Q=\partial f(\overline{x})$, the lower semicontinuous functions $f$ verifying the above subdifferential inclusion are positively homogeneous, up to some translation. Connections with the class of linearly well-conditioned functions are explored, and an application is given to the steepest descent equation. A parallel study is developed for an inequality on the directional derivative in place of the above subdifferential inclusion.

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

Abstract We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.

A variational approach via bipotentials for unilateral contact problems

We consider a unilateral contact model for nonlinearly elastic materials, under the small deformation hypothesis, for static processes. The contact is modeled with Signorini’s condition with zero gap and the friction is neglected on the potential contact zone. The behavior of the material is modeled by a subdifferential inclusion, the constitutive map being proper, convex, and lower semicontinuous. After describing the model, we give a weak formulation using a bipotential which depends on the constitutive map and its Fenchel conjugate. We arrive to a system of two variational inequalities whose unknown is the pair consisting of the displacement field and the Cauchy stress field. We look for the unknown into a Cartesian product of two nonempty, convex, closed, unbounded subsets of two Hilbert spaces. We prove the existence and the uniqueness of the weak solution based on minimization arguments for appropriate functionals associated with the variational system. How the proposed variational approach is related to previous variational approaches, is discussed too.