Evolutionary Variational Inequalities Research Articles

New existence results for evolutionary quasi-hemi-variational inequalities and their optimal control

This paper investigates evolutionary quasi-variational inequalities that involve multi-valued monotone and pseudo-monotone maps. The purpose of including the pseudo-monotone map is to encompass generalized derivatives of certain nonconvex locally Lipschitz functionals as a specific instance. Consequently, the considered evolutionary quasi-variational inequalities subsume evolutionary quasi-hemi-variational inequalities as a particular case. We present an existence result wherein the key novelty lies in the variational selection that separates the monotone and pseudo-monotone components, thus eliminating the conventional assumption that their sum is monotone. While we continue to employ the Minty formulation, its necessity is limited to evolutionary monotone variational inequalities alone. Furthermore, we offer new applications to optimal control problems governed by evolutionary quasi-variational inequalities.

Optimal Control Problems for Evolutionary Variational Inequalities with Volterra-Type Operators

Optimal Control Problems for Evolutionary Variational Inequalities with Volterra-Type Operators

On the Approximation of Some Evolutionary Inclusions in Banach Spaces

This paper deals with an original application of method of finite difference approximations for evolutionary variational inequalities in Banach spaces. The main result of the study is the suggestion of a procedure and the demonstration of a theorem that, with a different approach, respect to the well known methods present in literature, proves some properties of the solutions of evolutionary variational inequalities.

Evolution equations with complete irreversibility and energy conservation

This paper is concerned with the initial-boundary value problem for an evolutionary variational inequality complying with three intrinsic properties: complete irreversibility, unilateral equilibrium of an energy and an energy conservation law, which cannot generally be realized in dissipative systems such as standard gradient flows. Main results consist of well-posedness in a strong formulation, qualitative properties of strong solutions (i.e., comparison principle and the three properties mentioned above) and long-time dynamics of strong solutions (more precisely, convergence to an equilibrium). The well-posedness will be proved based on a minimizing movement scheme without parabolic regularization, which will also play a crucial role for proving qualitative and asymptotic properties of strong solutions. Moreover, the variational inequality under consideration will be characterized as a singular limit of some (generalized) gradient flows.

A sustainable dynamic closed-loop supply chain network equilibrium for collectibles markets

In this paper, we present a sustainable closed-loop supply chain network equilibrium problem for collectible items in a time-dependent framework. In particular, we consider a network consisting of manufacturers, retailers, demand markets, and an online second-hand platform involved in forward and reverse logistics competition. Manufacturers as well as retailers try to control the amount of emissions generated in transaction processes. We analyze the optimal behaviors of all the decision-makers, and we give the governing closed-loop supply chain network equilibrium conditions. Then, we introduce the associated evolutionary variational inequality and the related infinite-dimensional projected dynamical system, which are convenient to solve such a problem. Finally, using a discrete-time approximation procedure, we solve a numerical example considering eBay as online platform.

Existence of a Generalized Solution for the Fractional Contact Problem

In this paper, we take into consideration the mathematical analysis of time-dependent quasistatic processes involving the contact between a solid body and an extremely rigid structure, referred to as a foundation. It is assumed that the constitutive law is fractional long-memory viscoelastic. The contact is considered to be bilateral and is modeled around Tresca’s law. We establish the existence of the generalized solution’s result. The proof is supported by the surjectivity of the multivalued maximum monotone operator, Rothe’s semidiscretization method, and arguments for evolutionary variational inequality.

Ensemble approximations for constrained dynamical systems using Liouville equation

For a class of state-constrained dynamical systems described by evolution variational inequalities, we study the time evolution of a probability measure which describes the distribution of the state over a set. In contrast to smooth ordinary differential equations, where the evolution of this probability measure is described by the Liouville equations, the flow map associated with the nonsmooth differential inclusion is not necessarily invertible and one cannot directly derive a continuity equation to describe the evolution of the distribution of states. Instead, we consider Lipschitz approximation of our original nonsmooth system and construct a sequence of measures obtained from Liouville equations corresponding to these approximations. This sequence of measures converges in weak-star topology to the measure describing the evolution of the distribution of states for the original nonsmooth system. This allows us to approximate numerically the evolution of moments (up to some finite order) for our original nonsmooth system, using a solver that uses finite order moment approximations of the Liouville equation. Our approach is illustrated with the help of two academic examples.

Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models

Abstract We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and a symmetric deviatoric stress tensor. This stress tensor is transported via the Zaremba-Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray-Hopf-type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the nondiffusive limit in the relative energy inequality satisfied by generalized solutions for nonzero stress diffusion.

Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces

We introduce and investigate a notion of multivalued lambda -dissipative probability vector field (MPVF) in the Wasserstein space mathcal {P}_2({textsf {X} }) of Borel probability measures on a Hilbert space {textsf {X} }. Taking inspiration from the theories of dissipative operators in Hilbert spaces and of Wasserstein gradient flows for geodesically convex functionals, we study local and global well posedness of evolution equations driven by dissipative MPVFs. Our approach is based on a measure-theoretic version of the Explicit Euler scheme, for which we prove novel convergence results with optimal error estimates under an abstract stability condition, which do not rely on compactness arguments and also hold when {textsf {X} } has infinite dimension. We characterize the limit solutions by a suitable Evolution Variational Inequality (EVI), inspired by the Bénilan notion of integral solutions to dissipative evolutions in Banach spaces. Existence, uniqueness and stability of EVI solutions are then obtained under quite general assumptions, leading to the generation of a semigroup of nonlinear contractions.

The method of extremal shift in control problems for evolution variational inequalities under disturbances

<p style='text-indent:20px;'>The problems of designing feedback control algorithms for parabolic and hyperbolic variational inequalities are considered. These algorithms should preserve given properties of solutions of inequalities under the action of unknown disturbances. Solving algorithms that are stable with respect to informational noises are constructed. The algorithms are based on the method of extremal shift well-known in the theory of guaranteed control.</p>

Optimal Control of Plasticity with Inertia

The paper is concerned with an optimal control problem governed by the equations of elasto plasticity with linear kinematic hardening and the inertia term at small strain. The objective is to optimize the displacement field and plastic strain by controlling volume forces. The idea given in [10] is used to transform the state equation into an evolution variational inequality (EVI) involving a certain maximal monotone operator. Results from [27] are then used to analyze the EVI. A regularization is obtained via the Yosida approximation of the maximal monotone operator, this approximation is smoothed further to derive optimality conditions for the smoothed optimal control problem.

Lagrange multipliers for evolution problems with constraints on the derivatives

The existence of generalized Lagrange multipliers is proved for a class of evolution problems for linear differential operators of various types subject to constraints on the derivatives. Those Lagrange multipliers and the respective solutions are stable for the vanishing of the coercive parameter and are naturally associated with evolution variational inequalities with time-dependent convex sets of gradient type. These results are applied to the sandpile problem, to superconductivity problems, to flows of thick fluids, to problems with the biharmonic operator, and to first order vector fields of subelliptic type.

Cooperation in traffic network problems via evolutionary split variational inequalities

<p style='text-indent:20px;'>In this paper, we construct an evolutionary (time-dependent) split variational inequality problem and show how to reformulate equilibria of the dynamic traffic network models of two cities as such problem. We also establish existence result for the proposed model. Primary numerical results of equilibria illustrate the validity and applicability of our results.</p>

A λ-convexity based proof for the propagation of chaos for weakly interacting stochastic particles

In this work we give a proof of the mean-field limit for λ-convex potentials using a purely variational viewpoint. Our approach is based on the observation that all evolution equations that we study can be written as gradient flows of functionals at different levels: in the set of probability measures, in the set of symmetric probability measures on N variables, and in the set of probability measures on probability measures. This basic fact allows us to rely on Γ-convergence tools for gradient flows to complete the proof by identifying the limits of the different terms in the Evolutionary Variational Inequalities (EVIs) associated to each gradient flow. The λ-convexity of the confining and interaction potentials is crucial for the unique identification of the limits and for deriving the EVIs at each description level of the interacting particle system.

Optimal control of an abstract evolution variational inequality with application to homogenized plasticity

The paper is concerned with an optimal control problem governed by a state equation in form of a generalized abstract operator differential equation involving a maximal monotone operator. The state equation is uniquely solvable, but the associated solution operator is in general not G\^ateaux-differentiable. In order to derive optimality conditions, we therefore regularize the state equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation. We show convergence of global minimizers for regularization parameter tending to zero and derive necessary and sufficient optimality conditions for the regularized problems. The paper ends with an application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.

Convergence analysis and error estimates of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time

This paper is presented for the convergence analysis of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time, which arises from the theory of viscoelastic materials with edge friction. First, the existence and uniqueness of the solutions for the evolutionary variational inequality of the second-order in time are proved, which are mainly based on the fixed point theorem. Second, the convergence analysis of the interpolating element-free Galerkin method is presented for them. The error estimates show that the convergence order depends not only on the number of basis functions in the interpolating moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates.

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 nonsmooth approach for the modelling of a mechanical rotary drilling system with friction

<p style='text-indent:20px;'>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.

Evolutionary Variational Inequalities with Volterra Type Operators

In this paper, we consider the initial-value problem for parabolic variational inequalities (subdifferential inclusions) with Volterra type operators. We prove the existence and the uniqueness of the solution. Furthermore, the estimates of the solution are obtained. The results are achieved using the Banach's fixed point theorem (the principle of compression mappings). The motivation for this work comes from the evolutionary variational inequalities arising in the study of frictionless contact problems for linear viscoelastic materials with long-term memory. Also, such kind of problems have their application in constructing different models of the injection molding processes.

Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces

Aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X, d,m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N -dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N -dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger’s energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.