Evolutionary variational inequalities on the Hellinger-Kantorovich and spherical Hellinger-Kantorovich spaces

Gradient flow based phase-field modeling using separable neural networks

This paper first develops a multitiered supply chain network equilibrium model with fixed demands and proves that the governing equilibrium conditions satisfy a finite-dimensional variational inequality. The paper then establishes that the static supply chain network model with its governing equilibrium conditions can be reformulated as a transportation network equilibrium model over an appropriately constructed abstract network or supernetwork. This identification provides a new interpretation of equilibrium in supply chain networks with fixed demands in terms of path flows. The equivalence is then further exploited to construct a dynamic supply chain network model with time-varying demands (and flows) using an evolutionary (time-dependent) variational inequality formulation. Recent theoretical results in the unification of projected dynamical systems and evolutionary variational inequalities are presented and then applied to formulate dynamic numerical supply chain network examples and to compute the curves of equilibria. An example with step-wise time-dependent demand is also given for illustration purposes.

It is well known that the time-dependent spatial price equilibrium problem can be transformed into and studied as an evolutionary variational inequality. However, in some situations, control policies may be imposed to the end of regulating the amounts of production and consumption. As a consequence, the problem becomes a time-dependent spatial price equilibrium control problem and is formulated as an evolutionary inverse variational inequality. The existence of solutions is then investigated and a numerical example is also provided.

Evolution equations with complete irreversibility and energy conservation

This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge's fixed point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.

In this paper we present an evolutionary variational inequality model of vaccination strategies games in a population with a known vaccine coverage profile over a certain time interval. The population is considered to be heterogeneous, namely its individuals are divided into a finite number of distinct population groups, where each group has different perceptions of vaccine and disease risks. Previous game theoretical analyses of vaccinating behaviour have studied the strategic interaction between individuals attempting to maximize their health states, in situations where an individual's health state depends upon the vaccination decisions of others due to the presence of herd immunity. Here we extend such analyses by applying the theory of evolutionary variational inequalities (EVI) to a (one parameter) family of generalized vaccination games. An EVI is used to provide conditions for existence of solutions (generalized Nash equilibria) for the family of vaccination games, while a projected dynamical system is used to compute approximate solutions of the EVI problem. In particular we study a population model with two groups, where the size of one group is strictly larger than the size of the other group (a majority/minority population). The smaller group is considered much less vaccination inclined than the larger group. Under these hypotheses, considering that the vaccine coverage of the entire population is measured during a vaccine scare period, we find that our model reproduces a feature of real populations: the vaccine averse minority will react immediately to a vaccine scare by dropping their strategy to a nonvaccinator one; the vaccine inclined majority does not follow a nonvaccinator strategy during the scare, although vaccination in this group decreases as well. Moreover we find that there is a delay in the majority's reaction to the scare. This is the first time EVI problems are used in the context of mathematical epidemiology. The results presented emphasize the important role played by social heterogeneity in vaccination behaviour, while also highlighting the valuable role that can be played by EVI in this area of research.

In this chapter, we continue the study of evolutionary variational inequalities started in Chapter 4. The di.erence between the problems studied there and those studied here lies in the fact that the variational inequalities presented in this chapter do not involve a viscosity term. Their study is more complicated, since it cannot be done based on the unique solvability of time-dependent elliptic variational inequalities in velocities. We start with the study of evolutionary variational inequalities involving a di.erentiable functional j, for which we prove an existence and uniqueness result. The proof is based on the study of a sequence of evolutionary variational inequalities with viscosity, compactness, and lower semicontinuity arguments. Then, we extend this result to a class of evolutionary variational inequalities for which the function j can be approached, in a sense that will be described below, by a family of di.erentiable functionals. We complete our results with a convergence result that shows that the solution of the evolutionary variational inequality with viscosity converges to the solution of the corresponding inviscid evolutionary variational inequality, as the viscosity converges to zero. Finally, we present an existence result for evolutionary quasivariational inequalities, obtained by using a time discretization method. The results presented in this chapter will be applied in the study of quasistatic antiplane frictional contact problems with elastic materials. As usual, everywhere in this chapter X is a real Hilbert space with the inner product (·,·)X and the norm $$||\cdot||\ X$$ , and [0, T] denotes the time interval of interest, T > 0. Moreover, X is assumed to be separable everywhere in Section 5.4 of this chapter.

A class of evolution variational inequalities (EVIs), which comprises ordinary differential equations (ODEs) coupled with variational inequalities (VIs) associated with time-varying set-valued mappings, is proposed in this paper. We first study the conditions for existence and uniqueness of solutions. The central idea behind the proof is to rewrite the system dynamics as a differential inclusion which can be decomposed into a single-valued Lipschitz map, and a time-dependent maximal monotone operator. Regularity assumptions on the set-valued mapping determine the regularity of the resulting solutions. Complementarity systems with time-dependence are studied as a particular case. We then use this result to study the problem of designing state feedback control laws for output regulation in systems described by EVIs. The derivation of control laws for output regulation is based on the use of internal model principle, and two cases are treated: First, a static feedback control law is derived when full state feedback is available, In the second case, only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed. As applications, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. Another application is seen in designing control inputs for regulation in power converters.

Evolution variational inequalities and projected dynamical systems with application to human migration

We study the asymptotic behavior of a discrete-in-time minimizing movement scheme for square lattice interfaces when both the lattice spacing and the time step vanish. The motion is assumed to be driven by minimization of a weighted random perimeter functional with an additional deterministic dissipation term. We consider rectangular initial sets and lower order random perturbations of the perimeter functional. In case of stationary, α-mixing perturbations we prove a stochastic homogenization result for the interface velocity. We also provide an example which indicates that only stationary, ergodic perturbations might not yield a spatially homogenized limit velocity for this minimizing movement scheme.

As noted by the second author in the context of unstable two-phase porous medium flow, entropy solutions of Burgers’ equation can be recovered from a minimizing movement scheme involving the Wasserstein metric in the limit of vanishing time step size (Otto, Commun Pure Appl Math, 1999). In this paper, we give a simpler proof by verifying that the anti-derivative is a viscosity solution of the associated Hamilton Jacobi equation.

The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction–diffusion equations of the form [see formula in PDF] with parameters Λ, Σ > 0 and no-flux boundary condition [see formula in PDF] which is built on their gradient-flow-like structure in the space [see formula in PDF] of finite nonnegative Radon measures on [see formula in PDF], endowed with the recently introduced Hellinger-Kantorovich distance HKΛ,Σ. It is proved that, under natural general assumptions on [see formula in PDF] and [see formula in PDF], the Minimizing Movement scheme [see formula in PDF] for [see formula in PDF] yields weak solutions to the above equation as the discrete time step size τ ↓ 0. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance HKΛ,Σ, which will play an important role in this context, is established in the general setting of a separable Hilbert space; that result will constitute a starting point for the study of the differentiability of HKΛ,Σ along absolutely continuous curves which will be carried out in a subsequent paper.

The aim of this paper is to present the relation between an evolutionary variational inequality with long-term memory and Lagrange multipliers. More precisely, we study the oligopolistic market equilibrium problem in which the profit function depends also on previous events of the market by means of a long-term memory which takes into account the previous states of the equilibrium. Moreover, thanks to the variational formulation, we are able to show existence and regularity results for equilibrium solutions. Then, we apply the infinite dimensional duality theory through which we obtain the existence of Lagrange multipliers which are great utility in order to understand the behaviour of the market. Finally, an example is provided, which allows to analyse the influence of the long-term memory on the equilibrium solution.

Double-layered dynamics: A unified theory of projected dynamical systems and evolutionary variational inequalities

In this paper, we make explicit the connection between projected dynamical systems on Hilbert spaces and evolutionary variational inequalities. We give a novel formulation that unifies the underlying constraint sets for such inequalities, which arise in time-dependent traffic network, spatial price equilibrium, and a variety of financial equilibrium problems. We emphasize the importance of the results in applications and provide a traffic network numerical example in which we compute the curve of equilibria.