Time-Dependent Mean-Field Games in the Subquadratic Case

In this paper we consider time-dependent mean-field games with subquadratic Hamiltonians and power-like local dependence on the measure. We establish existence of classical solutions under a certain set of conditions depending on both the growth of the Hamiltonian and the dimension. This is done by combining regularity estimates for the Hamilton-Jacobi equation based on the Gagliardo-Nirenberg interpolation inequality with polynomial estimates for the Fokker-Planck equation. This technique improves substantially the previous results on the regularity of time-dependent mean-field games.     

A general characterization of the mean field limit for stochastic differential games

The mean field limit of large-population symmetric stochastic differential games is derived in a general setting, with and without common noise, on a finite time horizon. Minimal assumptions are imposed on equilibrium strategies, which may be asymmetric and based on full information. It is shown that approximate Nash equilibria in the n-player games admit certain weak limits as n tends to infinity, and every limit is a weak solution of the mean field game (MFG). Conversely, every weak MFG solution can be obtained as the limit of a sequence of approximate Nash equilibria in the n-player games. Thus, the MFG precisely characterizes the possible limiting equilibrium behavior of the n-player games. Even in the setting without common noise, the empirical state distributions may admit stochastic limits which cannot be described by the usual notion of MFG solution.     

一类含临界耦合非线性项的奇异椭圆方程组的正解

本文研究了一类含临界指数与耦合非线性项的奇异椭圆方程组. 利用变分方法与极大值原理, 通过证明对应的能量泛函满足局部的 (PS)c 条件, 得到了这类方程组正解的存在性, 推广了单个方程与方程组中的相应结果.     

Absolute optimal solution for a compact and convex game

This paper investigates the existence of absolute optimal solutions for a partition P in continuous and quasiconcave games. We show that the P-consistency property introduced in the paper, together with the quasiconcavity and continuity of payoffs, permits the existence of P-absolute optimal solutions in games with compact and convex strategy spaces. The P-consistency property is a general condition that cannot be dispensed with for the existence of P-absolute optimal solutions. We also characterize the existence of P-absolute optimal solutions by providing necessary and sufficient conditions. Moreover, we suggest an algorithm for efficiently computing P-absolute optimal solutions.     

Periodic solutions of abstract functional differential equations with state‐dependent delay

In this paper, we are concerned with the existence of solutions of systems determined by abstract functional differential equations with infinite and state‐dependent delay. We establish the existence of mild solutions and the existence of periodic solutions. Our results are based on local Lipschitz conditions of the involved functions. We apply our results to study the existence of periodic solutions of a partial differential equation with infinite and state‐dependent delay. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal control of generalized multiobjective games with application to traffic networks modeling

The purpose of this paper is to study some new results on the existence and convergence of the solutions to controlled systems of generalized multiobjective games, controlled systems of traffic networks, and optimal control problems (OCPs). First, we introduce the controlled systems of generalized multiobjective games and establish the existence of the solutions for these systems using Browder-type fixed point theorem in the noncompact case and the $C_i$-quasi-concavity. Results on the convergence of controlled systems of the solutions for such problems using the auxiliary solution sets and the extended $C_i$-convexity of the objective functions are studied. Second, we investigate OCPs governed by generalized multiobjective games. The existence and convergence of the solutions to these problems are also obtained. Finally, as a real-world application, we consider the special case of controlled systems of traffic networks. Many examples are given for the illustration of our results.     

Doubly Nonlinear Degenerate Parabolic Systems with Coupled Nonlinear Boundary Conditions

In this paper, we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions. We first prove a local existence result by the regularization method. Next, we construct a weak comparison principle. Then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained.     

Some remarks on mean field games

In this article, we study three aspects of mean field games (MFG). The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of “noise” in discrete space models and the formulation of the Master Equation in this case. Finally, we show how MFG reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.     

A multistage pursuit-evasion game that admits a gaussian random process as a maximin control policy

In this paper we consider the existence and structure of both minimax and maximin policies for the special class of LQG pursuit-evasion games which is characterized by (i) a blind evader; and (ii) a pursuer who can make use of noise corrupted state measurements. The particular class of games which we consider has been studied previously by other investigators who have shown that pure strategies exist for both players. The major contribution of our paper is the delineation of the existence and structure of a mixed strategy for the evader in this class of games. This new maximin strategy is defined by a gaussian measure, which can be determined explicitly by the method of least favorable prior distributions. We show that the validity of the pure solutions determined previously is limited by the duration of the game, due to the existence of a ‘pure solution conjugate point’; further, we prove that our new strategies are valid solutions which extend the possible duration of the game beyond the limit imposed by the pure solution conjugate point. We believe that our paper constitutes the first report on the existence of a mixed strategy for an LQG game, and the first report on the role conjugate points play in the transition between pure strategies and mixed strategies.     

Phase-field systems with nonlinear coupling and dynamic boundary conditions

We consider phase-field systems of Caginalp type on a three-dimensional bounded domain. The order parameter fulfills a dynamic boundary condition, while the (relative) temperature is subject to a homogeneous boundary condition of Dirichlet, Neumann or Robin type. Moreover, the two equations are nonlinearly coupled through a quadratic growth function. Here we extend several results which have been proven by some of the authors for the linear coupling. More precisely, we demonstrate the existence and uniqueness of global solutions. Then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors. We also discuss the convergence of given solutions to a single equilibrium.     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 3: applications

This is the third part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so‐called energy method. In the above sense the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 2: abstract quasilinear theory

This is the second part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so‐called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non‐linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

The solvability of some chemotaxis systems

In this paper, we study the local existence and uniqueness of classical solutions to a wide class of systems of chemotaxis equations. These systems are essentially quasi-linear strongly coupled partial differential equations. We also study the maximal interval of existence in time of solutions. The results are illustrated in application to a number of partial differential equation models arising in biology.     

Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations

The paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations. We carry out continuation of local solutions to maximal solutions in a functional analytic setting. For maximal solutions we establish global existence via exponential boundedness and by a contraction argument, adapted to derive uniform existence time. Moreover, within the setting of dual Banach spaces, we derive results on continuous dependence with respect to time and initial state.To achieve generality the paper is organized top down, in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold.     

Existence of pulses in excitable media with nonlocal coupling

We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been crucial in previous existence proofs, by a PDE oriented approach, relying on exponential weights, Fredholm theory, and commutator estimates.     

Set-valued nonlinear analogues of the shapley value

One of Shapley's classic axioms deals with the addition of the games. We consider the family of axiom systems depending on various operations over the set of cooperative games. The existence and the uniqueness of the correspondinga priori set-valued solutions are studied. The family of solutions (satisfying the proposed system of axioms) contains, as particular cases, the Shapley value and the -core. The modification of the proposed system of axioms is studied, prekernel being the maximal solution satisfying this modification.We would like to thank two anonymous referees for helpful comments.     

The dynamics ofn weakly coupled identical oscillators

Summary We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.     

Representation Formulas for Solutions of the HJI Equations with Discontinuous Coefficients and Existence of Value in Differential Games

In this paper, we study the Hamilton-Jacobi-Isaacs equation of zerosum differential games with discontinuous running cost. For such class of equations, the uniqueness of the solutions is not guaranteed in general. We prove principles of optimality for viscosity solutions where one of the players can play either causal strategies or only a subset of continuous strategies. This allows us to obtain nonstandard representation formulas for the minimal and maximal viscosity solutions and prove that a weak form of the existence of value is always satisfied. We state also an explicit uniqueness result for the HJI equations for piecewise continuous coefficients, in which case the usual statement on the existence of value holds.     

Nontrivial periodic solutions for strong resonance Hamiltonian systems via a local linking theorem

In this paper, we first establish an existence result of critical points for a class of functionals defined on Hilbert spaces by using a local linking idea. Then as an application of the existence result, we obtain the existence of periodic solutions of strong resonance Hamiltonian systems which are asymptotically linear both at infinity and at origin.     

Pareto equilibria for generalized constrained multiobjective games in FC-spaces without local convexity structure

In this paper, we further study a class of generalized constrained multiobjective games where the number of players may be finite or infinite, the strategy sets may be general FC-spaces without local convexity structure, and all payoff functions get their values in infinite-dimensional topological vector spaces. By using an existence theorem of maximal elements for a family of set-valued mappings in FC-spaces due to the author, an existence theorem of solutions for a system of generalized vector quasivariational inclusions is first proved in general FC-spaces. By applying the existence result of solutions of the system of generalized vector quasivariational inclusions, some existence theorems of (weak) Pareto equilibria for the generalized constrained multiobjective games are established in noncompact product FC-spaces. Some special cases of our results are also discussed. Our results are new and different from the corresponding known results in the literature.