New relaxations of the Nash equilibrium concept are shown to exist in any strategic game with discontinuous payoff functions. The new concepts are used (1) to show the equivalence between Reny’s better-reply security condition [28] and Simon-Zame’s endogenous tie-breaking rule equilibrium concept [32], (2) to obtain conditions for the existence of approximated equilibria in a class of discontin...

Abstract In this work, we apply the hypercircle method to Discontinuous Galerkin (DG) approximations of second order diffusion problems featuring inhomogeneous Dirichlet and Neumann boundary conditions. We focus on interior penalty discontinuous (IPDG) in primal variational formulation produce a Prager–Synge theorem for such DG methods. Using method, derive an posteriori error estimator terms e...

We consider a class of optimal control problems defined on a stratified domain. Namely, we assume that the state space R admits a stratification as a disjoint union of finitely many embedded submanifolds Mi. The dynamics of the system and the cost function are Lipschitz continuous restricted to each submanifold. We provide conditions which guarantee the existence of an optimal solution, and stu...

In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin (HDG) method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the “hybridized” versions of several of the most commonly-used finite element methods, including mixed, no...

To avoid the order reduction when third order implicit-explicit Runge-Kutta time discretization is used together with the local discontinuous Galerkin (LDG) spatial discretization, for solving convection-diffusion problems with time-dependent Dirichlet boundary conditions, we propose a strategy of boundary treatment at each intermediate stage in this paper. The proposed strategy can achieve opt...

We study the McKean–Vlasov equation on the finite tori of length scale L in d–dimensions. We (re)derive the necessary and sufficient conditions for the existence of a phase transition – first uncovered in [13] and [20]. Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, θ of the interaction parameter. We show (there is ...

This paper presents a new stability analysis of Networked Control Systems (NCSs), where the sampling and the constant network-induced delays are taken into account. The new method is inspired by discontinuous Lyapunov functionals that were recently introduced for sampled-data systems in [1] (in the framework of impulsive system representation) and in [2] (in the framework of input delay approac...

A local discontinuous Galerkin method for solving Korteweg-de Vries (KdV) type equations with non-homogeneous boundary effect is developed. We provide a criterion for imposing appropriate boundary conditions for general KdV type equations. The discussion is then focused on the KdV equation posed on the negative half plane, which arises in the modelling of transition dynamics in the plasma sheat...

In this paper, a m algorithm for solving piecewiselinear networks that include elements with discontinuous characteristics is presented. The algorithm is based on modifications of the Katzenelson algorithm that traces the solution curve through regions where boundary conditions are satisfied. Jumps in the network output variables, caused by state changes of elements with discontinuous character...