On the Euler equation approach to discrete--time nonstationary optimal control problems
David González-Sánchez Onésimo Hernández-Lerma
Journal of Dynamics & Games 2014, 1(1): 57-78 doi: 10.3934/jdg.2014.1.57
We are concerned with deterministic and stochastic nonstationary discrete--time optimal control problems in infinite horizon. We show, using Gâteaux differentials, that the so--called Euler equation and a transversality condition are necessary conditions for optimality. In particular, the transversality condition is obtained in a more general form and under milder hypotheses than in previous works. Sufficient conditions are also provided. We also find closed--form solutions to several (discounted) stationary and nonstationary control problems.
keywords: infinite horizon time--varying systems Discrete--time control systems Euler equation transversality condition.
Stability of the replicator dynamics for games in metric spaces
Saul Mendoza-Palacios Onésimo Hernández-Lerma
Journal of Dynamics & Games 2017, 4(4): 319-333 doi: 10.3934/jdg.2017017

In this paper we study the stability of the replicator dynamics for symmetric games when the strategy set is a separable metric space. In this case the replicator dynamics evolves in a space of measures. We study stability criteria with respect to different topologies and metrics on the space of probability measures. This allows us to establish relations among Nash equilibria (of a certain normal form game) and the stability of the replicator dynamics in different metrics. Some examples illustrate our results.

keywords: Evolutionary games population games replicator dynamics space of finite signed measures
A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models
Alejandra Fonseca-Morales Onésimo Hernández-Lerma
Journal of Dynamics & Games 2017, 4(3): 195-203 doi: 10.3934/jdg.2017012

Pareto optimality and Nash equilibrium are two standard solution concepts for cooperative and non-cooperative games, respectively. At the outset, these concepts are incompatible-see, for instance, [7] or [10]. But, on the other hand, there are particular games in which Nash equilibria turn out to be Pareto-optimal [1], [4], [6], [18], [20]. A class of these games has been identified in the context of discrete-time potential games [13]. In this paper we introduce several classes of deterministic and stochastic potential differential games [12] in which open-loop Nash equilibria are also Pareto optimal.

keywords: Nash equilibrium Pareto optimal differential games potential games

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