Abstract
The purpose of this paper is to study the existence of solutions of a Hamilton-Jacobi equation in a minimax discrete-time case and to show different characterizations for a real number called the critical value, which plays a central role in this work. We study the behavior of solutions of this problem using tools of game theory to obtain a “fixed point” of the Lax operator associated, considering some facts of weak KAM theory to interpret these solutions as discrete viscosity solutions. These solutions represent the optimal payoff of a zero-sum game of two players, with increasingly long time payoffs. The developed techniques allow us to study the behavior of an infinite time game without using discount factors or average actions.
Highlights
The Hamilton-Jacobi equation is a very important tool for the study of Lagrangian systems and control theory
In the case of convex Lagrangians, Fathi, Mather, and Mane have perfected several techniques to understand the solutions of this equation ([1,2,3,4,5,6,7])
These techniques can be translated to a discrete case where, instead of a Lagrangian, there is a function V(x, y) of two variables in the configuration space
Summary
The Hamilton-Jacobi equation is a very important tool for the study of Lagrangian systems and control theory. The functions in the previous result are called fixed points of the Lax operator with critical value c These fixed points are solutions of the following discrete-time Hamilton-Jacobi equation associated to V: y1 ∈N. We will consider an infinite horizon problem, which will help us to find a different characterization of critical value c To this end, we define the lower Peierls barrier hk− as follows, given a real number k and an initial state (x, y) ∈ M × N: hk−. If M and N are compact metric spaces, V : M × N×M×N → R is a Lipschitz function, and u ∈ C(M×N, R) is a fixed point of the Lax operator Ln with critical value c ∈ R, c = sup SU (u) = inf SL (u) ,. We will show that −hc− is a critical solution and a solution of the Hamilton-Jacobi equation
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