Viscosity Solutions of Hamilton-Jacobi-Bellman Equations for Nonzero-Sum Differential Games with Safety Consideration

In this paper, the solutions of a class of infinite-horizon, nonzero-sum differential games are studied in the viscosity solution sense while considering the system's safety. The value functions, which are solutions of Hamilton-Jacobi-Bellman equations, are assumed to be continuous so that the safety constraint defined by the control barrier function can be incorporated. We present a new idea for solving nonzero-sum differential games by approximating their closed-loop solutions via a collection of static noncooperative games. Novel algorithms based on fixed-point iteration and static noncooperative games are proposed to approximate the solutions of nonzero-sum differential games, both with and without the safety constraint. The cost function values, evaluated at the Nash equilibrium points of unconstrained static noncooperative games and at the generalized Nash equilibrium points of constrained static noncooperative games, converge to the solutions of the associated nonzero-sum differential games under the proposed fixed-point iteration scheme. The approximation errors of the proposed algorithms are analyzed. Numerical examples are provided at the end to demonstrate the effectiveness of the proposed methods.