On polynomial feedback Nash equilibria for two-player scalar differential games

In this paper, two-player scalar differential games are thoroughly studied, in the presence of polynomial dynamics and focusing on the notion of solution provided by polynomial feedback Nash equilibria. It is well-known that such strategies are related to the solution of coupled partial differential equations, namely the so-called Hamilton-Jacobi-Isaacs equations. Herein, we firstly prove a somewhat negative result, stating that, for a generic choice of the parameters, two-player scalar polynomial differential games do not admit polynomial Nash equilibria. Then, we focus on the class of Linear-Quadratic (LQ) games and we propose an algorithm that, by borrowing techniques from algebraic geometry, allows to recast the problem of computing all stabilizing Nash feedback strategies into that of finding the zero of a single polynomial function in a scalar variable. This permits a comprehensive characterization-in terms of number and values-of the set of solutions to the associated game.