Feedback Nash Equilibria in Differential Games with Impulse Control

TL;DR

This paper analyzes feedback Nash equilibria in a class of two-player differential games with mixed control types, deriving conditions for equilibrium existence and characterizing solutions in linear-quadratic cases.

Contribution

It provides the first analytical characterization of feedback Nash equilibria in linear-quadratic differential games with impulse control, including a verification theorem and bounds on interventions.

Findings

Derived conditions for equilibrium existence using HJB and QVIs.

Established an upper bound on the number of impulses by Player 2.

Provided analytical solutions for linear-quadratic game with impulse control.

Abstract

We study a class of deterministic finite-horizon two-player nonzero-sum differential games where players are endowed with different kinds of controls. We assume that Player 1 uses piecewise-continuous controls, while Player 2 uses impulse controls. For this class of games, we seek to derive conditions for the existence of feedback Nash equilibrium strategies for the players. More specifically, we provide a verification theorem for identifying such equilibrium strategies, using the Hamilton-Jacobi-Bellman (HJB) equations for Player 1 and the quasi-variational inequalities (QVIs) for Player 2. Further, we show that the equilibrium number of interventions by Player 2 is upper bounded. Furthermore, we specialize the obtained results to a scalar two-player linear-quadratic differential game. In this game, Player 1's objective is to drive the state variable towards a specific target value, and Player 2 has a similar objective with a different target value. We provide, for the first time, an analytical characterization of the feedback Nash equilibrium in a linear-quadratic differential game with impulse control. We illustrate our results using numerical experiments.