Approximate Feedback Nash Equilibria in Constrained Differential Games via Model Predictive Control with Upper Bound Guarantees

2026-10-03

AI Summary

This study addresses the computational complexity and difficulty of online adaptation associated with feedback Nash equilibria (FNE) in constrained differential games by proposing a model predictive control (MPC)-based approximation framework. The method approximates infinite-horizon FNE trajectories through finite-horizon open-loop and auxiliary games, while jointly optimizing the prediction horizon and terminal cost. Furthermore, analytical upper bounds on state deviations are derived for unconstrained scenarios to guarantee closed-loop performance. Numerical experiments demonstrate that the proposed approach significantly outperforms existing baseline methods in both constraint handling and approximation accuracy.

Abstract

Physical human--machine interaction and other multi-agent control settings require decision-making policies that adapt online. Differential games provide a principled framework where each agent optimizes an individual objective while anticipating the other's response. The relevant solution concept in many applications is the feedback Nash equilibrium (FNE), which yields time-consistent state-feedback strategies. However, computing the FNE is demanding and becomes intractable when state and input constraints must be enforced, motivating the need for approximate methods. This paper presents a Model Predictive Control (MPC) approach that approximates infinite-horizon FNE trajectories through repeated solution of finite-horizon open-loop games. An auxiliary-game formulation is introduced that selects prediction horizons and terminal costs to approximate the feedback-game optimality conditions. The approach is extended to incorporate hard constraints via a constrained open-loop game formulation. For the unconstrained setting, an analytic upper bound on the state-trajectory deviation between the MPC-induced and FNE trajectories is derived, enabling quantitative performance certification. Numerical examples illustrate the effectiveness of the proposed method compared with baseline approaches from the literature.

Problem

• Differential games
• Feedback Nash equilibrium
• Constrained systems
• Multi-agent control

Innovation

• Model Predictive Control
• Feedback Nash Equilibrium
• Differential Games
• Upper Bound Guarantees
• Constrained Open-loop Game