Linear-Quadratic Dynamic Games as Receding-Horizon Variational Inequalities

TL;DR

This paper analyzes linear-quadratic dynamic games, revealing their connection to receding-horizon variational inequalities, and provides explicit solutions and stability results for both open-loop and closed-loop Nash equilibria.

Contribution

It establishes a novel link between linear-quadratic dynamic games and receding-horizon variational inequalities, offering explicit solutions and stability guarantees.

Findings

Unconstrained open-loop Nash equilibrium matches a linear quadratic regulator.

Receding-horizon solutions are asymptotically stable.

Problem is equivalent to a non-symmetric variational inequality.

Abstract

We consider dynamic games with linear dynamics and quadratic objective functions. We observe that the unconstrained open-loop Nash equilibrium coincides with a linear quadratic regulator in an augmented space, thus deriving an explicit expression of the cost-to-go. With such cost-to-go as a terminal cost, we show asymptotic stability for the receding-horizon solution of the finite-horizon, constrained game. Furthermore, we show that the problem is equivalent to a non-symmetric variational inequality, which does not correspond to any Nash equilibrium problem. For unconstrained closed-loop Nash equilibria, we derive a receding-horizon controller that is equivalent to the infinite-horizon one and ensures asymptotic stability.