Suboptimal open-loop solution of a Stackelberg linear-quadratic differential game with cheap control of a follower: analytical/numerical study

TL;DR

This paper analyzes the open-loop solutions of a Stackelberg linear-quadratic differential game with a small control cost for the follower, focusing on singular perturbation analysis and asymptotic behavior of solutions.

Contribution

It introduces an analytical and numerical framework for solving a singularly perturbed Stackelberg game with cheap control, providing asymptotic approximations and suboptimal control strategies.

Findings

Asymptotic analysis of the boundary-value problem reveals the solution's behavior as control cost approaches zero.

Derived asymptotically suboptimal controls for players in the cheap control regime.

Illustrative example demonstrates the application to a supply chain problem with a retailer.

Abstract

A two-player finite horizon linear-quadratic Stackelberg differential game is considered. The feature of this game is that the control cost of a follower in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. The open-loop solution of this game is studied. Using the game's solvability conditions, obtaining such a game's solution is reduced to the solution of a proper boundary-value problem. Due to the smallness of the follower's control cost, this boundary-value problem is singularly perturbed. The asymptotic behaviour of the solution to this problem is analysed. Based on this analysis, the asymptotic behaviour of the open-loop optimal players' controls and the optimal values of the cost functionals is studied. Using these results, asymptotically suboptimal players' controls are designed. An illustrative example of a supply chain problem with a small control cost of a retailer is presented.