A new feedback form of open-loop Stackelberg strategy in a general linear-quadratic differential game

Yu Li; Kok Lay Teo; Shuhua Zhang

doi:10.3934/jimo.2022105

Article Contents

2023, Volume 19, Issue 5: 3706-3723. Doi: 10.3934/jimo.2022105

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A new feedback form of open-loop Stackelberg strategy in a general linear-quadratic differential game

1.
Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China
2.
School of Mathematical Sciences, Sunway University, Malaysia

^*Corresponding author: Yu Li
^*Corresponding author: Yu Li

Received: December 2021

Revised: May 2022

Early access: June 2022

Published: May 2023

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Abstract

In this paper, we consider a general form of linear-quadratic Stackelberg deterministic differential game model, which consists of one leader and one follower. Each of their utility functions includes all possible squared terms, cross terms and single terms of states and controls of the two players, and constant terms. The time-consistent state feedback form of Stackelberg equilibrium strategy is obtained. Its explicit expression is in terms of the solutions of three decoupled symmetric Riccati differential equations. These decoupled symmetric Riccati differential equations are independent of the state and can be solved backward in time one by one. The proposed model and theory are applied to some classical Stackelberg games.

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Mathematics Subject Classification: Primary: 91A65, 49N10; Secondary: 90B06.

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Figure 1. $ c_e = 0.5, 1.0, 1.5, 2.0 $ from left to right and from up to down

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Figure 2. The optimal polices under promotion

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Figure 3. Zoom in at $ t_s $ and $ t_f $

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Figure 4. The optimal polices under promotion

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Figure 5. Zoom in at $ t_s $ and $ t_f $

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References

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