Open-loop Nash equilibrium for networked control systems with asymmetric information

Jiao Liu; Xianping Wu; Yue Sun

doi:10.3934/naco.2026010

Article Contents

2026, Volume 19: 35-49. Doi: 10.3934/naco.2026010

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Open-loop Nash equilibrium for networked control systems with asymmetric information

1.
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, GuangDong, 510520, China
2.
College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, ShanDong, 266590, China

^*Corresponding author: Xianping Wu
^*Corresponding author: Xianping Wu

Received: September 30, 2025

Revised: January 14, 2026

Early access: February 6, 2026

Published online: February 6, 2026

The second author is supported by [National Natural Science Foundation of China (Nos. 12201129 and 12471351)].

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Abstract

This paper focuses on the open-loop Nash equilibrium for networked control systems (NCSs) with asymmetric information. In this NCSs model, player 1 shares its observations and historical control inputs with player 2, whereas player 2 does not disclose any of its information to player 1. Using the maximum principle, we obtain an explicit analytical Nash equilibrium by decoupling solving the forward and backward stochastic difference equations (FBSDEs), while the optimal gain matrices are given by coupled Riccati equations. The asymmetric information leads to coupled forward and backward Riccati equations, which complicates the calculation. To address it, a forward iteration algorithm is employed to obtain a suboptimal solution for the open-loop Nash equilibrium strategy with asymmetric information. Numerical results illustrate that incorporating the Nash equilibrium into NCSs can improve the system performance.

Keywords:

Open-loop Nash equilibrium,
forward and backward stochastic difference equations,
networked control system,
Riccati equations.

Mathematics Subject Classification: Primary: 93E20, 91A28.

Citation:

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Figure 1. Trajectories of $ P_{k}^{1} $, $ P_{k}^{2} $ and $ P_{k} $

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Figure 2. Performance of $ \Sigma_{k|k}^{1} $ and $ \Sigma_{k|k} $

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