Adaptive LQ Differential Games with Asymmetric Information

Abstract This paper investigates the stochastic adaptive linear-quadratic(LQ) two-person zero-sum differential game with asymmetric information. Compared with most existing literature, the study incorporates “parameter uncertainty” and “limited decision-making capability” into the dynamic game framework, thereby extending the applicable scope of traditional models. The information acquired by the two players is asymmetric, with an inclusion relation between their information sets. We analyze the adaptive problem where the system parameter matrix is unknown to both players. Based on the least squares estimator, the certainty equivalence principle, the stochastic filtering theory and the diminishing excitation technique, we construct a pair of adaptive strategies. It is proven that, when the system matrix is controllable and the algebraic Riccati equation associated with the game system admits a desired real symmetric solution, the Lyapunov equation, Cauchy–Schwarz inequality and Itô formula are adopted to prove that the designed adaptive strategies can guarantee global stability of the system and realize the Nash equilibrium. Moreover, the terminal objective payoff under the proposed adaptive strategies is derived by virtue of theories related to inner product spaces. Data availability No datasets were generated or analysed during the current study. References Isaacs R (1965) Differential games. Wiley, New York Bagchi A (1984) Stackelberg differential games in economic models. Springer, Berlin Smith JM (1982) Evolution and the theory of games. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511806292 Fudenberg D, Tirole J (1991) Game theory. MIT, Cambridge Engwerda J (2005) LQ dynamic optimization and differential games. Wiley, New York Başar T, Olsder GJ (1999) Dynamic noncooperative game theory, 2nd edn. SIAM Bernhard P (1979) LQ, two-person, zero-sum differential games: necessary and sufficient conditions. J Optim Theory Appl 27:51–69. https://doi.org/10.1007/BF00933325 Wang GC, Yu ZY (2012) A partial information non-zero sum differential game of backward stochastic differential equations with applications. Automatica 48:342–352. https://doi.org/10.1016/j.automatica.2011.11.010 Huang JH, Wang GC, Xiong J (2009) A maximum principle for partial information backward stochastic control problems with applications. SIAM J Control Optim 48:2106–2117. https://doi.org/10.1137/080738465 Wang GC, Wu Z, Xiong J (2015) A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information. In: 2020 39th Chinese control conference (CCC), pp 966–971. https://doi.org/10.1109/TAC.2015.2411871 Zheng YY, Shi JT (2020) A linear quadratic Stackelberg game of backward stochastic differential equations with partial information. SIAM J Control Optim 60:2904–2916. https://doi.org/10.23919/CCC50068.2020.9189372 Wang GC, Xiao H, Xiong J (2018) A kind of LQ non-zero sum differential game of backward stochastic differential equation with asymmetric information. Automatica 97:346–352. https://doi.org/10.1016/j.automatica.2018.08.019 Chang DJ, Xiao H (2014) Linear quadratic nonzero sum differential games with asymmetric information. Math Probl Eng 2014:262314. https://doi.org/10.1155/2014/262314 Wu S (2022) Linear-quadratic non-zero sum backward stochastic differential game with overlapping information. IEEE Trans Autom Control 68:1800–1806. https://doi.org/10.1109/TAC.2022.3159490 Shi JT, Wang GC, Xiong J (2020) Stochastic linear quadratic Stackelberg differential game with overlapping information. ESAIM Control Optim Calc Var 26:83. https://doi.org/10.1051/cocv/2020006 Li Y, Guo L (2011) Towards a theory of stochastic adaptive differential games. In: Proceedings of the 50th IEEE conference on decision and control and European control conference, pp 5041–5046. https://doi.org/10.1109/CDC.2011.6160768 Yuan S, Guo L (2016) Stochastic adaptive dynamical games. Sci Sin Math 46:1367–1382 Zhang R, Guo L (2019) Controllability of Nash equilibrium in game-based control systems. IEEE Trans Autom Control 64:4180–4187. https://doi.org/10.1109/TAC.2019.2893150 Zhang R, Guo L (2019) Controllability of stochastic game-based control systems. SIAM J Control Optim 57:3799–3826. https://doi.org/10.1137/18M120854 Liu F, Yu Z (2022) Controllability Gramian for stochastic game-based systems. IEEE Trans Autom Control 68:6036–6050. https://doi.org/10.1109/TAC.2022.3232181 Zhang R, Guo L (2021) Stabilizability of game-based control systems. SIAM J Control Optim 59:3999–4023. https://doi.org/10.1137/20M133587X Rizvi SAA, Lin Z (2018) Output feedback q-learning for discrete-time linear zero-sum games with application to the h-infinity control. Automatica 95:213–221. https://doi.org/10.1016/j.automatica.2018.05.027 Rizvi SAA, Lin Z (2020) Output feedback adaptive dynamic programming for linear differential zero-sum games. Automatica 122:109272. https://doi.org/10.1016/j.automatica.2020.109272 Liu N, Guo L (2023) Stochastic adaptive linear quadratic differential games. IEEE Trans Autom Control 69:1066–1073. https://doi.org/10.1109/TAC.2023.3274863 Liu N, Guo L (2024) Adaptive stabilization of noncooperative stochastic differential games. SIAM J Control Optim 62:1317–1342. https://doi.org/10.1137/22M1530549 Tian XQ, Liu SJ, Yang X (2024) Stochastic adaptive linear quadratic nonzero-sum differential games. Appl Math Comput 477:128803. https://doi.org/10.1016/j.amc.2024.128803 Liu N, Tan S, Tao Y, Lü J (2025) Adaptive non-cooperative differential games with a regulator. Automatica 175:112201. https://doi.org/10.1016/j.automatica.2025.112201 Guo L (1996) Self-convergence of weighted least-squares with applications to stochastic adaptive control. IEEE Trans Autom Control 41:79–89. https://doi.org/10.1109/9.481609 Duncan TE, Guo L, Pasik-Duncan B (1999) Adaptive continuous-time linear quadratic Gaussian control. IEEE Trans Autom Control 44:1653–1662. https://doi.org/10.1109/9.788532 Abou-Kandil H, Freiling G, Ionescu V, Jank G (2003) Matrix Riccati equations in control and systems theory. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8081-7 Chen HF, Guo L (1987) Optimal adaptive control and consistent parameter estimates for armax model with quadratic cost. SIAM J Control Optim 25:845–867. https://doi.org/10.1137/0325047 Duncan TE, Pasik-Duncan B (1990) Adaptive control of continuous-time linear stochastic systems. Math Control Signals Syst 3:45–60. https://doi.org/10.1007/BF02551355 Guo L (2020) Time-varying stochastic systems: stability and adaptive theory, 2nd edn. Science Press, Beijing Chen HF, Guo L (1991) Identification and stochastic adaptive control. Birkhäuser, Boston Chow Y, Teicher H (2008) Probability theory: independence, interchangeability, martingales. Springer, Berlin Acknowledgements This work was supported by the National Natural Science Foundation of China [Grant Numbers 12061020, 12461054, 71961003]; the Science and Technology Program of Guizhou Province [Grant Number QKH-LH(2017)7223] and the Doctoral Foundation Project of Guizhou University [Grant Number (2019)49]. Author information Authors and Affiliations Contributions S.Y. and Y.Y. wrote the main manuscript text. All authors reviewed the manuscript. Corresponding author Additional information Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. About this article Cite this article Yan, S., Yang, Y. Adaptive LQ Differential Games with Asymmetric Information. Dyn Games Appl (2026). https://doi.org/10.1007/s13235-026-00713-2 Received: Accepted: Published: Version of record: DOI: https://doi.org/10.1007/s13235-026-00713-2 Keywords - Adaptive control - Least squares - Nash equilibrium - Stochastic differential games - Uncertain stochastic systems - Asymmetric information