Non-autonomous Linear Quadratic Non-cooperative Differential Games with Continuous Updating

Ildus Kuchkarov; Ovanes Petrosian; Yin Li

doi:10.21638/11701/spbu31.2022.11

Authors

Ildus Kuchkarov Saint Petersburg State University
Ovanes Petrosian Saint Petersburg State University; National Research University Higher School of Economics
Yin Li St. Petersburg State University; Harbin Institute of Technology

DOI:

https://doi.org/10.21638/11701/spbu31.2022.11

Abstract

The subject of this paper is a non-autonomous linear quadratic case of a differential game model with continuous updating. This class of differential games is essentially new where it is assumed that, at each time instant, players have or use information about the game structure defined on a closed time interval with a fixed duration. During the interval information about motion equations and payoff functions of players updates. It is non-autonomy that simulates this effect of updating information. A linear quadratic case for this class of games is particularly important for practical problems arising in the engineering of human-machine interaction. Here we define the Nash equilibrium as an optimality principle and present an explicit form of Nash equilibrium for the linear quadratic case. Also, the case of dynamic updating for the linear quadratic differential game is studied and uniform convergence of Nash equilibrium strategies and corresponding trajectory for a case of continuous updating and dynamic updating is demonstrated.

Keywords:

differential games with continuous updating, Nash equilibrium, linear quadratic differential games, non-autonomous

Downloads

Download data is not yet available.

References

Basar, T. and Olsder, G. J. (1995). Dynamic noncooperative game theory. Academic Press, London

Bellman, R. (1957). Dynamic Programming. Princeton University Press: Princeton, NJ

Bemporad, A., Morari, M., Dua, V. and Pistikopoulos, E. (2002). The explicit linear quadratic regulator for constrained systems. Automatica, 38(1), 3–20

Eisele, T. (1982). Nonexistence and nonuniqueness of open-loop equilibria in linear-quadratic differential games. Journal of Optimization Theory and Applications, 37(4), 443–468

Engwerda, J. (2005). LQ Dynamic Optimization and Differential Games, Willey: New York

Filippov, A. (2004). Introduction to the theory of differential equations (in Russian), Editorial URSS: Moscow

Goodwin, G., Seron, M. and Dona, J. (2005). Constrained Control and Estimation: An Optimisation Approach, Springer-Verlag: London

Gromova, E. V. and Petrosian, O. L. (2016). Control of information horizon for cooperative differential game of pollution control, 2016 International Conference Stability and Oscillations of Nonlinear Control Systems: Pyatnitskiy

Hempel, A., Goulart, P. and Lygeros, J. (2015). Inverse parametric optimization with an application to hybrid system control. IEEE Transactions on Automatic Control, 60(4), 1064–1069

Isaacs, R. (1965). Differential Games, John Wiley and Sons: New York

Kuchkarov, I. and Petrosian, O. (2019). On class of linear quadratic non-cooperative differential games with continuous updating. Lecture Notes in Computer Science, 11548, 635–650

Kuchkarov, I. and Petrosian, O. (2020). Open-loop based strategies for autonomous linear quadratic game models with continuous updating. In: Kononov, A., M. Khachay, V. A. Kalyagin and P. Pardalos (eds.). Mathematical Optimization Theory and Operations Research, pp. 212–230. Springer International Publishing: Cham

Kwon, W., Bruckstein, A. and Kailath, T. (1982). Stabilizing state-feedback design via the moving horizon method. 21 IEEE Conference on Decision and Control

Kwon, W. and Han, S. (2005). Receding Horizon Control: Model Predictive Control for State Models, Springer-Verlag: London

Kwon, W. and Pearson, A. (1977). A modified quadratic cost problem and feedback stabilization of a linear system. IEEE Transactions on Automatic Control, 22(5), 838–842

Mayne, D. and Michalska, H. (1990). Receding horizon control of nonlinear systems. IEEE Transactions on Automatic Control, 35(7), 814–824

Petrosian, O. L. (2016a). Looking forward approach in cooperative differential games. International Game Theory Review, (18), 1–14

Petrosian, O. L. (2016b). Looking forward approach in cooperative differential games with infinite-horizon. Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, (4), 18–30

Petrosian, O. L. and Barabanov, A. E. (2017). Looking forward approach in cooperative differential games with uncertain-stochastic dynamics. Journal of Optimization Theory and Applications, 172, 328–347

Petrosian, O. L., Nastych, M. A. and Volf, D. A. (2018). Non-cooperative differential game model of oil market with looking forward approach. Frontiers of Dynamic Games, Game Theory and Management, St. Petersburg, 2017 (Petrosyan L. A., V. V. Mazalov and N. Zenkevich eds), Birkhuser: Basel

Petrosian, O. L., Nastych, M. A. and Volf, D. A. (2017). Differential game of oil market with moving informational horizon and non-transferable utility. 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V. F. Demyanov)

Petrosian, O., Shi, L., Li, Y. and Gao, H. (2019). Moving information horizon approach for dynamic game models, Mathematics, 7(12), 1–31

Petrosian, O. and Tur, A. (2019). Hamilton-jacobi-bellman equations for non-cooperative differential games with continuous updating. In: Mathematical Optimization Theory and Operations Research, pp. 178–191

Petrosyan, L. A. and Murzov, N. V. (1966). Game-theoretic problems in mechanics. Lithuanian Mathematical Collection, (3), 423–433

Pontryagin, L. S. (1996). On theory of differential games. Successes of Mathematical Sciences, 26, 4(130), 219–274

Rawlings, J. and Mayne, D. (2009). Model Predictive Control: Theory and Design. Nob Hill Publishing, LLC: Madison

Shaw, L. (1979). Nonlinear control of linear multivariable systems via state-dependent feedback gains. IEEE Transactions on Automatic Control, 24(1), 108–112

Shevkoplyas, E. (2014). Optimal solutions in differential games with random duration. Journal of Mathematical Sciences, 199(6), 715–722

Wang, L. (2005). Model Predictive Control System Design and Implementation Using MATLAB. Springer-Verlag: London

Yeung, D. W. K. and Petrosian, O. (2017). Cooperative stochastic differential games with information adaptation. International Conference on Communication and Electronic Information Engineering

Yeung, D. W. K. and Petrosian, O. (2017). Infinite horizon dynamic games: A new approach via information updating. International Game Theory Review, 19, 1–23