Abstract
We consider a finite-horizon two-person zero-sum differential game in which the system dynamics is described by a linear differential equation with a Caputo fractional derivative and the goals of the players’ control are to minimize and maximize a quadratic terminal-integral cost function, respectively. We present conditions for the existence of a game value and obtain formulas for players’ optimal feedback control strategies with memory of motion history. The results are based on the construction of a solution to an appropriate Hamilton–Jacobi equation with fractional coinvariant derivatives under a natural right-end boundary condition.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
V. I. Zhukovskii and A. A. Chikrii, Linear-Quadratic Differential Games (Naukova Dumka, Kiev, 1994) [in Russian].
T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, 2nd ed. (SIAM, Philadelphia, PA, 1999).
J. Engwerda, LQ Dynamic Optimization and Differential Games (Wiley, Chichester, 2005).
Y. C. Ho, A. E. Bryson, and S. Baron, “Differential games and optimal pursuit-evasion strategies,” IEEE Trans. Autom. Control 10 (4), 385–389 (1965).
P. Bernhard, “Linear-quadratic, two-person, zero-sum differential games: Necessary and sufficient conditions,” J. Optim. Theory Appl. 27 (1), 51–69 (1979).
J. Yong, Differential Games: A Concise Introduction (World Scientific, Hackensack, NJ, 2015).
N. N. Krasovskii, Control of a Dynamical System: The Problem of the Minimum of a Guaranteed Result (Nauka, Moscow, 1985) [in Russian].
N. N. Krasovskii and V. E. Tret’yakov, Control Problems with a Guaranteed Result (Sredne-Ural. Knizhn. Izd., Sverdlovsk, 1986) [in Russian].
V. Ya. Turetskii, “Solubility condition for a linear-quadratic game,” Autom. Remote Control. 50 (8), 1045–1052 (1989).
T. Başar and P. Bernhard, \({{H}^{\infty }}\)-Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, 2nd ed. (Birkhäuser, Boston, MA, 1995).
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science, New York, 1993).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006).
K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (Springer, Berlin, 2010).
A. A. Chikrii and I. I. Matichin, “Game problems for fractional-order linear systems,” Proc. Steklov Inst. Math. 268, Suppl. 1, S54–S70 (2010).
M. Mamatov and K. Alimov, “Differential games of persecution of frozen order with separate dynamics,” J. Appl. Math. Phys. 6 (3), 475–487 (2018).
A. I. Machtakova and N. N. Petrov, “On a linear group pursuit problem with fractional derivatives,” Proc. Steklov Inst. Math. 319, Suppl. 1, S175–S187 (2022).
M. I. Gomoyunov, “Extremal shift to accompanying points in a positional differential game for a fractional-order system,” Proc. Steklov Inst. Math. 308, Suppl. 1, S83–S105 (2020).
M. I. Gomoyunov, “Differential games for fractional-order systems: Hamilton–Jacobi–Bellman–Isaacs equation and optimal feedback strategies,” Mathematics 9 (14), 1667 (2021).
Y. You, “Quadratic integral games and causal synthesis,” Trans. Am. Math. Soc. 352 (6), 2737–2764 (2000).
M. I. Gomoyunov, “Dynamic programming principle and Hamilton–Jacobi–Bellman equations for fractional-order systems,” SIAM J. Control Optim. 58 (6), 3185–3211 (2020).
A. V. Kim, Functional Differential Equations: Application of \(i\)-Smooth Calculus (Springer, Dordrecht, 1999).
N. Yu. Lukoyanov, Functional Hamilton–Jacobi Equations and Control Problems with Hereditary Information (Ural. Fed. Univ., Yekaterinburg, 2011) [in Russian].
M. I. Gomoyunov, “Value functional and optimal feedback control in linear-quadratic optimal control problem for fractional-order system,” Math. Control Relat. Fields (2023). https://doi.org/10.3934/mcrf.2023002
M. I. Gomoyunov, “Optimal control problems with a fixed terminal time in linear fractional-order systems,” Arch. Control Sci. 30 (4), 721–744 (2020).
Kolmogorov A. N. and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover, Mineola, NY, 1999).
Funding
This work was supported by the Russian Science Foundation, project no. 21-71-10070, https://rscf.ru/en/project/21-71-10070/.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Translated by I. Ruzanova
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gomoyunov, M.I., Lukoyanov, N.Y. On Linear-Quadratic Differential Games for Fractional-Order Systems. Dokl. Math. 108 (Suppl 1), S122–S127 (2023). https://doi.org/10.1134/S1064562423600689
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562423600689