Abstract
We consider a non-stationary problem of bringing the trajectory of a conflict-controlled process to a given terminal set that varies over time. The basic method for research is the method of resolving functions. While the main scheme of the method uses scalar resolving functions, this paper uses matrix functions of a diagonal form with different elements on the diagonal. This circumstance makes it possible to cover more general game situations. A number of schemes for solving the problem are proposed. Sufficient conditions for the termination of the game in the finite time in the class of quasi and stroboscopic strategies are obtained. The way is indicated for expanding the possibilities and effective application of the method of resolving functions.
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References
Isaacs, R. (1965). Differential games. John Wiley.
Pontryagin, L. S. (1988). Selected scientific papers (Vol. 2). Nauka.
Pschenichnyi, B. N., & Ostapenko, V. V. (1992). Differential games. Naukova Dumka.
Krasovskii, N. N. (1970). Game problems on the encounter of motions. Nauka.
Chikrii, A. A. (2013). Conflict-controlled processes. Springer Science and Business Media.
Aubin, J., & Frankowska, H. (1990). Set-valued analysis. Birkhauser.
Locke, A. S. (1955). Guidance. D. Van Nostrand Company.
Siouris, G. M. (2004). Missile guidance and control systems. Springer.
Shouchuan, H. S., & Papageorgion, N. S. (1997). Handbook of multi-valued analysis theory (vol. 1). Springer.
Chikrii, A. A. (2020). Conflict situations involving controlled object Groups. Part.1. Collision avoidance. Journal of Automation and Information Sciences, 52(7), 19–37.
Pontryagin, L. S. (1971). A linear differential evasion game. Trudy MIAN, 112, 30–63. (in Russian).
Chikrii, A. A. (1976). Linear problem of avoiding several pursuers. Engineering Cybernetics, 14(4), 38–42.
Pschenichnyi, B. N. (1976). Simple pursuit by several objects. Kibernetika, 3, 115–116. (in Russian).
Chikrii, A. A., Matychyn, I., Gromaszek, K., & Smolarc, L. (2011). Control of fractional-order dynamic systems under uncertainty, modelling and optimization (pp. 3–56). Publ. Lublin, Univ. of Technology.
Vlasenko, L. A., Rutkas, A. G., & Chikrii, A. A. (2020). On a differential Game in a stochastic system. Proceedings of the Steklov Institute of Mathematics, 309(1), 185–198.
Kacprzyk, J. (1997). Multistage fuzzy control: A prescriptive approach. John Wiley & Sons Inc.
Lodwick, W. A., & Kacprzyk, J. (Eds.) (2010). Fuzzy optimization, STUDFUZ 254. Springer.
Hajek, O. (1975). Pursuit games (p. 12). Academic Press.
Grigorenko, N. L. (1990). Mathematical methods of control of several dynamic processes. Izdat. Mosc. Gos. Univ.
Blagodatskikh, A. I., & Petrov, N. N. (2009). Conflict interaction of groups of controlled objects. Udmurtskiy Universitet.
Chikrii, A. A., & Chikrii, G. T. (2015). Matrix resolving functions in game problems of dynamics. Proceedings of the Steklov Institute of Mathematics, 291, 56–65.
Petrov, N. N. (2021). Matrix resolving functions in a linear problem of group pursuit with multiple captures. Trudy Instituta Matematiki i Mekhaniki Ur O RAN, 27(2), 185–196.
Gantmakher, F. R. (1967). Theory of matrices. Nauka.
Gaishun, I. V. (1999). Introduction into theory of linear non-stationary systems. Institute of Mathematics of NAS of Belarus.
Nikolskii, M. S. (1984). First direct method of L.S. Pontryagin in differential games. Izdat. Mosc. Gos. University.
Nikolskii, M. S. (1992). Stroboscopic strategies and first direct method of L.S. Pontryagin in quasi-linear differential games of pursuit-evasion. Problems of Control and Information Theory, 11(5), 373–377.
Aumann, R. J. (1965). Integrals of set-valued functions. Journal of Mathematical Analysis and Applications, 12, 1–12.
Natanson, I. P. (1974). Theory of functions of real variable. Nauka.
Joffe, A. D., & Tikhomirov, V. M. (1974). Theory of extreme problems. Nauka.
Chikrii, A. A. (2010). An analytical method in dynamic pursuit games. Proceedings of the Steklov Institute of Mathematics, 271, 69–85.
Kuntsevich, V. M., et al. (Eds.) (2018). Control systems: Theory and applications. In Automation, control and robotics. River Publishers.
Kondratenko, Y. P., et al. (Eds.) (2021). Advanced control systems: Theory and applications. In Automation, control and robotics. River Publishers.
Kondratenko, Y. P., et al. (Eds.) (2022). Recent developments in control systems. In Automation, control and robotics. River Publishers.
Chikrii, G. Ts. (2021). Principle of time stretching for motion control in condition of conflict. Advanced control systems: Theory and applications. In Automation, control and robotics (pp. 53–82). River Publishers.
Chikrii, G. T. (2009). On one problem of approach for damped oscillations. Journal of Automation and Information Sciences, 41(10), 1–9.
Acknowledgements
This work was partially supported by the National Research Foundation of Ukraine. Grant 2020.02/0121 “Analytic methods and machine learning in control theory and decision making in conditions of conflict and uncertainty”.
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Chikrii, A.A., Chikrii, G.T. (2023). Matrix Resolving Functions in Game Dynamic Problems. In: Kondratenko, Y.P., Kreinovich, V., Pedrycz, W., Chikrii, A., Gil-Lafuente, A.M. (eds) Artificial Intelligence in Control and Decision-making Systems. Studies in Computational Intelligence, vol 1087. Springer, Cham. https://doi.org/10.1007/978-3-031-25759-9_5
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