Abstract
In this paper, we consider an antagonistic differential game of two persons with dynamics described by a differential equation with simple motions and an integral terminal payment functional. In this game, there exists a price function, which is a generalized (minimax or viscous) solution of the corresponding Hamilton–Jacobi equation. For the case where the terminal function and the Hamiltonian are piecewise linear and the dimension of the phase space is equal to 2, we propose a finite algorithm for the exact construction of the price function. The algorithm consists of the sequential solution of elementary problems arising in a certain order. The piecewise linear price function of a differential game is constructed by gluing piecewise linear solutions of elementary problems. Structural matrices are a convenient tool of representing such functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Bardi and L. Evans, “On Hopf’s formulas for solutions of Hamilton–Jacobi equations,” Nonlin. Anal. Th. Meth. Appl., 8, No. 11, 1373–1381 (1984).
M. G. Crandall and P. L. Lions, “Viscosity solutions of Hamilton–Jacobi equations,” Trans. Am. Math. Soc., 377, No. 1, 1–42 (1983).
E. Hopf, “Generalized solutions of nonlinear equations of first order,” J. Math. Mech., 14, No. 6, 951–973 (1965).
L. V. Kamneva and V. S. Patsko, “Construction of the maximal stable bridge in games with simple motions on the plane,” Tr. Inst. Mat. Mekh., 20, No. 4, 128–142 (2014).
N. N. Krasovsky and A. I. Subbotin, Positional Differential Games, Nauka, Moscow (1974).
N. N. Krasovsky and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York (1988).
M. Pachter and Y. Yavin, “Simple-motion pursuit-evasion differential games. Part 1. Stroboscopic strategies in collision-course guidance and proportional navigation,” J. Optim. Theory Appl., 51, No. 1, 95–127 (1986).
L. A. Petrosjan, Differential Games of Pursuit, World Scientific, Singapore (1993).
B. N. Pshenichny and M. I. Sagaidak, “On differential games with fixed time,” Kibernetika, No. 2, 54–63 (1970).
L. G. Shagalova, “A piecewise linear minimax solution of the Hamilton–Jacobi equation,” IFAC Proc. Vols., 31, No. 13, 193–197 (1998).
L. G. Shagalova, “Piecewise linear price function of a differential game with simple motions,” Vestn. Tambov. Univ. Ser. Estestv. Tekhn. Nauki, 12, No. 4, 564–565 (2007).
A. I. Subbotin, Minimax Inequalities and Hamilton–Jacobi Equations, Nauka, Moscow (1991).
A. I. Subbotin, Generalized Solutions of First Order PDEs. The Dynamical Optimization Perspective, Birkhäuser, Boston (1995).
A. I. Subbotin and L. G. Shagalova, “Piecewise linear solution of the Cauchy problem for the Hamilton–Jacobi equation,” Dokl. Ross. Akad. Nauk, 325, No. 5, 144–148 (1992).
V. I. Ukhobotov and I. V. Tseunova, “Differential games with simple motions,” Vestn. Chelyabinsk. Univ., 11, 84–96 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 168, Proceedings of the International Conference “Geometric Methods in the Control Theory and Mathematical Physics” Dedicated to the 70th Anniversary of Prof. S. L. Atanasyan, 70th Anniversary of Prof. I. S. Krasil’shchik, 70th Anniversary of Prof. A. V. Samokhin, and 80th Anniversary of Prof. V. T. Fomenko. Ryazan State University named for S. Yesenin, Ryazan, September 25–28, 2018. Part I, 2019.
Rights and permissions
About this article
Cite this article
Shagalova, L.G. Piecewise Linear Price Function of a Differential Game with Simple Dynamics and Integral Terminal Price Functional. J Math Sci 262, 878–886 (2022). https://doi.org/10.1007/s10958-022-05867-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-05867-z
Keywords and phrases
- differential game
- simple motion
- price function
- Hamilton–Jacobi equation
- generalized solution
- minimax solution
- algorithm