Abstract
We consider the class of differential games with random duration. We show that a problem with random game duration can be reduced to a standard problem with an infinite time horizon. A Hamilton-Jacobi-Bellman-type equation is derived for finding optimal solutions in differential games with random duration. Results are illustrated by an example of a game-theoretic model of nonrenewable resource extraction. The problem is analyzed under the assumption of Weibull-distributed random terminal time of the game.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gavrilov, L.A. and Gavrilova, N.S., Biologiya prodolzhitel’nosti zhizni (Biology of Lifetime), Moscow: Nauka, 1991.
Dunford, N. and Schwartz, J.T., Linear Operators, vol. 1: General Theory, New York: Interscience, 1958. Translated under the title Lineinye operatory. Obshchaya teoriya, Moscow: Inostrannaya Literatura, 1962.
Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Nauka, 1976.
Matveevskii, V.R., Nadezhnost’ tekhnicheskikh sistem (Reliability of Engineering Systems), Moscow: Mosk. Gos. Inst. Elektron. Mat., 2002.
Petrosyan, L.A., Differentsial’nye igry presledovaniya (Differential Pursuit Games), Leningrad: Leningr. Gos. Univ., 1977.
Petrosyan, L.A. and Danilov, N.V., Kooperativnye differentsial’nye igry i ikh prilozheniya (Cooperative Differential Games and Their Applications), Tomsk: Tomsk. Univ., 1985.
Petrosyan, L.A. and Murzov, N.V., Game-theoretic Problems of Mechanics, Litovsk. Mat. Sb., 1966, no. 6(3), pp. 423–433.
Petrosyan, L.A. and Shevkoplyas, E.V., Cooperative Differential Games with Random Duration, Vestn. S.-Peterburg. Gos. Univ., 2000, ser. 1, no. 4, pp. 18–23.
Shevkoplyas, E.V., Cooperative Differential Games with Random Duration, Cand. Sci. (Phys.-Math.) Dissertation, St. Petersburg, 2004.
Shevkoplyas, E.V., On the Construction of the Characteristic Function in Cooperative Differential Games with Random Duration, in Proc. Int. Seminar “Control Theory and Theory of Generalized Solutions of Hamilton-Jacobi Equations,” Yekaterinburg: Uralsk. Univ., 2006, no. 1, pp. 285–293.
Chang, F.R., Stochastic Optimization in Continuous Time, Cambridge: Cambridge Univ. Press, 2004.
Dockner, E.J., Jorgensen, S., van Long, N., and Sorger, G., Differential Games in Economics and Management Science, Cambridge: Cambridge Univ. Press, 2000.
Haurie, A., A Multigenerational Game Model to Analyze Sustainable Development, Ann. Oper. Res., 2005, vol. 137, no. 1, pp. 369–386.
Henley, E.J. and Kumamoto, H., Reliability Engineering and Risk Assessment, Englewood Cliffs: Prentice-Hall, 1981.
Karp, L., Non-constant Discounting in Continuous Time, J. Econ. Theory, 2007, vol. 132, pp. 557–568.
Marín-Solano, J. and Navas, J., Non-constant Discounting in Finite Horizon: The Free Terminal Time Case, J. Econ. Dynam. Control, 2009, vol. 33, pp. 666–675.
Yaari, M.E., Uncertain Lifetime, Life Insurance, and the Theory of the Consumer, Rev. Econ. Stud., 1965, vol. 32, no. 2, pp. 137–150.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.V. Shevkoplyas, 2009, published in Matematicheskaya Teoriya Igr i Prilozheniya, 2009, No. 2, pp. 98–118.
Rights and permissions
About this article
Cite this article
Shevkoplyas, E.V. The Hamilton-Jacobi-Bellman equation for a class of differential games with random duration. Autom Remote Control 75, 959–970 (2014). https://doi.org/10.1134/S0005117914050142
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117914050142