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Hamilton–Jacobi equations and two-person zero-sum differentialgames with unbounded controls∗

Published online by Cambridge University Press:  23 January 2013

Hong Qiu and
Jiongmin Yong
Hong Qiu
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Weihai 264209, Shandong, P.R. ChinaDepartment of Mathematics, University of Central Florida, Orlando, 32816 FL, USA. Jiongmin.Yong@ucf.edu
Jiongmin Yong
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, 32816 FL, USA. Jiongmin.Yong@ucf.edu

Abstract

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A two-person zero-sum differential game with unbounded controls is considered. Underproper coercivity conditions, the upper and lower value functions are characterized as theunique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacsequations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upperand lower value functions coincide, leading to the existence of the value function of thedifferential game. Due to the unboundedness of the controls, the corresponding upper andlower Hamiltonians grow super linearly in the gradient of the upper and lower valuefunctions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobiequations involving such kind of Hamiltonian is proved, without relying on theconvexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivityconditions guaranteeing the finiteness of the upper and lower value functions are sharp insome sense.

Keywords

Two-person zero-sum differential gamesunbounded controlHamilton–Jacobi equationviscosity solution

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013, pp. 404 - 437
Copyright
© EDP Sciences, SMAI, 2013

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