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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. China Department 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
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Abstract

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
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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References

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997).
Bardi, M. and Da Lio, F., On the Bellman equation for some unbounded control problems. NoDEA 4 (1997) 491–510. Google Scholar
Barles, G., Existence results for first order Hamilton-Jacobi equations. Ann. Inst. Henri Poincaré 1 (1984) 325–340. Google Scholar
Biton, S., Nonlinear monotone semigroups and viscosity solutions. Ann. Inst. Henri Poincaré 18 (2001) 383–402. Google Scholar
Crandall, M.G. and Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations. Trans. AMS 277 (1983) 1–42. Google Scholar
Crandall, M.G. and Lions, P.L., On existence and uniqueness of solutions of Hamilton-Jacobi equations. Nonlinear Anal. 10 (1986) 353–370. Google Scholar
Crandall, M.G. and Lions, P.L., Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Ill. J. Math. 31 (1987) 665–688. Google Scholar
Da Lio, F., On the Bellman equation for infinite horizon problems with unblounded cost functional. Appl. Math. Optim. 41 (2000) 171–197. Google Scholar
Da Lio, F. and Ley, O., Uniqueness results for second-order Bellman-Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim. 45 (2006) 74–106. Google Scholar
Da Lio, F. and Ley, O., Convex Hamilton-Jacobi equations under superlinear growth conditions on data. Appl. Math. Optim. 63 (2011) 309–339. Google Scholar
Elliott, R.J. and Kalton, N.J., The existence of value in differential games. Amer. Math. Soc., Providence, RI. Memoirs of AMS 126 (1972). Google Scholar
Evans, L.C. and Souganidis, P.E., Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ. Math. J. 5 (1984) 773–797. Google Scholar
Fleming, W.H. and Souganidis, P.E., On the existence of value functions of two-players, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 (1989) 293–314. Google Scholar
Friedman, A. and Souganidis, P.E., Blow-up solutions of Hamilton-Jacobi equations. Commun. Partial Differ. Equ. 11 (1986) 397–443. Google Scholar
Garavello, M. and Soravia, P., Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. NoDEA 11 (2004) 271–298. Google Scholar
Ishii, H., Uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33 (1984) 721–748. Google Scholar
Ishii, H., Representation of solutions of Hamilton-Jacobi equations. Nonlinear Anal. 12 (1988) 121–146. Google Scholar
P.L. Lions, Generalized Solutions of Hamilton-Jacobi equations. Pitman, London (1982).
Lions, P.L. and Souganidis, P.E., Differential games, optimal conrol and directional derivatives of viscosity solutions of Bellman’s and Isaacs’ equations. SIAM J. Control Optim. 23 (1985) 566–583. Google Scholar
McEneaney, W., A uniqueness result for the Isaacs equation corresponding to nonlinear H ∞ control. Math. Control Signals Syst. 11 (1998) 303–334. Google Scholar
Rampazzo, F., Differential games with unbounded versus bounded controls. SIAM J. Control Optim. 36 (1998) 814–839. Google Scholar
Soravia, P., Equivalence between nonlinear ℋ∞ control problems and existence of viscosity solutions of Hamilton-Jacobi-Isaacs equations. Appl. Math. Optim. 39 (1999) 17–32. Google Scholar
Souganidis, P.E., Existence of viscosity solution of Hamilton-Jacobi equations. J. Differ. Equ. 56 (1985) 345–390. Google Scholar
Yong, J., Zero-sum differential games involving impusle controls. Appl. Math. Optim. 29 (1994) 243–261. Google Scholar
You, Y., Syntheses of differential games and pseudo-Riccati equations. Abstr. Appl. Anal. 7 (2002) 61–83. Google Scholar