Abstract
We prove the existence of a lower semicontinuous value function for Bolza problem in differential games with state-constraints. As a byproduct, we obtain a new estimation of trajectories of a control system by trajectories with state constraints. This result which could be interesting by itself enables us to build a suitable strategy for constrained differential games. We also characterize the value function by means of viscosity solutions and give conditions under which the value function is locally Lipschitz continuous.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
de Roquefort A (1991) Jeux différentiels et approximation numérique de fonctions valeur. RAIRO Math Model Numer Anal 25:517–560
Arisawa M, Lions PL (1996) Continuity of admissible trajectories for state constraints control problems. Discrete Cont Dyn Syst 2(3):297–305
Aubin J-P (1991) Viability Theory. Birkhäuser, Boston
Aubin J-P, Frankowska H (1990) Set-valued analysis. Birkhäuser, Boston
Bardi M, Bottacin S, Falcone M (1995) Convergence of discrete schemes for discontinuous value functions of pursuit-evasion games. New trends in dynamic games and applications. Ann Int Soc Dyn Games 3:273–304
Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi-Bellman equations. Systems and control: foundations and applications, vol xvii. Birkhäuser, Boston, p 570
Bardi M, Koike S, Soravia P (2000) Pursuit-evasion game with state constraints: dynamic programming and discrete-time approximations. Discrete Contin Dyn Syst 6(2):361–380
Barles G (1994) Solutions de viscosité des équations de Hamilton-Jacobi. (Viscosity solutions of Hamilton-Jacobi equations). Mathématiques & Applications (Paris). 17. vol ix. Springer, Paris, p 194
Bettiol P, Frankowska H (2006) Regularity of solution maps of differential inclusions for systems under state constraints. Set-Valued Anal (to appear)
Cardaliaguet P (1996) A differential game with two players and one target. SIAM J Control Optim 34(4):1441–1460
Cardaliaguet P (1997) Non smooth semi-permeable barriers, Isaacs equation and application to a differential game with one target and two players. Appl Math Opti 36:125–146
Cardaliaguet P, Quincampoix M, Saint-Pierre P (1999) Numerical methods for differential games. In: Bardi M, Raghavan TES, Parthasarathy T (eds) Stochastic and differential games : Theory and numerical methods, Annals of the international Society of Dynamic Games. Birkhäuser, Boston pp 177–247
Cardaliaguet P, Quincampoix M, Saint-Pierre P (2001) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615–1632
Cardaliaguet P, Plaskacz S (2000) Invariant solutions of differential games and Hamilton-Jacobi equations for time-measurable hamiltonians. SIAM J Control Optim 38(5):1501–1520
Evans LC, Souganidis PE (1984) Differential games and representation formulas for solutions of Hamilton–Jacobi equations. Indiana Univ Math J 282:487–502
Frankowska H, Plaskacz S, Rzezuchowski T (1995) Measurable viability theorems and the Hamilton-Jacobi-Bellman Equation. J Differ Equ 116(2):265–305
Frankowska H, Rampazzo F (2000) Filippov’s and Filippov-Wazewski’s theorems on closed domains. J Differ Equ 161(2):449–478
Isaacs R (1965) Differential Games. Wiley, New York
Krasovskii NN, Subbotin AI (1988) Game-theorical control problems. Springer, Berlin Heidelberg New York
Loreti P, Tessitore ME (1994) Approximation and regularity results on constrained viscosity solutions of Hamilton-Jacobi-Bellman equations. J Math Syst Estim Control 4(4):467–483
Osipov Ju S (1971) Alternative in a differential-difference Game. Soviet Math Dokl 12:619–624
Rozyev I, Subbotin AI (1988) Semicontinuous solutions of Hamilton–Jacobi equations. PMM USSR 52(2):141–146
Soner HM (1986) Optimal control problems with state-space constraints. SIAM J Control Optim 24:552–562, 1110–1122
Author information
Authors and Affiliations
Corresponding author
Additional information
Work supported by the European Community’s Human Potential Program under contract HPRN-CT-2002-00281, [Evolution Equations].
Rights and permissions
About this article
Cite this article
Bettiol, P., Cardaliaguet, P. & Quincampoix, M. Zero-sum state constrained differential games: existence of value for Bolza problem. Int J Game Theory 34, 495–527 (2006). https://doi.org/10.1007/s00182-006-0030-9
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-006-0030-9