International Journal of Game Theory

, Volume 34, Issue 4, pp 495–527 | Cite as

Zero-sum state constrained differential games: existence of value for Bolza problem

  • Piernicola BettiolEmail author
  • Pierre Cardaliaguet
  • Marc Quincampoix
Original Article


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.


Differential games Bolza problem State constraints Viability Theory 

AMS Classification

49N70 90D25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. de Roquefort A (1991) Jeux différentiels et approximation numérique de fonctions valeur. RAIRO Math Model Numer Anal 25:517–560Google Scholar
  2. Arisawa M, Lions PL (1996) Continuity of admissible trajectories for state constraints control problems. Discrete Cont Dyn Syst 2(3):297–305Google Scholar
  3. Aubin J-P (1991) Viability Theory. Birkhäuser, BostonGoogle Scholar
  4. Aubin J-P, Frankowska H (1990) Set-valued analysis. Birkhäuser, BostonGoogle Scholar
  5. 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 3:273–304Google Scholar
  6. 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 570Google Scholar
  7. 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):361–380CrossRefGoogle Scholar
  8. 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 194Google Scholar
  9. Bettiol P, Frankowska H (2006) Regularity of solution maps of differential inclusions for systems under state constraints. Set-Valued Anal (to appear)Google Scholar
  10. Cardaliaguet P (1996) A differential game with two players and one target. SIAM J Control Optim 34(4):1441–1460CrossRefGoogle Scholar
  11. 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 36:125–146CrossRefGoogle Scholar
  12. 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 177–247Google Scholar
  13. Cardaliaguet P, Quincampoix M, Saint-Pierre P (2001) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615–1632CrossRefGoogle Scholar
  14. Cardaliaguet P, Plaskacz S (2000) Invariant solutions of differential games and Hamilton-Jacobi equations for time-measurable hamiltonians. SIAM J Control Optim 38(5):1501–1520CrossRefGoogle Scholar
  15. Evans LC, Souganidis PE (1984) Differential games and representation formulas for solutions of Hamilton–Jacobi equations. Indiana Univ Math J 282:487–502Google Scholar
  16. Frankowska H, Plaskacz S, Rzezuchowski T (1995) Measurable viability theorems and the Hamilton-Jacobi-Bellman Equation. J Differ Equ 116(2):265–305CrossRefGoogle Scholar
  17. Frankowska H, Rampazzo F (2000) Filippov’s and Filippov-Wazewski’s theorems on closed domains. J Differ Equ 161(2):449–478CrossRefGoogle Scholar
  18. Isaacs R (1965) Differential Games. Wiley, New YorkGoogle Scholar
  19. Krasovskii NN, Subbotin AI (1988) Game-theorical control problems. Springer, Berlin Heidelberg New YorkGoogle Scholar
  20. 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):467–483Google Scholar
  21. Osipov Ju S (1971) Alternative in a differential-difference Game. Soviet Math Dokl 12:619–624Google Scholar
  22. Rozyev I, Subbotin AI (1988) Semicontinuous solutions of Hamilton–Jacobi equations. PMM USSR 52(2):141–146Google Scholar
  23. Soner HM (1986) Optimal control problems with state-space constraints. SIAM J Control Optim 24:552–562, 1110–1122Google Scholar

Copyright information

© Springer Verlag 2006

Authors and Affiliations

  • Piernicola Bettiol
    • 1
    Email author
  • Pierre Cardaliaguet
    • 2
  • Marc Quincampoix
    • 2
  1. 1.SISSA/ISASTriesteItaly
  2. 2.Département de MathématiquesUniversité de Bretagne OccidentaleBrestFrance

Personalised recommendations