Differential Games Through Viability Theory: Old and Recent Results

  • Pierre Cardaliaguet
  • Marc Quincampoix
  • Patrick Saint-Pierre
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 9)


This article is devoted to a survey of results for differential games obtained through Viability Theory. We recall the basic theory for differential games (obtained in the 1990s), but we also give an overview of recent advances in the following areas: games with hard constraints, stochastic differential games, and hybrid differential games.We also discuss several applications.


Differential Game Differential Inclusion Admissible Control Viability Theory Stochastic Differential Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Aubin J.-P. Contingent Isaacs’ Equations of a Differential Game, in Differential Games and Applications, Eds. T. Baäar & P. Bernhard, Lecture Notes in Control and Information Sciences, Springer-Verlag (1989).Google Scholar
  2. [2]
    Aubin J.-P. Victory and Defeat in Differential Games, in Modeling and Control of Systems, Proceedings of the Bellman Continuum, June 1988, Ed. A. Blaquiãre, Lecture Notes in Control and Information Sciences, Springer-Verlag (1989).Google Scholar
  3. [3]
    Aubin J.-P. & Frankowska H. Set-valued analysis. Birkhäuser, Boston (1990).Google Scholar
  4. [4]
    Aubin J.-P. & Da Prato G. Stochastic viability and invariance. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)17, no. 4, 595–613 (1990).Google Scholar
  5. [5]
    Aubin J.-P. Differential games: a viability approach. SIAM J. Control Optim. 28, no. 6, 1294–1320 (1991).CrossRefMathSciNetGoogle Scholar
  6. [6]
    Aubin J.-P. Viability Theory. Birkhäuser, Boston (1991).MATHGoogle Scholar
  7. [7]
    Aubin J.-P. Impulse Differential Inclusions and Hybrid Systems: A Viability Approach, Lecture Notes, University of California at Berkeley (1999).Google Scholar
  8. [8]
    Aubin J.-P. & Da Prato G. Stochastic Nagumo’s Viability Theorem, Stochastic Analysis and Applications, 13, 1–11 (1995).MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Aubin J.-P. Dynamic Economic Theory:AViabilityApproach, Springer-Verlag, Berlin and New York (1997).Google Scholar
  10. [10]
    Aubin J.-P. & Da Prato G. The Viability Theorem for Stochastic Differential Inclusions, Stochastic Analysis and Applications, 16, 1–15 (1998).MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Aubin J.-P., Da Prato G. & Frankowska H. Stochastic invariance for differential inclusions. Set-valued analysis in control theory. Set-Valued Anal. 8, no. 1–2, 181–201 (2000).MATHCrossRefGoogle Scholar
  12. [12]
    Aubin J.-P., Pujal D. & Saint-Pierre P. Dynamic Management of Portfolios with Transaction Costs under Tychastic Uncertainty, Preprint (2001).Google Scholar
  13. [13]
    Aubin J.-P. & Shi Shuzhong, Eds Stochastic and viability approaches to dynamical uncertainty in economics and finance, Presses de l’Université de Beijing (2001).Google Scholar
  14. [14]
    Aubin J.-P. & Saint-Pierre P. An introduction to Viability Theory and management of renewable resources, in ch.2, Progress of Artificial Intelligence in Sustainability Science, Kropp J. & Scheffran J. (Eds.) Nova Science Publ. Inc., New York (2004).Google Scholar
  15. [15]
    Aubin J.-P., Lygeros J., Quincampoix. M., Sastry S. & Seube N. Impulse differential inclusions: a viability approach to hybrid systems. IEEE Trans. Automat. Control 47, no. 1, 2–20 (2002).CrossRefMathSciNetGoogle Scholar
  16. [16]
    Bardi, M., Capuzzo-Dolcetta I. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems and Control: Foundations and Applications. Birkhäuser, Boston (1997).Google Scholar
  17. [17]
    Bardi, M., Jensen R. A geometric characterization of viable sets for controlled degenerate diffusions. Calculus of variations, nonsmooth analysis and related topics. Set-Valued Anal. 10, no. 2–3, 129–141 (2002).Google Scholar
  18. [18]
    Barles G. Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications (Paris), 17. Springer-Verlag, Paris (1994).Google Scholar
  19. [19]
    Bettiol P., Cardaliaguet P. & Quincampoix M. Zero-sum state constrained differential games: Existence of value for Bolza problem, Preprint (2004).Google Scholar
  20. [20]
    Bonneuil N. & Saint-Pierre P. The Hybrid Guaranteed Capture Basin Algorithm in Economics, Preprint (2004).Google Scholar
  21. [21]
    Buckdahn R., Peng S., Quincampoix M. & Rainer C. Existence of stochastic control under state constraints. C. R. Acad. Sci. Paris Sér. I Math. 327, no. 1, 17–22 (1998).MATHMathSciNetGoogle Scholar
  22. [22]
    Cardaliaguet P., Quincampoix M. & Saint-Pierre P. Some Algorithms for Differential Games with two Players and one Target, RAIRO Mathematical Modeling and Numerical Analysis 28, no. 4, 441–461 (1994).MATHMathSciNetGoogle Scholar
  23. [23]
    Cardaliaguet P. A differential game with two players and one target, SIAM J. Control and Optimization 34, no. 4, 1441–1460 (1996).MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Cardaliaguet P. 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 (1997).MATHMathSciNetGoogle Scholar
  25. [25]
    Cardaliaguet P., Quincampoix M. & Saint-Pierre P. Numerical methods for differential games, in “Stochastic and differential games: Theory and numerical methods”, pp. 177–247. Annals of the international Society of Dynamic Games, M. Bardi, T.E.S. Raghavan, T. Parthasarathy Eds. Birkhäuser (1999).Google Scholar
  26. [26]
    Cardaliaguet P. & Plaskacz S. Invariant solutions of differential games and Hamilton-Jacobi equations for time-measurable hamiltonians, SIAM Journal on Control and Optim. 38, no. 5, 1501–1520 (2000).MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Cardaliaguet P., Quincampoix M. & Saint-Pierre P. Pursuit differential games with state constraints., SIAM J. Control and Optimization 39, no. 5, 1615–1632 (2001).MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    Cardaliaguet P., Quincampoix M. & Saint-Pierre P. Differential games with state-constraints. ISDG2002, Vol. I, II (St. Petersburg), 179–182, St. Petersburg State Univ. Inst. Chem., St. Petersburg (2002).Google Scholar
  29. [29]
    Cruck E. Target problems under state constraints for nonlinear controlled impulsive systems, J. Math. Anal. Appl. 270, no. 2, 636–656 (2002).CrossRefMathSciNetGoogle Scholar
  30. [30]
    Crßck E., Quincampoix, M. & Saint-Pierre P., Existence of a value for Pursuit-Evasion Games with Impulsive Dynamics, Preprint (2003).Google Scholar
  31. [31]
    Crßck E. & Saint-Pierre P. Non linear Impulse Target Problems under State Constraint: A Numerical Analysis based on Viability Theory. Set-Valued Analysis. Set-Valued Analysis. 12, no. 4, 383–416 (2004).Google Scholar
  32. [32]
    Da Prato G. & Frankowska H. A stochastic Filippov Theorem, Stochastic Calculus 12, 409–426 (1994).MATHGoogle Scholar
  33. [33]
    Doyen L. & Seube N. Control of uncertain systems under bounded chattering. Dynam. Control 8, no. 2, 163–176 (1998).MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    Evans L.C. & Souganidis P. E. Differential games and representation formulas for solutions of Hamilton-Jacobi Equations Indiana Univ. Math. J. 282, 487–502 (1984).MATHMathSciNetGoogle Scholar
  35. [35]
    Filippova T.F., Kurzhanski A. B., Sugimoto K. & Valyi I. Ellipsoidal calculus, singular perturbations and the state estimation problems for uncertain systems. J. Math. Systems Estim. Control 6, no. 3, 323–338 (1996).MATHMathSciNetGoogle Scholar
  36. [36]
    Fleming W. & Souganis P. On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38, no. 2, 293–314 (1989).MATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    Frankowska H. Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation, SIAM J. on Control and Optimization 31, no. 1, 257–272 (1993).MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    Gao Y., Lygeros J., Quincampoix M. & Seube N. Approximate stabilisation of uncertain hybrid systems Proceedings of Computation and control. 6th international workshop, HSCC 2003, Springer-Verlag, Berlin, Lect. Notes Comput. Sci. 2623, 203–215 (2003).Google Scholar
  39. [39]
    Gautier S. & Thibault L. Viability for constrained stochastic differential equations. Differential Integral Equations 6, no. 6, 1395–1414 (1993).MATHMathSciNetGoogle Scholar
  40. [40]
    Haddad G. Monotone trajectories of differential inclusions with memory, Isr. J. Math. 39, 83–100 (1981).MATHMathSciNetGoogle Scholar
  41. [41]
    Haddad G. Monotone viable trajectories for functional differential inclusions, J. Diff. Eq. 42, 1–24 (1981).MATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    Isaacs R. Differential GamesWiley, New York (1965).Google Scholar
  43. [43]
    Jachimiak W. Stochastic invariance in infinite dimension, Preprint (1998).Google Scholar
  44. [44]
    Kisielewicz M. Viability theorem for stochastic inclusions, Discussiones Mathematicae, Differential Inclusions 15, 61–74 (1995).MathSciNetGoogle Scholar
  45. [45]
    Krasovskii N.N. & Subbotin A.I. Game-Theorical Control Problems Springer-Verlag, New York (1988).Google Scholar
  46. [46]
    Kurzhanski A. B. & Filippova T.F. On the theory of trajectory tubes—a mathematical formalism for uncertain dynamics, viability and control. Advances in nonlinear dynamics and control: a report from Russia, 122–188, Progr. Systems Control Theory, 17, Birkhäuser, Boston (1993).Google Scholar
  47. [47]
    Moitie G., Quincampoix M. & Veliov V. Optimal control of uncertain discrete systems with imperfect measurements, IEEE Trans. Automat. Control 47, no. 11, 1909–1914 (2002).CrossRefMathSciNetGoogle Scholar
  48. [48]
    Plaskacz S. & Quincampoix M. Value-functions for differential games and control systems with discontinuous terminal cost. SIAM J. Control and Optim. 39, no. 5, 1485–1498 (2001).Google Scholar
  49. [49]
    Pujal D. & Saint-Pierre P. L’Algorithme du bassin de capture appliqué pour évaluer des options européennes, américaines ou exotiques, Revue de l’Association Française de Finance, Paris (2004).Google Scholar
  50. [50]
    Quincampoix M. Differential inclusions and target problems, SIAM J. Control and Optimization 30, 324–335 (1992).Google Scholar
  51. [51]
    Quincampoix M. & Saint-Pierre An algorithm for viability kernels in Hölderian case: Approximation by discrete viability kernels, J. Math. Syst. Est. and Control, Summary: 115–120 (1995).Google Scholar
  52. [52]
    Quincampoix M. & Veliov V. Viability with a target: theory and applications, in Applications of mathematics in engineering, 47–54, Heron Press, Sofia, Bulgaria (1998).Google Scholar
  53. [53]
    Quincampoix M. & Veliov V. Open-loop viable control under uncertain initial state information. Set-Valued Anal. 7, no. 1, 55–87 (1999).MATHMathSciNetGoogle Scholar
  54. [54]
    Quincampoix M. & Veliov V. Solution tubes to differential equations within a collection of sets. Control and Cybernetics 31, 3 (2002).MathSciNetGoogle Scholar
  55. [55]
    Quincampoix M. & Veliov V. Optimal control of uncertain systems with incomplete information for the disturbance, Siam J. Control Opti. 43, no. 4, 1373–1399 (2005).MATHCrossRefMathSciNetGoogle Scholar
  56. [56]
    Quincampoix M. & Seube N. Stabilization of uncertain Control Systems through Piecewise Constant Feedback. J. Math. Anal. Appl. no. 196, 240–255 (1998).Google Scholar
  57. [57]
    Rigal S. A set evolution approach to the control of uncertain systems with discrete time measurement, Preprint (2004).Google Scholar
  58. [58]
    Saint-Pierre P. Approximation of the viability kernel, Applied Mathematics& Optimisation, 29, 187–209 (1994).MATHCrossRefMathSciNetGoogle Scholar
  59. [59]
    Saint-Pierre P. Evaluation of crisis, reversibility, alert management for constrained dynamical systems using impulse dynamical systems. Numerical methods and applications, 255–263, Lecture Notes in Comput. Sci., 2542, Springer, BerlinCrossRefGoogle Scholar
  60. [60]
    Saint-Pierre P. Viable Capture Basin for Studying Differential and Hybrid Games, (to appear) International Game Theory Review, World Scientific Publishing Company, Singapore (2003).Google Scholar
  61. [61]
    Saint-Pierre P. Hybrid Kernels and Capture Basins for Impulse Constrained Systems, Proceedings of Hybrid Systems (2003).Google Scholar
  62. [62]
    Saint-Pierre P. The Guaranteed Hybrid Kernel Algorithm applied to evaluate barrier options in Finance, Proceeding MTNS04 (2004).Google Scholar
  63. [63]
    Saint-Pierre P. Viable capture basin for studying differential and hybrid games: application to finance. Int. Game Theory Rev. 6, no. 1, 109–136 (2004).MATHCrossRefMathSciNetGoogle Scholar
  64. [64]
    Saint-Pierre P. Approximation of capture basins for hybrid systems, in Proceedings of the 2001 European Control Conference, Porto, Portugal, September 4–7 (2001).Google Scholar
  65. [65]
    Seube N., Moitie R. & Leitmann G. Viability analysis of an aircraft flight domain for take-off in a windshear. Lyapunov’s methods in stability and control. Math. Comput. Modelling 36, no. 6, 633–641 (2002).MATHMathSciNetGoogle Scholar
  66. [66]
    Seube N., Moitie R. & Leitmann G. Aircraft take-off in windshear: a viability approach. Set-valued analysis in control theory. Set-Valued Anal. 8, no. 1–2, 163–180 (2000).MATHCrossRefMathSciNetGoogle Scholar
  67. [67]
    Serea O.-S. Discontinuous differential games and control systems with supremum cost. J. Math. Anal. Appl. 270, no. 2, 519–542 (2002).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • Pierre Cardaliaguet
    • 1
  • Marc Quincampoix
    • 1
  • Patrick Saint-Pierre
    • 2
  1. 1.Laboratoire de Mathématiques Unité CNRS UMR 6205Université de BretagneBrestFrance
  2. 2.Centre de Recherche Viabilité, Jeux, ContrÔleUniversité Paris IX-Dauphine Place du Maréchal de Lattre de TassignyParis Cedex 16France

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