Sub- and Super-optimality Principles and Construction of Almost Optimal Strategies for Differential Games in Hilbert Spaces

Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 11)


We prove sub- and superoptimality principles of dynamic programming and show how to use the theory of viscosity solutions to construct almost optimal strategies for two-player, zero-sum differential games driven by abstract evolution equations in Hilbert spaces.


Hilbert Space Optimal Strategy Viscosity Solution Differential Game Admissible Strategy 
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  1. 1.
    Berkovitz, L.: The existence of value and saddle point in games of fixed duration. SIAM J. Control Optim. 23(2), 172–196 (1985). Erratum and addendum: SIAM J. Control Optim. 26(3), 740–742 (1988)Google Scholar
  2. 2.
    Borwein, J.M., Preiss, D.: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Am. Math. Soc. 303(2), 517–527 (1987)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Clarke, F.H., Ledyaev, Yu.S., Sontag, E.D., Subbotin, A.I.: Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control 42(10), 1394–1407 (1997)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Crandall, M.G., Lions, P.L.: Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions. J. Funct. Anal. 65(3), 368–405 (1986)MATHMathSciNetGoogle Scholar
  5. 5.
    Crandall, M.G., Lions, P.L.: Hamilton-Jacobi equations in infinite dimensions. IV. Hamiltonians with unbounded linear terms. J. Funct. Anal. 90, 237–283 (1990)MATHMathSciNetGoogle Scholar
  6. 6.
    Crandall, M.G., Lions, P.L.: Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and B-continuous solutions. J. Funct. Anal. 97, 417–465 (1991)Google Scholar
  7. 7.
    Crandall, M.G., Lions, P.L.: Hamilton-Jacobi equations in infinite dimensions. VI. Nonlinear A and Tataru’s method refined. In: Evolution equations, control theory, and biomathematics (Han sur Lesse, 1991), pp. 51–89, Lecture Notes in Pure and Appl. Math. 155, Dekker, New York (1994)Google Scholar
  8. 8.
    Elliott, R.J., Kalton, N.J.: The existence of value in differential games. Memoirs of the American Mathematical Society, 126. American Mathematical Society, Providence (1972)Google Scholar
  9. 9.
    Evans, L.C., Souganidis, P.E.: Differential games and representation formulas for solutions of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33, 773–797 (1984)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fabbri, G., Gozzi, F., Świȩch, A.: Verification theorem and construction of ε-optimal controls for control of abstract evolution equations. J. Convex Anal. 17(2), 611–642 (2010)MATHMathSciNetGoogle Scholar
  11. 11.
    Fleming, W.H., Souganidis, P.E.: On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38(2), 293–314 (1989)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ghosh, M.K., Shaiju, A.J.: Existence of value and saddle point in infinite-dimensional differential games. J. Optim. Theory Appl. 121(2), 301–325 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ishii, H.: Viscosity solutions for a class of Hamilton–Jacobi equations in Hilbert spaces. J. Funct. Anal. 105, 301–341 (1992)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kelome, D., Świȩch, A.: Perron’s method and the method of relaxed limits for “unbounded” PDE in Hilbert spaces. Studia Math. 176(3), 249–277 (2006)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kocan, M., Soravia, P., Świȩch, A.: On differential games for infinite-dimensional systems with nonlinear, unbounded operators. J. Math. Anal. Appl. 211(2), 395–423 (1997)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Li, X.Y., Yong, J.M.: Optimal control theory for infinite-dimensional systems. Birkhauser, Boston, Cambridge, (1995)Google Scholar
  17. 17.
    Ramaswamy, M., Shaiju, A.J.: Construction of approximate saddle point strategies for differential games in a Hilbert space. J. Optim. Theory Appl. 141(2), 349–370 (2009)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Renardy, M.: Polar decomposition of positive operators and problem od Crandall and Lions. Appl. Anal. 57(3), 383–385 (1995)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Shaiju, A.J.: Infinite horizon differential games for abstract evolution equations. Comput. Appl. Math. 22(3), 335–357 (2003)MathSciNetGoogle Scholar
  20. 20.
    Soravia, P.: Feedback stabilization and H control of nonlinear systems affected by disturbances: The differential games approach. Dynamics, bifurcations, and control (Kloster Irsee, 2001), 173–190, Lecture Notes in Control and Inform. Sci. 273, Springer, Berlin (2002)Google Scholar
  21. 21.
    Świȩch, A.: Sub- and superoptimality principles of dynamic programming revisited. Nonlinear Anal. 26(8), 1429–1436 (1996)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Świȩch, A.: Another approach to the existence of value functions of stochastic differential games. J. Math. Anal. Appl. 204(3), 884–897 (1996)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Tataru, D.: Viscosity solutions of Hamilton–Jacobi equations with unbounded linear terms. J. Math. Anal. Appl. 163, 345–392 (1992)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Tataru, D.: Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: A simplified approach. J. Differ. Equ. 111(1), 123–146 (1994)MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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