On Grid Optimal Feedbacks to Control Problems of Prescribed Duration on the Plane

  • Nina N. Subbotina
  • Timofey B. Tokmantsev
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 11)


We consider optimal control problems of prescribed duration. A new numerical method is suggested to solve the problems of controlling until a given instant. The solution is based on a generalization of the method of characteristics for Hamilton–Jacobi–Bellman equations. Constructions of optimal grid synthesis are suggested and numerical algorithms solving the problems on the plane are created. Efficiency of the grid feedback is estimated. Results of simulations using the numerical algorithms are exposed.


Optimal Control Problem Viscosity Solution Jacobi Equation Bellman Equation Adaptive Grid 
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  1. 1.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston (1997)MATHCrossRefGoogle Scholar
  2. 2.
    Bellman, R.: Dynamic Programming. University Press, Princeton (1957)MATHGoogle Scholar
  3. 3.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  4. 4.
    Demyanov, V.F., Vasiliev, L.V.: Nondifferentiable Optimization. Springer-Optimization Software, New York (1985)MATHGoogle Scholar
  5. 5.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (1993)MATHGoogle Scholar
  6. 6.
    Krasovskii, N.N.: The Theory of Control of Motion. Nauka, Moscow (1968) (in Russian)Google Scholar
  7. 7.
    Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York (1988)MATHGoogle Scholar
  8. 8.
    Kurzhanski, A.B., Valyi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1996)Google Scholar
  9. 9.
    Melikyan, A.A.: Generalized Characteristics of First Order PDEs: Applications in Optimal Control and Differential Games. Birkhäuser, Boston (1998)MATHGoogle Scholar
  10. 10.
    Osipov, Yu.S., Kryazhimskii, A.V.: Inverse Problems for Ordinary Differential Equations: Dynamical Solutions. Gordon and Breach, Boston (1995)MATHGoogle Scholar
  11. 11.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: Mathematical Theory of Optimal Processes. Interscience, New York (1962)MATHGoogle Scholar
  12. 12.
    Subbotin, A.I.: Generalized Solutions of First Order of PDEs:The Dynamical Optimization Perspectives. Birkhäuser, Boston (1995)Google Scholar
  13. 13.
    Subbotin, A.I., Chentsov, A.G.: Optimization of Guaranty in Control Problems. Nauka, Moscow (1981) (in Russian)Google Scholar
  14. 14.
    Subbotina, N.N.: The Method of Characteristics for Hamilton–Jacobi Equations and Applications to Dynamical Optimization. Journal of Mathematical Sciences 135(3), 2955–3091, Springer, New York (2006)Google Scholar
  15. 15.
    Blagodatskikh, V.I.: The maximum principle for differential inclusions. Proceedings of the Steklov Insitute of Mathematics 166, 23–43 (1984)MATHMathSciNetGoogle Scholar
  16. 16.
    Crandall, M.G. and Lions, P.-L.: Viscosity Solutions of Hamilton–Jacobi Equations. Transactions of the American Mathematical Society. 277(1), 1–42 (1983)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ganebny, S.A., Kumkov, S.S., Patsko, V.S., Pyatko, S.G.: Robust control in game problems with linear dynamics. Preprint, Institute of Mathematics and Mechanics UrB RAS, Ekaterinburg (2005) (in Russian)Google Scholar
  18. 18.
    Guseinov, Kh.G., Ushakov, V.N.: The construction of differential inclusions with prescribed properties. Differential equations. 36(4), 488–496 (2000)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Kleimenov, A.F.: On the theory of hierarchical differential two-person games. Preprint (Institute of Mathematics and Mechanics USC, Academy of Sciences USSR, Sverdlovsk, 1985) (in Russian)Google Scholar
  20. 20.
    Rockafellar, R.T., Wets, R.J-B.: Variational Analysis, Springer (1997)Google Scholar
  21. 21.
    Souganidis, P.E.: Approximation schemes for viscosity solutions of Hamilton–Jacobi equations. Journal of Differential Equations 59, 1–43 (1985)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Subbotina, N.N., Kolpakova, E.A.: On a Linear -Quadratic Optimal Control Problem with Geometric Restriction on Controls. International Journal of Tomography and Statistics 5(W07), 62–67 (2007)Google Scholar
  23. 23.
    Subbotina, N.N, Subbotin, A.I., and Tret’jakov, V.E.: Stochastic and deterministic control. Differential inequalities. Lect. Notes in Control and Information Science, Stochastic Optimization 81, 728–737, Springer, Berlin (1986)Google Scholar
  24. 24.
    Subbotina, N.N., Tokmantsev, T.B.: A numerical method for the minimax solution of the Bellman equation in the Cauchy problem with additional restrictions. Proceedings of the Steklov Institute of Mathematics, Supp.l, S222–S229 (2006)Google Scholar
  25. 25.
    Subbotina, N.N., Tokmantsev, T.B.: On the Efficiency of Optimal Grid Synthesis in Optimal Control Problems with Fixed Terminal Time. Journal of Differential Equations 45(11), 1686–1698 (2009)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Tarasyev, A.M.: Approximation schemes for constructing minimax solutions of Hamilton–Jacobi equations. Journal of Applied Mathematics and Mechanics 58(2), 207–221 (1985)CrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of Mathematics and Mechanics UrB RASEkaterinburgRussia

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