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Viscous Solutions of the Hamilton–Jacobi–Bellman Equation on Time Scales

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We introduce a concept of viscous solution of the Bellman equation on time scales and establish conditions for the existence and uniqueness of this solution.

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References

  1. 1.

    R. Agarwal, M. Bohner, A. Boichuk, and O. Strakh, “Fredholm boundary-value problems for perturbed systems of dynamic equations on time scales,” Math. Methods Appl. Sci., DOI: 10.1002/mma.3356, 2014.

  2. 2.

    M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser Boston, Boston (2001).

  3. 3.

    M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Boston (2003).

  4. 4.

    L. Bourdin and L. Trelat, “General Cauchy–Lipschitz theory for ∆-Cauchy problems with Caratheodory dynamics on time scales,” J. Difference Equat. Appl., 20, No. 4, 526–547 (2014).

  5. 5.

    L. Bourdin and L. Trelat, “Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales,” SIAM J. Control Optim., 51, No. 5, 3781–3813 (2013).

  6. 6.

    I. Capuzzo Dolcetta, “On discrete approximation of the Hamilton–Jacobi equation of dynamic programming,” Appl. Math. Optim., 10, 367–377 (1983).

  7. 7.

    I. Capuzzo Dolcetta and H. Ishii, “Approximate solutions of the Bellman equation of deterministic control theory,” Appl. Math. Optim., 11, 161–181 (1984).

  8. 8.

    M. G. Crandal and P. L. Lions, “Viscosity solutions of Hamilton–Bellman–Jacobi equations,” Trans Amer. Math. Soc., 277, 1–45 (1983).

  9. 9.

    W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solution, Springer, New York (2006).

  10. 10.

    R. L. Gonzales and M. M. Tidball, “On discrete time approximation of the Hamilton–Jacobi equation of dynamics programming,” INRIA Rapports de Recherch., No. 1375 (1991).

  11. 11.

    K. Hall and R. Oberste-Vorth, “Totally discrete and Eulerian time scales,” Difference Equations, Special Functions, and Orthogonal Polynomials, World Scientific, Singapore (2007), pp. 462–470.

  12. 12.

    S. Hilger, Ein Maßkettenkalkül mit Anwendungen auf Zentrums, PhD Thesis, Universität Würzburg (1988).

  13. 13.

    L. Lastivka and O. Lavrova, “The method of dynamic programming for systems of differential equations on time scales,” Bull. Kyiv Shevchenko Nat. Univ., No. 2, 71–76 (2014).

  14. 14.

    E. B. Lee and L. Markus, Foundations of Optimal Control Theory, Wiley, New York (1967).

  15. 15.

    A. M. Samoilenko and M. O. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore (1995).

  16. 16.

    Z. Zhan, W. Wei, and H. Xu, “Hamilton–Jacobi–Bellman equations on time scales,” Math. Comput. Modelling, 49, 2019–2028 (2009).

  17. 17.

    Z. Zhan and W. Wei, “On existence of optimal control governed by a class of the first-order linear dynamic systems on time scales,” Appl. Math. Comput., 215, No. 6, 2070–2081 (2009).

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Correspondence to V. Ya. Danilov.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 7, pp. 933–950, July, 2017.

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Danilov, V.Y., Lavrova, O.E. & Stanzhyts’kyi, O.M. Viscous Solutions of the Hamilton–Jacobi–Bellman Equation on Time Scales. Ukr Math J 69, 1085–1106 (2017). https://doi.org/10.1007/s11253-017-1417-4

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