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On the Solvability of Stochastic Helmholtz Problem

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We consider inverse problems for differential systems in the presence of random perturbations. By the method of additional variables, we establish sufficient conditions for the representation of stochastic differential equations of the second order in the form of stochastic Lagrange equations, as well as for the representation of stochastic differential equations of the first order in the form of stochastic canonical equations. The obtained results are illustrated by specific examples.

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References

  1. N. P. Erugin, “Construction of the entire set of systems of differential equations with a given integral curve,” Prikl. Mat. Mekh., 10, Issue 6, 659–670 (1952).

    MathSciNet  Google Scholar 

  2. A. S. Galiullin, Methods for the Solution of the Inverse Problems of Dynamics [in Russian], Nauka, Moscow (1986).

    MATH  Google Scholar 

  3. A. S. Galiullin, “Construction of a force field according to a given family of trajectories,” Differents. Uravn., 17, No. 8, 1487–1489 (1981).

    MathSciNet  MATH  Google Scholar 

  4. I. A. Mukhametzyanov and R. G. Mukharlyamov, Equations of Program Motions [in Russian], Izd. RUDN, Moscow (1986).

    Google Scholar 

  5. R. G. Mukharlyamov, “Construction of the systems of differential equations of motion of mechanical systems,” Differents. Uravn., 39, No. 3, 343–353 (2003).

    MathSciNet  MATH  Google Scholar 

  6. R. G. Mukharlyamov, “Simulation of control processes, stability, and stabilization of systems with program connections,” Izv. Ros. Akad. Nauk. Teor. Sist. Upravl., No. 1, 15–28 (2015).

  7. S. S. Zhumatov, “Asymptotic stability of implicit differential systems in the vicinity of a program manifold,” Ukr. Mat. Zh., 66, No. 4, 558–565 (2014); English translation: Ukr. Math. J., 66, No. 4, 625–632 (2014).

  8. S. S. Zhumatov, “Exponential stability of a program manifold of indirect control systems,” Ukr. Mat. Zh., 62, No. 6, 784–790 (2010); English translation: 62, No. 6, 907–915 (2010).

  9. S. S. Zhumatov, “Stability of a program manifold of control systems with locally quadratic relations,” Ukr. Mat. Zh., 61, No. 3, 418–424 (2009); English translation: 61, No. 3, 500–509 (2009).

  10. M. I. Tleubergenov and G. T. Ibraeva, “On the solvability of the main inverse problem for stochastic differential systems,” Ukr. Mat. Zh., 71, No 1, 139–144 (2019); English translation: 71, No 1, 157–165 (2019).

  11. M. I. Tleubergenov and G. T. Ibraeva, “On the stochastic problem of reconstruction with indirect control,” Differents. Uravn., 53, No. 10, 1418–1428 (2017).

    MATH  Google Scholar 

  12. M. I. Tleubergenov and D. T. Azhymbaev, “On construction of force function in the presence of random perturbations,” in: T. S. Kalmenov, E. D. Nursultanov, M. V. Ruzhansky, and M. A. Sadybekov (editors), Springer Proc. in Mathematics & Statistics, Vol. 216 (2017), pp. 443–450.

  13. O. Misiats, O. Stanzhytskyi, and N. Yip, “Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains,” J. Theor. Probab., 29, No. 3, 996–1026 (2016).

    Article  MathSciNet  Google Scholar 

  14. A. S. Galiullin, Helmholtz Systems [in Russian], Izd. RUDN, Moscow (1995).

    Google Scholar 

  15. H. Helmholtz, “Über die physikalische Bedeutung des Prinzips der kleinsten Wirkung,” Borchardt-Crelle Journal für Mathematik, 100, 137–166, 213–222 (1886).

    MathSciNet  MATH  Google Scholar 

  16. A. Mayer, “Die existenzbeingungen eines kinetischen potentiales,” Ber. Verhand. Kgl. Sachs. Ges. Wiss. Leipzig, 48, 519–529 (1896).

    Google Scholar 

  17. G. K. Suslov, “On the kinetic Helmholtz potential,” Mat. Sb., 19, No. 1, 197–210 (1896).

    Google Scholar 

  18. R. M. Santilli, Foundations of Theoretical Mechanics. I. The Inverse Problem in Newtonian Mechanics, Springer, New York (1978).

  19. R. M. Santilli, Foundation of Theoretical Mechanics. II. Birkhoffian Generalization of Hamiltonian Mechanics, Springer, Berlin–Heidelberg (1983).

    MATH  Google Scholar 

  20. S. A. Budochkina and V. M. Savchin, “An operator equation with the second time derivative and Hamilton-admissible equations,” Dokl. Math., 94, No. 2, 487–489 (2016).

    Article  MathSciNet  Google Scholar 

  21. V. M. Savchin and S. A. Budochkina, “Nonclassical Hamilton’s actions and the numerical performance of variational methods for some dissipative problems,” in: Comm. Comp. Inform. Sci., Springer, Cham, Vol. 678 (2016), pp. 624–634.

  22. V. M. Savchin and S. A. Budochkina, “Invariance of functionals and related Euler–Lagrange equations,” Russ. Math. (Izv. VUZ), 61, No. 2, 49–54 (2017).

    Article  Google Scholar 

  23. V. M. Filippov, V. M. Savchin, and S. G. Shorokhov, “Variational principles for nonpotential operators,” in: VINITI Series in Contemporary Problems of Mathematics. Latest Acievements [in Russian], Vol. 40, VINITI, Moscow (1990) pp. 3–178.

  24. M. I. Tleubergenov, On the Inverse Problem of Newtonian Mechanics in the Probability Statement [in Russian], Deposited at VINITI 14.02.92, No. 499-V92 (1992).

  25. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam (1981).

    MATH  Google Scholar 

  26. M. I. Tleubergenov and D. T. Azhymbaev, “On the stochastic Helmholtz problem for the Birkhoff systems,” in: Proc. of the Eighth Internat. Sci. Conf. “Problems of Differential Equations, Analysis, and Algebra” (2018), pp. 126–131.

  27. V. S. Pugachev and I. N. Sinitsyn, Stochastic Differential Systems. Analysis and Filtration [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  28. P. Sagirow, Stochastic Methods in the Dynamics of Satellites, International Center for Mechanical Sciences, Udine, Courses and Lectures, No. 57 (1974).

  29. I. N. Sinitsyn, “On the fluctuations of gyros in gimbal suspensions,” Izv. Akad. Nauk SSSR, Ser. Mekh. Tverd. Tela, No. 3, 23–31 (1976).

  30. M. F. Shulgin, Some Differential Equations of Analytic Dynamics and Their Integration [in Russian], Izd. Sredneaziat. Gos. Univ., Tashkent (1958).

  31. P. Appell, Traité de Mécanique Rationnelle. Tome Deuxième. Dynamique des Systèmes. Mécanique Analytique, Gauthier-Villars, Paris (1953).

  32. R. L. Stratonovich, “New form of representation of stochastic integrals and equations,” Vestn. MGU, Ser. 1, Mat. Mekh., No. 1, 3–11 (1964).

  33. K. Ito, “On a stochastic differential equation,” Mem. Amer. Math. Soc., 4, 51–89 (1951).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Mathematical Modelling, Pushkin Str., 125, Almaty, Kazakhstan, 050010

    M. I. Tleubergenov

  2. Zhubanov Aktyubinsk Regional State University, Moldagulova Ave., 34, Aktobe, Kazakhstan, 030000

    D. T. Azhymbaev

Authors
  1. M. I. Tleubergenov
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  2. D. T. Azhymbaev
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Correspondence to M. I. Tleubergenov.

Additional information

Translated from Neliniini Kolyvannya, Vol. 22, No. 3, pp. 398–405, July–September, 2019.

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Tleubergenov, M.I., Azhymbaev, D.T. On the Solvability of Stochastic Helmholtz Problem. J Math Sci 253, 297–305 (2021). https://doi.org/10.1007/s10958-021-05229-1

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