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**On the solvability of the main inverse problem for stochastic differential systems.**
*(English.
Russian original)*
Zbl 1433.34031

Ukr. Math. J. 71, No. 1, 157-165 (2019); translation from Ukr. Mat. Zh. 71, No. 1, 139-145 (2019).

Summary: By using the quasiinversion method, we establish necessary and sufficient conditions for the solvability of the main (according to Galiullin’s classification) inverse problem in the class of first-order Itô stochastic differential equations with random perturbations from the class of processes with independent increments with diffusion degenerated with respect to a part of variables and given properties depending on a part of variables.

### MSC:

34A55 | Inverse problems involving ordinary differential equations |

34F05 | Ordinary differential equations and systems with randomness |

34C45 | Invariant manifolds for ordinary differential equations |

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\textit{M. I. Tleubergenov} and \textit{G. T. Ibraeva}, Ukr. Math. J. 71, No. 1, 157--165 (2019; Zbl 1433.34031); translation from Ukr. Mat. Zh. 71, No. 1, 139--145 (2019)

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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). |

[2] | A. S. Galiullin, Methods for the Solution of the Inverse Problems of Dynamics [in Russian], Nauka, Moscow (1986). · Zbl 0658.70001 |

[3] | A. S. Galiullin, “Construction of a force field on the basis of a given family of trajectories,” Differents. Uravn., 17, No. 8, 1487-1489 (1981). · Zbl 0505.70004 |

[4] | I. A. Mukhametzyanov and R. G. Mukharlyamov, Equations of Program Motions [in Russian], University of Friendship of Peoples, Moscow (1986). |

[5] | R. G. Mukharlyamov, “On the construction of systems of differential equations of motion of mechanical systems,” Differents. Uravn., 39, No. 3, 343-353 (2003). · Zbl 1174.70324 |

[6] | R. G. Mukharlyamov, “Simulation of control processes, stability, and stabilization of systems with program connections,” Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., No. 1, 15-28 (2015). |

[7] | R. G. Mukharlyamov, M. Amabili, R. Garziera, and K. Riabova, “Stability of nonlinear vibrations of doubly cured shallow shells,” Vestn. Ros. Univ. Druzhby Narodov, Ser. Mat. Inform., No. 2, 53-63 (2016). |

[8] | 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). · Zbl 1322.34016 |

[9] | 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). · Zbl 1224.34202 |

[10] | 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). · Zbl 1224.93021 |

[11] | 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). · Zbl 06688859 |

[12] | V. M. Savchin and S. A. Budochkina, “Nonclassical Hamilton’s actions and the numerical performance of variational methods for some dissipative problems,” in: Communications in Computer and Information Science, Springer, Cham, Vol. 678 (2016), pp. 624-634. · Zbl 06892112 |

[13] | V. M. Savchin and S. A. Budochkina, “Invariance of functionals and related Euler-Lagrange equations,” Russian Math., 61, No. 2, 49-54 (2017). · Zbl 1370.35027 |

[14] | M. I. Tleubergenov, “On the inverse problem of dynamics in the presence of random perturbations,” Izv. MN-AN RK, Ser. Fiz.-Mat., No. 3, 55-61 (1998). |

[15] | M. I. Tleubergenov, “An inverse problem for stochastic differential systems,” Different. Equat., 37, No. 5, 751-753 (2001). · Zbl 0991.60046 |

[16] | M. I. Tleubergenov, “On the inverse stochastic problem of closure,” Dokl. MN-AN RK, No. 1, 53-60 (1999). |

[17] | G. T. Ibraeva and M. I. Tleubergenov, “On the main inverse problem for differential systems with diffusion degenerated in a part of variables,” Mat. Zh., 4, No. 4(14), 86-92 (2004). · Zbl 1133.60335 |

[18] | G. T. Ibraeva and M. I. Tleubergenov, “Main inverse problem for differential systems with degenerate diffusion,” Ukr. Mat. Zh., 65, No. 5, 712-716 (2013); English translation: Ukr. Math. J., 65, No. 5, 787-792 (2013). · Zbl 1290.60058 |

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

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