The inverse problem for singular perturbed system with many-sheeted slow surfaces

L. I. Kononenko

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 81-88

Résumé

We consider a singularly perturbed system of ordinary differential equations with small parameter, which describes a problem of chemical kinetics. We examine the system by using the method of integral manifolds that serves as a convenient tool for studying multidimensional singularly perturbed systems of differential equations and makes it possible to lower the dimension of the system. The integral manifold consists of sheets; for the small parameter $\varepsilon=0$, it is a slow surface. We formulate the direct and inverse problems for the system. The direct problem is as follows: given the right-hand sides of the system, find a solution to the system or prove its existence. The inverse problem is to find the unknown right-hand sides of the system of differential equations from some data on a solution of the direct problem. First, we consider the degenerate case, in which the small parameter $\varepsilon$ equals zero, and some restrictions are imposed on the dimension of the slow and fast variables, on the class of the right-hand sides that are assumed polynomial (with degree $1$), and on the number of sheets of the slow surface. Then we pass to the nondegenerate case $\varepsilon\neq 0$. In the case of a single sheet of the slow surface, the existence and uniqueness theorem was previously proven for a solution of the inverse problem. In this paper, we consider a system whose slow surface consists of several sheets. We prove an existence and uniqueness theorem for a solution to such a system. The proof is based on the result previously obtained for a system whose slow surface consists of a single sheet.

Bibliographie

[1] Kononenko, L. I., “The Identification Problem for a Nondegenerate System of Ordinary Differential Equations with Fast and Slow Variables”, Mathematical Notes of NEFU, 28:2 (2021), 3–15 (In Russian) | DOI

[2] Gutman, A. E. and Kononenko, L. I., “Formalization of Inverse Problems and its Applications”, Siberian Journal of Pure and Applied Mathematics, 17:4 (2017), 49–56 (in Russian) | DOI | Zbl

[3] Gutman, A. E. and Kononenko, L. I., “The Inverse Problem of Chemical Kinetics as a Composition of Binary Correspondences”, Siberian Electronic Mathematical Reports, 15 (2018), 48–53 (In Russian) | DOI | Zbl

[4] Mitropolsky, Yu. A. and Lykova, O. B., Integral Manifolds in Nonlinear Mechanics, Nauka, M., 1963, 512 pp. (In Russian) | MR

[5] Vasil'eva, A. V. and Butuzov, V. F., Singularly Perturbed Equations in Critical Cases, Moscow State University, M., 1978, 106 pp. (In Russian) | MR

[6] Goldstein V. M. and Sobolev V. A., Qualitative Analysis of Singularly Perturbed Systems, Sobolev Institute of Mathematics, Novosibirsk, 1988 (In Russian)

[7] Kononenko, L. I., “On the Smoothness of Slow Surfaces of Singularly Perturbed Systems”, Sibirskii Zhurnal Industrial'noi Matematiki, 5:2 (2002), 109–125 (In Russian) | Zbl

[8] Kononenko, L. I., “Slow Surfaces in Problems of Chemical Kinetics”, Mathematical Notes of YSU, 19:2 (2012), 49–67 (In Russian) | Zbl

[9] Tikhonov, A. N., “On Independence of Solutions to Differential Equations on a Small Parameter”, Matematicheskii Sbornik, 22(64):2 (1948), 193–204 (In Russian) | Zbl

[10] Lavrent'ev, M. M., Romanov, V. G. and Shishatskii, S. P., Ill-posed Problems of Mathematical Physics and Analysis, Nauka, M., 1980, 287 pp. (In Russian) | MR

[11] Romanov, V. G., “Inverse Problems for Hyperbolic Systems”, Numerical Methods in Mathematical Physics, Geophysics and Optimal Control, Nauka, Novosibirsk, 1978, 75–83 (In Russian) | MR

[12] Romanov, V. G. and Slinyucheva L. I., “Inverse Problem for Linear Hyperbolic Systems of the First Order”, Math Problems Geophysics, 3, Izdatelstvo VTs SO AN SSSR, Novosibirsk, 1972, 187–215 (In Russian)

[13] Kozhanov, A. I., “Nonlinear Loaded Equations and Inverse Problems”, Computational Mathematics and Mathematical Physics, 44:4 (2004), 657–675 | MR | Zbl

[14] Kabanikhin, S. I., Inverse and Ill-posed Problems, Novosibirsk, 2010, 458 pp. (In Russian)

[15] Anikonov, Yu. E., “Some Questions in the Theory of Inverse Problems for Kinetic Equations”, Inverse Problems of Mathematical Physics, Akad. Nauk SSSR Sibirsk. Otdel., Vychisl. Tsentr, Novosibirsk, 1985, 28–41 (In Russian) | MR | Zbl

[16] Golubyatnikov, V. P., “An Inverse Problem for the Hamilton–Jacobi Equation on a Closed Manifold”, Siberian Mathematical Journal, 38:2 (1997), 235–238 | DOI | MR

[17] Kononenko, L. I., “Identification Problem for Singular Systems with Small Parameter in Chemical Kinetics”, Siberian Electronic Mathematical Reports, 13 (2016), 175–180 (In Russian) | DOI | Zbl

[18] Gutman, A. E. and Kononenko, L. I., “Binary Correspondences and the Inverse Problem of Chemical Kinetics”, Vladikavkaz Mathematical Journal, 20:3 (2018), 37–47 ; кн. 1, т. 2, 2000 | DOI | MR | Zbl

[19] Reshetnyak, Yu. G., Mathematical Analysis Course, v. 1, Novosibirsk, 1999; т. 2, 2000 (In Russian)