On the potentiality of a class of operators relative to local bilinear forms. (English) Zbl 1477.49054

Budochkina, Svetlana A.; Dekhanova, Ekaterina S.

Ural Math. J. 7, No. 1, 26-37 (2021).

Summary: The inverse problem of the calculus of variations (IPCV) is solved for a second-order ordinary differential equation with the use of a local bilinear form. We apply methods of analytical dynamics, nonlinear functional analysis, and modern methods for solving the IPCV. In the paper, we obtain necessary and sufficient conditions for a given operator to be potential relative to a local bilinear form, construct the corresponding functional, i.e., found a solution to the IPCV, and define the structure of the considered equation with the potential operator. As a consequence, similar results are obtained when using a nonlocal bilinear form. Theoretical results are illustrated with some examples.

MSC:

49N45	Inverse problems in optimal control
49J05	Existence theories for free problems in one independent variable
49J15	Existence theories for optimal control problems involving ordinary differential equations

Keywords:

inverse problem of calculus of variations; local bilinear form; potential operator; conditions of potentiality

References:

[1]	Berkovich L. M., “The generalized Emden-Fowler equation”, Symmetry in Nonlinear Mathematical Physics, 1 (1997), 155-163 · Zbl 0952.34030
[2]	Budochkina S. A., Savchin V. M., “On direct variational formulations for second order evolutionary equations”, Eurasian Math. J., 3:4 (2012), 23-34 · Zbl 1280.47054
[3]	Budotchkina S. A., Savchin V. M., “On indirect variational formulations for operator equations”, J. Funct. Spaces Appl., 5:3 (2007), 231-242 · Zbl 1162.47057
[4]	Filippov V. M., Variatsionnye printsipy dlya nepotentsial’nykh operatorov [Variational Principles for Nonpotential Operators], PFU, Moscow, 1985, 206 pp. (in Russian) · Zbl 0699.35011
[5]	Filippov V. M., Savchin V. M., Budochkina S. A., “On the existence of variational principles for differential-difference evolution equations”, Proc. Steklov Inst. Math., 283 (2013), 20-34 · Zbl 1296.49033
[6]	Filippov V. M., Savchin V. M., Shorokhov S. G., “Variational principles for nonpotential operators”, J. Math. Sci., 68:3 (1994), 275-398 · Zbl 0835.58012
[7]	Galiullin A. S., Obratnye zadachi dinamiki [Inverse Problems of Dynamics], Nauka, Moscow, 1981, 144 pp. (in Russian) · Zbl 0516.70002
[8]	Galiullin A. S., Gafarov G. G., Malayshka R. P., Khvan A. M., Analiticheskaya dinamika sistem Gel’mgol’tsa, Birkgofa, Nambu [Analytical dynamics of Helmholtz, Birkhoff, Nambu systems], Advances in Physical Sciences, Moscow, 1997, 324 pp. (in Russian)
[9]	Popov A. M., “Potentiality conditions for differential-difference equations”, Differ. Equ., 34:3 (1998), 423-426 · Zbl 0947.34067
[10]	Popov A. M., “Inverse problem of the calculus of variations for systems of differential-difference equations of second order”, Math. Notes, 72:5 (2002), 687—691 · Zbl 1035.49033
[11]	Santilli R. M., Foundations of Theoretical Mechanics, I: The Inverse Problems in Newtonian Mechanics, Springer-Verlag, Berlin—Heidelberg, 1977, 266 pp.
[12]	Santilli R. M., Foundations of Theoretical Mechanics, II: Birkhoffian Generalization of Hamiltonian Mechanics., Springer-Verlag, Berlin-Heidelberg, 1983, 370 pp. · Zbl 0536.70001
[13]	Savchin V. M., Matematicheskie metody mekhaniki beskonechnomernykh nepotentsial’nykh sistem [Mathematical Methods of Mechanics of Infinite-Dimensional Nonpotential Systems], PFU, Moscow, 1991, 237 pp. (in Russian) · Zbl 0925.70160
[14]	Savchin V. M., “An operator approach to Birkhoff”s equations”, Vestnik RUDN. Ser. Math., 1995, no. 2 (2), 111-123 · Zbl 0969.37035
[15]	Savchin V. M., Budochkina S. A., “On the structure of a variational equation of evolution type with the second t-derivative”, Differ. Equ., 39:1 (2003), 127-134 · Zbl 1229.34096
[16]	Tleubergenov M. I., Ibraeva G. T., “On the solvability of the main inverse problem for stochastic differential systems”, Ukrainian Math. J., 71:1 (2019), 157—165 · Zbl 1433.34031
[17]	Tleubergenov M. I., Ibraeva G. T., “On inverse problem of closure of differential systems with degenerate diffusion”, Eurasian Math. J., 10:2 (2019), 93-102 · Zbl 1463.34067
[18]	Tonti E., “On the variational formulation for linear initial value problems”, Ann. Mat. Pura Appl. (4), 95 (1973), 331—359 · Zbl 0278.49047
[19]	Tonti E., “Variational formulation for every nonlinear problem”, Internat. J. Engrg. Sci., 22:11-12 (1984), 1343—1371 · Zbl 0558.49022
[20]	Tonti E., “Extended variational formulation”, Vestnik RUDN. Ser. Math., 1995, no. 2 (2), 148-162 · Zbl 0965.35036
[21]	Tunitsky D. V., “On the inverse variational problem for one class of quasilinear equations”, J. Geom. Phys., 148 (2020), 103568 · Zbl 1433.49055