Rayleigh--Ritz operator in inverse problems for higher order multilinear nonautonomous evolution equations

A. V. Lakeev ; Yu. E. Linke ; V. A. Rusanov

Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 162-176.

Résumé

We study solvability questions for the problem on realization of operator functions for an invariant polylinear regulator of a higher-order differential system in an infinite-dimensional separable Hilbert space. This is a nonstationary coefficient-operator inverse problem for multilinear evolution equations whose dynamic order is higher than one (notice that nonautomonous hyperbolic systems belong to this class of problems). We analyze semiadditivity and continuity for a nonlinear Rayleigh–Ritz functional operator and obtain an analytic model of an invariant polylinear regulator. This model allows us to combine two bundles of trajectory curves induced by different invariant polylinear regulators in a differential system and obtain a family of admissible solutions of the initial differential system in terms of an invariant polylinear action. The obtained results can be applied in the general qualitative theory of nonlinear infinite-dimensional adaptive control systems described by higher-order multilinear nonautonomous differential systems (including neuromodelling).

Bibliographie

[1] A. Van der Shaft, “K teorii realizatsii nelineinykh sistem, opisyvaemykh differentsialnymi uravneniyami vysshego poryadka”, Teoriya sistem. Matematicheskie metody i modelirovanie, Per. s angl. sb. statei, eds. A. N. Kolmogorov, S. P. Novikov, Mir, M., 1989, 192–237

[2] A. B. Daneev, V. A. Rusanov, “K probleme postroeniya silnykh differentsialnykh modelei s minimalnoi operatornoi normoi. I, II”, Kibernetika i sistemnyi analiz, 2004, no. 1, 144–153 ; No 2, 170–178 | MR | Zbl | MR | Zbl

[3] A. V. Daneev, A. E. Kumenko, V. A. Rusanov, “Zadacha spektralnoi identifikatsii matematicheskoi modeli lineinoi dinamicheskoi sistemy upravleniya LA”, Izv. vuzov. Aviatsionnaya tekhnika, 1999, no. 1, 20–24 | MR

[4] A. B. Dmitriev, E. I. Druzhinin, “Identifikatsiya dinamicheskikh kharakteristik nepreryvnykh lineinykh modelei v usloviyakh polnoi parametricheskoi neopredelennosti”, Izv. RAN. Teoriya i sistemy upravleniya, 1999, no. 3, 44–52 | Zbl

[5] E. I. Druzhinin, “Postroenie strukturno ustoichivykh modelei dinamiki bolshikh kosmicheskikh konstruktsii po dannym letnykh ispytanii”, Dok. RAN, 479:3 (2017), 285–288 | DOI

[6] S. I. Kabanikhin, Obratnye i nekorrektnye zadachi, Sib. nauch. izd-vo, Novosibirsk, 2009, 458 pp.

[7] R. Kalman, P. Falb, M. Arbib, Ocherki po matematicheskoi teorii sistem, Mir, M., 1971, 400 pp. | MR

[8] A. V. Lakeev, Yu. E. Lipke, V. A. Rusanov, “K realizatsii polilineinogo regulyatora differentsialnoi sistemy vtorogo poryadka v gilbertovom prostranstve”, Differents. uravneniya, 53:8 (2017), 1098–1109 | DOI | Zbl

[9] A. V. Lakeev, Yu. E. Lipke, V. A. Rusanov, “K differentsialnoi realizatsii bilineinoi sistemy vtorogo poryadka v gilbertovom prostranstve”, Sib. zhurn. industr. matematiki, XXII:2 (2019), 27–36 | MR | Zbl

[10] A. V. Lipke, Yu. E. Lakeev, V. A. Rusanov, “Metricheskie svoistva operatora Releya-Rittsa”, Izv. vuzov. Matematika, 2022, no. 9, 54–63

[11] M. Mesarovich, Ya. Takakhara, Obschaya teoriya sistem: Matematicheskie osnovy, Mir, M., 1978, 312 pp.

[12] V. A. Rusanov, A. V. Daneev, Yu. E. Lipke, “K geometricheskim osnovam differentsialnoi realizatsii dinamicheskikh protsessov v gilbertovom prostranstve”, Kibernetika i sistemnyi analiz, 53:4 (2017), 71–83 | MR | Zbl

[13] V. A. Rusanov, A. V. Lakeev, Yu. E. Lipke, “K razreshimosti differentsialnoi realizatsii minimalnogo dinamicheskogo poryadka semeistva nelineinykh protsessov “vkhod-vykhod” v gilbertovom prostranstve”, Differents. uravneniya, 51:4 (2015), 524–537 | DOI | Zbl

[14] A. V. Savelev, “Dinamicheskie modeli nervnoi sistemy: identifikatsiya i printsipy organizatsii neirosetei”, Materialy XI Vserossiiskogo seminara “Neiroinformatika i ee prilozheniya”, Institut vychislitelnogo modelirovaniya SO RAN, Krasnoyarsk, 2003, 140–142

[15] A. V. Savelev, “Istochniki variatsii dinamicheskikh svoistv nervnoi sistemy na sinapticheskom urovne v neirokompyutinge”, Iskusstvennyi intellekt, 4, NAN Ukrainy, 2006, 323–338

[16] R. Engelking, Obschaya topologiya, Mir, M., 1986

[17] V. A. Yurko, “Vosstanovlenie differentsialnykh operatorov peremennykh poryadkov na zvezdoobraznom grafe po spektram”, Differents. uravneniya, 49:12 (2013), 1537–1548 | Zbl

[18] S. Brzycbczy, R. Poznanski, Mathematical Neuroscience, Academic Press, 2013

[19] Y. Chen, “A new one-parameter inhomogeneous differential realization of the spl(2,1) superalgebra”, Internat. J. Theoret. Phys., 51:12 (2012), 3763–3768 | DOI | MR | Zbl

[20] A. V. Daneev, A. V. Lakeyev, V. A. Rusanov, “Existence of a bilinear differential realization in the constructions of tensor product of Hilbert spaces”, WSEAS Transactions on Mathematics, 19 (2020), 99–107 | DOI

[21] A. V. Daneev, A. V. Lakeyev, V. A. Rusanov, P. A. Plesnyov, “Differential non-autonomous representation of the integrative activity of a neural population by a bilinear second-order model with delay”, Lecture Notes in Networks and Systems, 319, 2022, 191–199 | DOI

[22] A. H. Hasanov, V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer International Publishing AG, Berlin, 2017 ; 2021, 515 pp. | MR | Zbl

[23] Y. Komma, “Nonlinear semi-groups in Hilbert space Japan”, J. Math. Soc., 19:4 (1967), 493–507 | MR

[24] E. Music, “An integrated brain-machine interface platform with thousands of channels”, J. Med. Internet Res., 21:10 (2019), el6194 | DOI

[25] V. A. Rusanov, A. V. Lakeyev, A. V. Banshchikov, A. V. Daneev, “On the bilinear second order differential realization of a infinite-dimensional dynamical system: An approach based on extensions to $M_2$-operators”, Fractal and Fractional, 7:4, Special Issues: Nonlinear Functional Analysis and Applications (2023), 1–18 | DOI

[26] V. A. Rusanov, A. V. Lakeyev, Yu. E. Linke, V. A. Voronov, “On realization of dynamic systems: Assessment of fiducial accuracy in the process of adjustment of the realization matrix”, Far East Journal of Dynamical Systems, 25:1 (2014), 23–35 | MR | Zbl