Metric properties of the Rayleigh--Ritz operator

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

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2022), pp. 54-63.

Résumé

We consider a nonlinear functional Rayleigh–Ritz operator defined on a set of pairs of measurable functions and equal to the ratio of their modules if the denominator is nonzero and zero otherwise. We investigate the continuity of this operator with respect to the convergence of the measure. It is shown that the convergence of the operator value on the sequence of pairs to the value on the limit pair of functions requires not only convergence as its arguments, but also convergence as the carriers of the second argument to the carrier of its limit. The results obtained have applications in the theory of differential realization (in Hilbert space) of higher-order nonlinear dynamic models.

Détail

Keywords: space of measurable functions, convergence by measure, non-invariant metric, Rayleigh–Ritz operator.

Bibliographie

[1] Daneev A.V., Rusanov V.A., “Geometricheskii podkhod k resheniyu nekotorykh obratnykh zadach sistemnogo analiza”, Izv. vuzov. Matem., 2001, no. 10, 18–28 | Zbl

[2] Rusanov V.A., Antonova L.V., Daneev A.V., “Inverse problem of nonlinear systems analysis: A behavioral approach”, Adv. in Diff. Equat. and Control Proces., 10:2 (2012), 69–88 | MR | Zbl

[3] Novikov S.P., Taimanov I.A., Sovremennye geometricheskie struktury i polya, MTsNMO, M., 2014

[4] Daneev A.V., Rusanov V.A., Sharpinskii D.Yu., “Printsip maksimuma entropii v strukturnoi identifikatsii dinamicheskikh sistem. Analiticheskii podkhod”, Izv. vuzov. Matem., 2005, no. 11, 16–24 | Zbl

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

[6] Lakeev A.V., Linke Yu.E., Rusanov V.A., “K differentsialnoi realizatsii bilineinoi sistemy vtorogo poryadka v gilbertovom prostranstve”, Sib. zhurn. industrialnoi matem., 22:2 (2019), 27–36 | MR | Zbl

[7] Kantorovich L.V., Akilov G.P., Funktsionalnyi analiz, Nauka, M., 1977 | MR

[8] Lakeev A.V., Linke Yu.E., Rusanov V.A., “Ob odnom kriterii nepreryvnosti operatora Releya–Rittsa”, Vestn. Buryatsk. gos. un-ta, Matematika, informatika, 2018, no. 3, 3–13

[9] Dyachenko M.I., Ulyanov P.L., Mera i integral, Faktorial, M., 1998

[10] Rusanov V.A., Daneev A.V., Linke Yu.È., Plesnyov P.A., “Existence of a bilinear delay differential realization of nonlinear neurodynamic process in the constructions of entropic Rayleigh–Ritz operator”, Adv. in Dynamical Syst. and Appl., 15:2 (2020), 199–215

[11] Rusanov V.A., Lakeev A.V., Daneev A.V., Linke Yu.È., “On the differential realization theory of non-linear dynamic processes in Hilbert space”, Far East J. Math. Sci., 97:4 (2015), 495–532 | Zbl

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

[13] Rusanov V.A., Daneev A.V., Lakeyev A.V., Sizykh V.N., “Higher-order differential realization of polylinear-controlled dynamic processes in a Hilbert space”, Adv. Diff. Equat. and Control Proces., 19:3 (2018), 263–274 | Zbl

[14] Rusanov V.A., Daneev A.V., Linke Yu.E., “K optimizatsii protsessa yustirovki modeli differentsialnoi realizatsii mnogomernoi sistemy vtorogo poryadka”, Differents. uravneniya, 55:10 (2019), 1432–1438 | Zbl

[15] Rusanov V.A., Daneev A.V., Lakeyev A.V., Linke Yu.È., “An Inverse Problems for Nonlinear Evolution Equations: Criteria of Existence of an Invariant Polylinear Controller for a Second-Order Differential System in a Hilbert Space”, International J. Diff. Equat., 16:1 (2021), 1–10 | DOI | MR

[16] Rusanov V.A., Lakeev A.V., Linke Yu.E., “Suschestvovanie differentsialnoi realizatsii dinamicheskoi sistemy v banakhovom prostranstve v konstruktsiyakh rasshirenii do $M_p$-operatorov”, Differents. uravneniya, 49:3 (2013), 358–370 | MR | Zbl

[17] Rusanov V.A., Banshchikov A.V., Daneev A.V., Lakeyev A.V., “Maximum entropy principle in the differential second-order realization of a nonstationary bilinear system”, Adv. Diff. Equat. and Control Processes, 20:2 (2019), 223–248 | MR | Zbl

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

[19] Daneev A.V., Lakeev A.V., Rusanov V.A., “K suschestvovaniyu vpolne nepreryvnoi differentsialnoi realizatsii bilineinoi sistemy vtorogo poryadka”, Izv. Samarsk. nauchn. tsentra RAN, 23:4 (2021), 116–132

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