BSU Bulletin. Mathematics, Informatics
Bibliographic description:
, , ON BILINEAR DIFFERENTIAL REALIZATION OF A CONTINUAL BEAM OF TRAJECTORY CURVES IN THE CONSTRUCTIONS OF THE RAYLEIGH– RITZ OPERATOR // BSU Bulletin. Mathematics, Informatics. - 2020. №1. . - С. 11-27.
Title:
ON BILINEAR DIFFERENTIAL REALIZATION OF A CONTINUAL BEAM OF TRAJECTORY CURVES IN THE CONSTRUCTIONS OF THE RAYLEIGH– RITZ OPERATOR
Financing:
Codes:
Annotation:
We present functional and geometric conditions (necessary and sufficient) for the ex- istence of five non-stationary bilinear operators in a model of differential realization of a continuous bundle of controlled trajectory curves (dynamic processes of the power input / output type) in the class of the second order bilinear non-autonomous ordinary differential equations (including quasi-linear hyperbolic models) in a real parallel Hilbert space. The problem under consideration is a type of non-stationary nonlinear coefficient inverse problems for evolutionary equations in Hilbert space and is solved on the basis of a qualitative study of the continuity property of the Ray- leigh-Ritz functional operator. It is shown that the structure of the fundamental group of the image of this operator depends on the dimension of the projective space on which it acts. The results obtained are applied to the qualitative theory of non-linear structural identification of higher order multilinear non-stationary differential models.
Keywords:
nonlinear inverse problems; bilinear non-stationary differential realization; the Rayleigh-Ritz operator; the Poincaré group.
List of references:
Willems J.C. System Theoretic Models for the Analysis of Physical Systems. Ric. Aut. 1979. No. 10. Pp. 71–106.
Ahmed N. U. Optimization and Identification of Systems Governed by Evolution Equations on Banach Space. New York: John Wiley and Sons, 1988. 187 p.
Kantorovich L. V., Akilov G. P. Funktsionalnyi analiz [Functional Analysis]. Moscow: Nauka Publ., 1977. 744 p.
Reed M., Simon B. Methods of Modern Mathematical Physics. Vol. 1. Func- tional Analysis. New York: Academic Press, 1972. 344 p.
Daletsky Yu. L., Fomin S. V. Mery i differentsialnye uravneniya v beskonech- nomernykh prostranstvakh [Measures and Differential Equations in Infinite- Dimensional Spaces]. Moscow: Nauka Publ., 1983. 384 p.
Goldman N. L. Opredelenie koeffitsientov pri proizvodnoi po vremeni v kvazi- lineinykh parabolicheskikh uravneniyakh v prostranstvakh Geldera [Finding the Coef- ficient Multiplying the Time Derivative in Quasilinear Parabolic Equations in Hölder spaces]. Differential Equations. 2012. Vol. 48, no. 12. Pp. 1597–1606. DOI: org/10.1134/S0012266112120026].
Rusanov V. A., Daneev A. V., Lakeyev A. V., Linké Yu. E. On the Dif-ferential Realization Theory of Nonlinear Dynamic Processes in Hilbert Space. Far East Jour- nal of Mathematical Sciences. 2015. Vol. 97, no. 4. Pp. 495–532. DOI: org/10.17654/FJMSJun2015_495_532 ].
Rusanov V. A., Daneev A. V., Linké Yu. E. K geometricheskim osnovam dif- ferentsialnoi realizatsii dinamicheskikh protsessov v gilbertovom prostranstve [To the Geometrical Theory of Differential Realization of Dynamic Processes in a Hilbert Space]. Cybernetics and Systems Analysis. 2017. Vol. 53, No. 4. Pp. 71–83. DOI: 10.1007/s10559-017-9957-z .
Rusanov V. A., Daneev A. V., Lakeyev A. V., Sizykh V. N. Higher-Order Dif- ferent Realization of Polyline-Controlled Dynamic Processes in a Hilbert Space. Ad- vances in different Equations and Control Processes. 2018. Vol. 19, no. 3. Pp. 263– 274. DOI: org/10.17654/DE019030263 ].
Anikonov Yu. E., Neshchadim M. V. Ob analiticheskikh metodakh v teorii obratnykh zadach dlya giperbolicheskikh uravnenii [On Analytical Methods in the Theory of Inverse Problems for Hyperbolic Equations]. Journal of Applied and Indus- trial Mathematics. 2011. Vol. 14, no. 2. Pp. 28–33. DOI: org/10.1134/S1990478912010024.
Rusanov V. A., Banshchikov A. V., Daneev A. V., Lakeyev A. V. Maximum Entropy Principle in the Differential Second-Order Realization of a Nonstationary Bi- linear System. Advances in Differential Equations and Control Processes. 2019. Vol. 20, no. 2. Pp. 223–248. DOI: org/10.17654/DE020020223.
Lakeyev A.V., Linké Yu. E., Rusanov V. A. K realizatsii polilineinogo regu- lyatora differentsial'noi sistemy vtorogo poryadka v gilbertovom [Realization of a Polylinear Controller as a Second-Order Differential System in a Hilbert Space]. Dif- ferential Equations. 2017. Vol. 53, No. 8. Pp. 1098–1109. DOI: 10.1134/S037406411708012X.
Prasolov V. V. Elementy kombinatornoi i differentsialnoi topologii [Elements of Combinatorial and Differential Topology]. Moscow: MCCME Publ., 2014. 360 p.
Rusanov V. A., Daneev A. V., Lakeyev A. V., Linké Yu. E., Vetrov A. A. Sys- tem-Theoretical Foundation for Identification of Dynamic Systems. II. Far East Jour- nal of Mathematical Sciences. 2019. Vol. 116, no. 1. Pp. 25–68 DOI: org/10.17654/MS116010025 ].
Kirillov A. A. Elementy teorii predstavlenii [Elements of Representation The- ory]. Moscow: Nauka Publ., 1978. 344 p.
Rusanov V. A., Antonova L. V., Daneev A. V. Inverse Problem of Nonlinear Systems Analysis: A Behavioral Approach. Advances in Differential Equations and Control Processes. 2012. Vol. 10, no. 2. Pp. 69–88.
Rusanov V. A., Lakeyev A. V., Linké Yu. E. Sushchestvovanie dif- ferentsialnoi realizatsii dinamicheskoi sistemy v banakhovom prostranstve v kon- struktsiyakh rasshirenii do Mp-operatorov [Existence of a Differential Realization of a Dynamical System in a Banach Space in the Constructions of Extensions to Mp- operators]. Differential Equations. 2013. Vol. 49, no. 3. Pp. 358–370. DOI: 10.1134/S0012266113030105.
Rusanov V. A., Lakeyev A. V., Linké Yu. E. K razreshimosti differentsialnoi realizatsii minimalnogo dinamicheskogo poryadka semeistva nelineinykh protsessov “vkhod-vykhod” v gilbertovom prostranstve [Solvability of Differential Realization of Minimum Dynamic Order for a Family of Nonlinear Input-Output Processes in Hilbert Space]. Differential Equations. 2015. Vol. 51, no. 4. Pp. 524–537. DOI: 10.1134/S037406411504010X.
Chen Y. A New One-Parameter Inhomogeneous Differential Realization of the spl(2,1) Superalgebra. Intern. Journal of Theoretical Physics. 2012. Vol. 51, no. 12. P. 3763–3768. DOI: 10.1007/s10773-012-1261-0.
Rusanov V. A., Daneev A. V., Linké Yu. E. K optimizatsii protsessa yustirovki modeli differentsialnoi realizatsii mnogomernoi sistemy vtorogo poryadka [Adjustment Optimization for a Model of Differential Realization of a Multidimensional Second- Order System]. Differential Equations. 2019. Vol. 55, no. 10. Pp. 1432–1438. DOI: 10.1134/S0374064119100145.
Yoshida K. Functional Analysis. Springer Berlin Heidelberg, 1966. 458 p.
Fomenko A. T., Fuks D. B. Kurs gomotopicheskoi topologii [A Course in Homotopic Topology]. Moscow: Nauka Publ., 1989, 528 p.
Novikov S. P., Taimanov I. A. Sovremennye geometricheskie struktury i polya [Modern Geometric Structures and Fields]. Moscow: MCCME Publ., 2014. 584 p.
Lakeyev A.V., Linké Yu. E., Rusanov V. A. On a Criterion for the Continuity of The Rayleigh–Ritz Operator. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika i informatika. 2018. Vyp. 3. Pp. 3–13. DOI: 10.18101/2304-5728-2018-3- 3-13.
Kalman P. L. Falb & M. A. Arbib. Topics in Mathematical System Theory. New York: McGraw-Hill, 1969. 358 p.
Edwards R. Functional Analysis: Theory and Applications. New York: Holt, Rinehart and Winston, 1965. 781 p.
Gromov V. P. Analiticheskie resheniya differentsialno-operatornykh uravnenii v lokalno vypuklykh prostranstvakh [Analytical Solutions for Differential Operator Equations in Locally Convex Spaces]. Doklady Mathematics. 2004. Vol. 394, no. 3. Pp. 305–308.
Popper K. R. Conjecture and Refutations. London: Harper and Row, 1963. 486
Ahmed N. U. Optimization and Identification of Systems Governed by Evolution Equations on Banach Space. New York: John Wiley and Sons, 1988. 187 p.
Kantorovich L. V., Akilov G. P. Funktsionalnyi analiz [Functional Analysis]. Moscow: Nauka Publ., 1977. 744 p.
Reed M., Simon B. Methods of Modern Mathematical Physics. Vol. 1. Func- tional Analysis. New York: Academic Press, 1972. 344 p.
Daletsky Yu. L., Fomin S. V. Mery i differentsialnye uravneniya v beskonech- nomernykh prostranstvakh [Measures and Differential Equations in Infinite- Dimensional Spaces]. Moscow: Nauka Publ., 1983. 384 p.
Goldman N. L. Opredelenie koeffitsientov pri proizvodnoi po vremeni v kvazi- lineinykh parabolicheskikh uravneniyakh v prostranstvakh Geldera [Finding the Coef- ficient Multiplying the Time Derivative in Quasilinear Parabolic Equations in Hölder spaces]. Differential Equations. 2012. Vol. 48, no. 12. Pp. 1597–1606. DOI: org/10.1134/S0012266112120026].
Rusanov V. A., Daneev A. V., Lakeyev A. V., Linké Yu. E. On the Dif-ferential Realization Theory of Nonlinear Dynamic Processes in Hilbert Space. Far East Jour- nal of Mathematical Sciences. 2015. Vol. 97, no. 4. Pp. 495–532. DOI: org/10.17654/FJMSJun2015_495_532 ].
Rusanov V. A., Daneev A. V., Linké Yu. E. K geometricheskim osnovam dif- ferentsialnoi realizatsii dinamicheskikh protsessov v gilbertovom prostranstve [To the Geometrical Theory of Differential Realization of Dynamic Processes in a Hilbert Space]. Cybernetics and Systems Analysis. 2017. Vol. 53, No. 4. Pp. 71–83. DOI: 10.1007/s10559-017-9957-z .
Rusanov V. A., Daneev A. V., Lakeyev A. V., Sizykh V. N. Higher-Order Dif- ferent Realization of Polyline-Controlled Dynamic Processes in a Hilbert Space. Ad- vances in different Equations and Control Processes. 2018. Vol. 19, no. 3. Pp. 263– 274. DOI: org/10.17654/DE019030263 ].
Anikonov Yu. E., Neshchadim M. V. Ob analiticheskikh metodakh v teorii obratnykh zadach dlya giperbolicheskikh uravnenii [On Analytical Methods in the Theory of Inverse Problems for Hyperbolic Equations]. Journal of Applied and Indus- trial Mathematics. 2011. Vol. 14, no. 2. Pp. 28–33. DOI: org/10.1134/S1990478912010024.
Rusanov V. A., Banshchikov A. V., Daneev A. V., Lakeyev A. V. Maximum Entropy Principle in the Differential Second-Order Realization of a Nonstationary Bi- linear System. Advances in Differential Equations and Control Processes. 2019. Vol. 20, no. 2. Pp. 223–248. DOI: org/10.17654/DE020020223.
Lakeyev A.V., Linké Yu. E., Rusanov V. A. K realizatsii polilineinogo regu- lyatora differentsial'noi sistemy vtorogo poryadka v gilbertovom [Realization of a Polylinear Controller as a Second-Order Differential System in a Hilbert Space]. Dif- ferential Equations. 2017. Vol. 53, No. 8. Pp. 1098–1109. DOI: 10.1134/S037406411708012X.
Prasolov V. V. Elementy kombinatornoi i differentsialnoi topologii [Elements of Combinatorial and Differential Topology]. Moscow: MCCME Publ., 2014. 360 p.
Rusanov V. A., Daneev A. V., Lakeyev A. V., Linké Yu. E., Vetrov A. A. Sys- tem-Theoretical Foundation for Identification of Dynamic Systems. II. Far East Jour- nal of Mathematical Sciences. 2019. Vol. 116, no. 1. Pp. 25–68 DOI: org/10.17654/MS116010025 ].
Kirillov A. A. Elementy teorii predstavlenii [Elements of Representation The- ory]. Moscow: Nauka Publ., 1978. 344 p.
Rusanov V. A., Antonova L. V., Daneev A. V. Inverse Problem of Nonlinear Systems Analysis: A Behavioral Approach. Advances in Differential Equations and Control Processes. 2012. Vol. 10, no. 2. Pp. 69–88.
Rusanov V. A., Lakeyev A. V., Linké Yu. E. Sushchestvovanie dif- ferentsialnoi realizatsii dinamicheskoi sistemy v banakhovom prostranstve v kon- struktsiyakh rasshirenii do Mp-operatorov [Existence of a Differential Realization of a Dynamical System in a Banach Space in the Constructions of Extensions to Mp- operators]. Differential Equations. 2013. Vol. 49, no. 3. Pp. 358–370. DOI: 10.1134/S0012266113030105.
Rusanov V. A., Lakeyev A. V., Linké Yu. E. K razreshimosti differentsialnoi realizatsii minimalnogo dinamicheskogo poryadka semeistva nelineinykh protsessov “vkhod-vykhod” v gilbertovom prostranstve [Solvability of Differential Realization of Minimum Dynamic Order for a Family of Nonlinear Input-Output Processes in Hilbert Space]. Differential Equations. 2015. Vol. 51, no. 4. Pp. 524–537. DOI: 10.1134/S037406411504010X.
Chen Y. A New One-Parameter Inhomogeneous Differential Realization of the spl(2,1) Superalgebra. Intern. Journal of Theoretical Physics. 2012. Vol. 51, no. 12. P. 3763–3768. DOI: 10.1007/s10773-012-1261-0.
Rusanov V. A., Daneev A. V., Linké Yu. E. K optimizatsii protsessa yustirovki modeli differentsialnoi realizatsii mnogomernoi sistemy vtorogo poryadka [Adjustment Optimization for a Model of Differential Realization of a Multidimensional Second- Order System]. Differential Equations. 2019. Vol. 55, no. 10. Pp. 1432–1438. DOI: 10.1134/S0374064119100145.
Yoshida K. Functional Analysis. Springer Berlin Heidelberg, 1966. 458 p.
Fomenko A. T., Fuks D. B. Kurs gomotopicheskoi topologii [A Course in Homotopic Topology]. Moscow: Nauka Publ., 1989, 528 p.
Novikov S. P., Taimanov I. A. Sovremennye geometricheskie struktury i polya [Modern Geometric Structures and Fields]. Moscow: MCCME Publ., 2014. 584 p.
Lakeyev A.V., Linké Yu. E., Rusanov V. A. On a Criterion for the Continuity of The Rayleigh–Ritz Operator. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika i informatika. 2018. Vyp. 3. Pp. 3–13. DOI: 10.18101/2304-5728-2018-3- 3-13.
Kalman P. L. Falb & M. A. Arbib. Topics in Mathematical System Theory. New York: McGraw-Hill, 1969. 358 p.
Edwards R. Functional Analysis: Theory and Applications. New York: Holt, Rinehart and Winston, 1965. 781 p.
Gromov V. P. Analiticheskie resheniya differentsialno-operatornykh uravnenii v lokalno vypuklykh prostranstvakh [Analytical Solutions for Differential Operator Equations in Locally Convex Spaces]. Doklady Mathematics. 2004. Vol. 394, no. 3. Pp. 305–308.
Popper K. R. Conjecture and Refutations. London: Harper and Row, 1963. 486
Article.pdf