BSU Bulletin. Mathematics, Informatics
Bibliographic description:
, , ON REALIZATION OF MULTIPLE-LINEAR REGULATOR OF NON-STATIONARY HYPERBOLIC SYSTEM // BSU Bulletin. Mathematics, Informatics. - 2019. №3. . - С. 3-16.
Title:
ON REALIZATION OF MULTIPLE-LINEAR REGULATOR OF NON-STATIONARY HYPERBOLIC SYSTEM
Financing:
Работа выполнена при финансовой поддержке РФФИ (проекты: 19-01-00301, 19-08-00746).
Codes:
Annotation:
The article studies some qualitative issues of the existing a solution to inverse prob- lem of nonlinear infinite-dimensional system analysis, regarding the solvability of the operator realization of invariant multiple-linear regulator of non-stationary hy- perbolic system. The studied statement for precision mathematical modeling con- siders the case, when for two different beams (finite, countable, or even continuous) of nonlinear controlled dynamic processes of the “trajectory, program control” type, induced in a real separable Hilbert space by some given non-stationary hyperbolic system, but with different multiple-linear regulators, we obtain sufficient conditions for solvability of the problem of realizing operator functions of a general (invariant) multiple-linear regulator, in which presence in equation structure of a given hyper- bolic system the union of these dynamic beams represent a fixed family of its ad- missible solutions. The study was carried out in the light of modern ideas about the geometry of infinite-dimensional vector fields and based on a qualitative study of the semi-additivity of non-linear Rayleigh-Ritz functional operator.
Keywords:
qualitative theory of nonlinear differential realization; non-stationary hyperbolic system; Rayleigh-Ritz functional operator; multiple-linear regulator.
List of references:
Kabanikhin S. I. Obratnye i nekorrektnye zadachi [Inverse and Ill-posed Prob- lems]. Novsibirsk: Siberian Scientific Publishing Department, 2009. 458 p.
Kaiser E., Kutz J. N., Brunton S. L. Sparse Identification of Nonlinear Dynamics for Model Predictive Control in the Low-data Limit. Available at: https://arxiv.org/abs/1711.05501v2 (accessed 10.09.2019).
Chen Y. A New One-Parameter Inhomogeneous Differential Realization of the spl (2,1) Superalgebra. International Journal of Theoretical Physics. 2012. V. 51. No. 12. Pp. 37633768. DOI: 10.1007/s10773-012-1261-0.
Rusanov V. A., Daneev A. V., Lakeyev A. V., Linke Yu. E. On Solvability of the Identification-Inverse Problem for Operator-Functions of a Nonlinear Regulator of a Nonstationary Hyperbolic System. Advances in Differential Equations and Control Processes. 2015. V. 16. No. 2. Pp. 7184. DOI: 10.17654/DE016020071.
Rudy S. H., Brunton S. L., Proctor J. L., Kutz J. N. Data-Driven Discovery of Partial Differential Equations. Science Advances. 2017. V. 3. No. 4. Pp. 16. DOI: 10.1126/sciadv.1602614.
Rusanov V. A., Daneev A. V., Lakeyev A. V., Sizykh V. N. Higher-Order Dif- ferential Realization of Polylinear-Controlled Dynamic Processes in a Hilbert Space. Advances in Differential Equations and Control Processes. 2018. V. 19. No. 3. Pp. 263274. DOI: 10.17654/DE019030263.
Rusanov V. A., Daneev A. V., Linke Yu. E., Sizykh V. N., Voronov V. A. Sys- tem-Theoretical Foundation for Identification of Dynamic Systems. I. Far East Journal of Mathematical Sciences. 2018. V. 106. No. 1. Pp. 142. DOI: 10.17654/MS106010001.
Yosida K. Functional Analysis. Springer-Verlag, 1980.
Novikov S. P., Taimanov I. A. Sovremennye geometricheskie struktury i polya [Modern Geometric Structures and Fields]. Moscow, 2014. 624 p.
Ahmed N. U. Optimization and Identification of Systems Governed by Evolution Equations on Banach Space. New York: John Wiley and Sons, 1988. 187 p.
Daneev A. V., Rusanov V. A., Rusanov M. V. Ot realizatsii Kalmana-Mesarovicha k lineinoi modeli normalno-giperbolicheskogo tipa [From Kalman Mesarovic Realization to a Normal-Hyperbolic Linear Model]. Cybernetics and Systems Analysis, 2005. No. 6. Pp. 137157.
Daneev A. V., Rusanov V. A., Rusanov M. V., Sizyh V. N. K aposteriornomu modelirovaniyu nestatsionarnykh giperbolicheskikh sistem [On a Posteriori Modeling of Nonstationary Hyperbolic Systems]. Izvestiya Samarskogo nauchnogo tsentra RAN. 2018. V. 20. No. 1. Pp. 106–113. DOI: 10.24411/1990-5378-2018-00003.
Rusanov V. A., Daneev A. V., Linke Yu. E. K geometricheskim osnovam dif- ferentsialnoi realizatsii dinamicheskikh processov v gilbertovom prostranstve [On Geometrical Basis for Differential Realization of Dynamic Processes in a Hilbert Space]. Cybernetics and Systems Analysis. 2017. V. 53. No. 4. Pp. 71–83. DOI: 10.1007/s10559-017-9957-z.
Lakeyev A. V., Linke Yu. E., Rusanov V. A. Ob odnom kriterii nepreryvnosti operatora Releya-Rittsa [On a Criterion of the Continuity of Relay-Ritz Operator]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika i informatika. 2018. V. 3. Pp. 3–13. DOI: 10.18101/2304-5728-2018-3-3-13.
Rusanov V. A., Sharpinskii D. Yu. K teorii strukturnoi identifikatsii nelineinykh mnogomernykh sistem [Towards a Theory of Structural Identification of Nonlinear Multidimensional Systems]. Journal of Applied Mathematics and Mechanics. 2010. V. 74. V. 1. Pp. 119–132.
Kadets M. I., Mityagin B. S. Dopolnyaemye podprostranstva v banakhovykh prostranstvakh [Complementary Subspaces in Banach Spaces]. Russian Mathematical Surveys. 1973. V. 28. No. 6. Pp. 77–94.
Kaiser E., Kutz J. N., Brunton S. L. Sparse Identification of Nonlinear Dynamics for Model Predictive Control in the Low-data Limit. Available at: https://arxiv.org/abs/1711.05501v2 (accessed 10.09.2019).
Chen Y. A New One-Parameter Inhomogeneous Differential Realization of the spl (2,1) Superalgebra. International Journal of Theoretical Physics. 2012. V. 51. No. 12. Pp. 37633768. DOI: 10.1007/s10773-012-1261-0.
Rusanov V. A., Daneev A. V., Lakeyev A. V., Linke Yu. E. On Solvability of the Identification-Inverse Problem for Operator-Functions of a Nonlinear Regulator of a Nonstationary Hyperbolic System. Advances in Differential Equations and Control Processes. 2015. V. 16. No. 2. Pp. 7184. DOI: 10.17654/DE016020071.
Rudy S. H., Brunton S. L., Proctor J. L., Kutz J. N. Data-Driven Discovery of Partial Differential Equations. Science Advances. 2017. V. 3. No. 4. Pp. 16. DOI: 10.1126/sciadv.1602614.
Rusanov V. A., Daneev A. V., Lakeyev A. V., Sizykh V. N. Higher-Order Dif- ferential Realization of Polylinear-Controlled Dynamic Processes in a Hilbert Space. Advances in Differential Equations and Control Processes. 2018. V. 19. No. 3. Pp. 263274. DOI: 10.17654/DE019030263.
Rusanov V. A., Daneev A. V., Linke Yu. E., Sizykh V. N., Voronov V. A. Sys- tem-Theoretical Foundation for Identification of Dynamic Systems. I. Far East Journal of Mathematical Sciences. 2018. V. 106. No. 1. Pp. 142. DOI: 10.17654/MS106010001.
Yosida K. Functional Analysis. Springer-Verlag, 1980.
Novikov S. P., Taimanov I. A. Sovremennye geometricheskie struktury i polya [Modern Geometric Structures and Fields]. Moscow, 2014. 624 p.
Ahmed N. U. Optimization and Identification of Systems Governed by Evolution Equations on Banach Space. New York: John Wiley and Sons, 1988. 187 p.
Daneev A. V., Rusanov V. A., Rusanov M. V. Ot realizatsii Kalmana-Mesarovicha k lineinoi modeli normalno-giperbolicheskogo tipa [From Kalman Mesarovic Realization to a Normal-Hyperbolic Linear Model]. Cybernetics and Systems Analysis, 2005. No. 6. Pp. 137157.
Daneev A. V., Rusanov V. A., Rusanov M. V., Sizyh V. N. K aposteriornomu modelirovaniyu nestatsionarnykh giperbolicheskikh sistem [On a Posteriori Modeling of Nonstationary Hyperbolic Systems]. Izvestiya Samarskogo nauchnogo tsentra RAN. 2018. V. 20. No. 1. Pp. 106–113. DOI: 10.24411/1990-5378-2018-00003.
Rusanov V. A., Daneev A. V., Linke Yu. E. K geometricheskim osnovam dif- ferentsialnoi realizatsii dinamicheskikh processov v gilbertovom prostranstve [On Geometrical Basis for Differential Realization of Dynamic Processes in a Hilbert Space]. Cybernetics and Systems Analysis. 2017. V. 53. No. 4. Pp. 71–83. DOI: 10.1007/s10559-017-9957-z.
Lakeyev A. V., Linke Yu. E., Rusanov V. A. Ob odnom kriterii nepreryvnosti operatora Releya-Rittsa [On a Criterion of the Continuity of Relay-Ritz Operator]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika i informatika. 2018. V. 3. Pp. 3–13. DOI: 10.18101/2304-5728-2018-3-3-13.
Rusanov V. A., Sharpinskii D. Yu. K teorii strukturnoi identifikatsii nelineinykh mnogomernykh sistem [Towards a Theory of Structural Identification of Nonlinear Multidimensional Systems]. Journal of Applied Mathematics and Mechanics. 2010. V. 74. V. 1. Pp. 119–132.
Kadets M. I., Mityagin B. S. Dopolnyaemye podprostranstva v banakhovykh prostranstvakh [Complementary Subspaces in Banach Spaces]. Russian Mathematical Surveys. 1973. V. 28. No. 6. Pp. 77–94.
Article.pdf