On the theory of realization of strong differential models. I

Daneev, A. V.; Лакеев, А. В.; Русанов, В. А.; Rusanov, M. V.

doi:10.1134/s1990478907030039

Cited by 5 publications

(3 citation statements)

References 4 publications

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“…The ideas and methods presented may have the following theoretical and applied applications: -development of the theory of strong autonomous ( , ) A B -models [18,19] in an infinite-dimensional Hilbert space as a subclass of Banach ( , )…”

Section: Discussionmentioning

confidence: 99%

From Kalman—Mesarovic realization to a normal-hyperbolic linear model

2005

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517.926The principles of the methodology are formulated, which underlie the procedures of the Kalman-Mesarovic realization for dynamic systems with state equations in the class of autonomous linear differential equations in the normalized Frechet space. In this connection, the key approaches to the solution of classical problems of realization theory regarding linear dissipation models of normal-hyperbolic type are interpreted.

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Section: Discussionmentioning

confidence: 99%

From Kalman—Mesarovic realization to a normal-hyperbolic linear model

2005

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“…The geometric ideas of this theory for finite-dimensional spaces were proposed by A.V. Lakeev in[21].…”

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confidence: 99%

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

Русанов¹,

Daneev²,

Lakeyev³

et al. 2021

International Journal of Difference Equations

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The solvability of the problem of realizing operator functions of an invariant polylinear controller (IPL-controller) of a non-stationary differential system (D-system) of the second order, which allows for two bundles of dynamic processes of the "trajectory, control" type that are induced in this D-system by two different polylinear controllers, to unite these bundles, through the action of the IPL-controller, into a subfamily of admissible solutions of the given Dsystem. The problem under consideration belongs to the type of nonstationary coefficient-operator inverse problems for evolution equations, including hyperbolic ones, in a separable Hilbert space and is solved on the basis of a qualitative study of the properties of continuity and semiadditivity of the RayleighRitz functional operator. The results obtained have applications in the theory of nonlinear infinite-dimensional adaptive dynamical systems for a class of higher-order polylinear differential models.

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“…In this sense, the key approaches to solving problems of realization theory in terms of the Rayleigh-Ritz operator are defined.The mathematics for the expansion of matrix-valued operators is developed in the Kalman-Mesarovic realization problem [1, 2] for nonstationary (with coefficients from Lebesgue L p -spaces) multidimensional differential systems in [3,4]. The concepts (categories of the dictionary of mathematical simulation of dynamic systems -D-systems) such as M-operator, M-continuability, the Rayleigh-Ritz operator (a reminiscence of the canonical Rayleigh-Ritz quotient [5]) are defined and formalized.…”

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Kalman-Mesarovic nonstationary realization in terms of Rayleigh-Ritz operator

2007

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517.926The paper outlines the methodology underlying the Kalman-Mesarovic realization for dynamic systems with equations of state in the class of linear nonstationary ordinary differential equations. In this sense, the key approaches to solving problems of realization theory in terms of the Rayleigh-Ritz operator are defined.The mathematics for the expansion of matrix-valued operators is developed in the Kalman-Mesarovic realization problem [1, 2] for nonstationary (with coefficients from Lebesgue L p -spaces) multidimensional differential systems in [3,4]. The concepts (categories of the dictionary of mathematical simulation of dynamic systems -D-systems) such as M-operator, M-continuability, the Rayleigh-Ritz operator (a reminiscence of the canonical Rayleigh-Ritz quotient [5]) are defined and formalized. These structures allow approaching, from the general system-theoretic standpoint, the qualitative analysis of the existence properties for the realizations of D-systems in a special class of strong differential ( , )A B -models [6, 7] -an "optimal" mathematical structure of the realization equations for linear continuous controlled D-systems.The purpose of this study is to show that the structure of the Rayleigh-Ritz operator allows a more efficient solution of realization problems in the class of nonstationary ( , )A Â -models (as compared with signal functions (Theorem 1 [6]) and measures (Theorems 2 [6] and 1, 2 [7]) defined in terms of D-systems from Definition 1.8 [1]). FORMULATIONS OF STANDARD REALIZATION PROBLEMS. PRELIMINARIES ON THE RAYLEIGH-RITZ OPERATOR.Let t t T t t 0 1 0 1 < = , : [ , ] be an interval of the number axis R with a Lebesgue measure m, and let P ACL n p m AC T R L T R : ( , ) ( , , ) =´¢ m ; here AC L p , ¢ are standard [8] function spaces of absolutely continuous vectorfunctions and equivalence classes of all vector functions with an L p¢ -norm that are Bochner m-integrable [9] on T; we will use the notation P ACL without an indication that (i) P

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On the theory of realization of strong differential models. I

Cited by 5 publications

References 4 publications

From Kalman—Mesarovic realization to a normal-hyperbolic linear model

From Kalman—Mesarovic realization to a normal-hyperbolic linear model

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

Kalman-Mesarovic nonstationary realization in terms of Rayleigh-Ritz operator

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