Higher-Order Differential Realization of Polylinear-Controlled Dynamic Processes in a Hilbert Space

Existence of a Bilinear Delay Differential Realization of Nonlinear Neurodynamic Process in the Constructions of Entropic Rayleigh Ritz Operator

“…The second conclusion expresses Proposition 1. Let the dynamic bundle N fulfill the following condition: In order to go further, one can quite confidently point out the theoretical and systemic direction which will constitute the algebraic basis without the excessive building-up of technical means of algebraic geometry of the next stage of development of the qualitative theory of differential realization of higher orders [3,4,19], namely, the transition from the bilinear structure of nonlinear links to polylinear links. Methodologically, this transition consists in using a geometric language of tensor structures of the Fock spaces [20] and projective representations [10, p. 238] in the context of the study of metric properties of the RayleighRitz operators [15], by means of computer algebra of mathematical physics [17].…”

“…The current period of intensive development of the qualitative theory of differential realization in an infinitedimensional formulation is largely related to the creation of a new mathematical language  the entropic theory of extensions of M2-operators [2]. This theory has substantially reconstructed and strengthened theoretical and system foundations of QTDR and provided harmonic connection of purely geometric ideas of 2 Mextendibility with the methods of a posterior modeling of differential equations of higher orders [3,4] in infinite-dimensional spaces, with an emphasis on the application [5,6] and not on achieving maximum generality of presentation.…”

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

“…In this context, the proposed work continues research [2,10], while understanding the nature of hyperbolic systems (in the technical sense) helps to clarify and motivate the entire discussion. Its main goal is to study the problem of existence of coefficient operator-functions of an invariant polylinear controller (IPL-controller) of a non-stationary differential system (D-system) 1 of the second order; however, its results can be extended to stationary D-systems [11]. The IPL-controller assumes that the controllable D-system must contain in the class of its admissible solutions the union of two given bundles of dynamic processes, while each such bundle is unlimited in power (finite/countable/continual) and is induced in this system by its own "individual" polylinear controller.…”

Existence of a Bilinear Differential Realization in the Constructions of Tensor Product of Hilbert Spaces

“…Significant analytical difficulties can be faced when moving to constructing a differential realization with a dynamic order higher than the first [9]. This includes the non-nominal consideration of the structure of hyperbolic models [10], the representations of which cannot dispense with taking into account the nonlinearity of their dynamics, in particular, the bilinear structure of the implementation model [11,12], which is the focus of attention in this article.…”