Existence of a differential realization of a dynamical system in a Banach space in the constructions of extensions to M p -operators

Abstract

For families of continuous controlled dynamic input-output processes, we obtain and discuss necessary and sufficient conditions for their differential realization in the class of quasilinear nonstationary differential equations of state in a Banach space.

Solvability of differential realization of minimum dynamic order for a family of nonlinear input-output processes in Hilbert space

“…In this setting, in the realization solvability problem, there arise substantial analytical distinctions [5,6] from the time-varying case, in particular, those related to the detection of minimum dynamic order [6][7][8] of the system. The present paper is an extended version of [8]; here we define the principles of M p -extensibility [4] with stationary M p -operators.…”

“…The study carried out in the present paper can be viewed as a continuation of the analysis of solvability of the problem on a nonlinear differential realization of complex behavioristic [1, p. 16] systems with state equations in an infinite-dimensional state space, which was initiated in the papers [2][3][4] dealing with the qualitative analysis of problems concerning the existence of realizations for time-varying models. In the present paper, we consider similar problems for autonomous equations with a nonlinear positional control.…”

About Extension of Differential Realization of the Countable Beam of Nonlinear Processes Input-Output in a Hilbert Space

“…) *  (based on the analytical structure of the subspace  *  , given in Lemma 4 [3]. Thus, by the arbitrariness of the choice of the element  * , the lemma will be proved as soon as we discover:…”

“…where (x,u,u # (x))Π, x  Carathé odory solution (C-solution), u and u # (x)  software and positional control, (A,B,B # )L 2 (T,μ,L(X,X))L 2 (T,μ,L(Y,X))L 2 (T,μ,L(Z,X)); in purposes of terminological convenience triple of vectorfunctions (x,u,u # (x)) we will also call C-solution of equation (1) and triple of operator-functions (A,B,B # ), adhering the terminology from [2,3] we will call (A,B,B # ) 2 -model of differential system (1).…”

“…Thus, taking into account the presence of a linear isometry between L 2 (T, *  ,R) and  *  (Lemma 4 [3]) will be   =    * , where in the end by the arbitrariness of the choice of function   , we obtain L 2 (T, *  ,R)    L 2 (T, * ,R) or taking into account Corollary 1 L 2 (T, *  ,R) =   L 2 (T, * ,R).…”