Realization of a polylinear controller as a second-order differential system in a Hilbert space

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, Corollary 2 allows us to construct an algebra of sets [17] of dynamical processes with unit  Nj, all elements of which have a D-realization (2). In this case, the question of the "individual" characteristic feature of D-realization (4) for each individual sheaf Nj (j = 1, …, k) is especially simply (constructively) solved for oneelement bundles Nj = {(x, u, D1(x), …, Dn(x, …, x))j} by checking (see Theorem 2 [2]) the conditions:…”

“…we denote the product space with the topology induced by the norm ||(h0, …, hn)||H := (T ||(h0(), …, hn()) 2 || U μ(d)) 1/2 , (h0, …, hn)  H2;…”

“…Definition 1 [2,8]. Consider on  := AC 1 (T, X)  H2 a nonlinear operator :   L(Т, R), constructed according to the following rule:…”

“…Lemma 1. 2 Semiadditivity with a fixed weight of the RayleighRitz operator is a finite property for subsets of .…”

“…Inverse problems of evolution equations (IPEE), as a section of the differential realization of dynamical systems, currently represent a fairly extensive area of research [110]. 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].…”