The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The constructions of tensor products of the Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of M₂ -operators and the functional apparatus of the entropy Relay Ritz operator represent the basis of this theory. It has been shown that in the case of the finite bundle of the controlled trajectory curves the existence of the property of sub-linearity of the given operator allows one to obtain sufficient conditions of existence of such realizations. Side by side with solving the main problems, grounded are topological-group conditions of continuity of projectivization of the Relay Ritz operator with computing the fundamental group (Poincare group) of its compact image. The results obtained give incentives for the development of the quantitative theory of converse problems of higher-order multilinear evolution equations with  the operators of generalized delay describing, for example, differential modeling of nonlinear Van der Pol oscillators or Lorentz strange attractors.

In this paper, an inverse problem of determining a kernel in a one-dimensional integro-differential time-fractional diffusion equation with initial-boundary and overdetermination conditions is investigated. An auxiliary problem equivalent to the problem is introduced first. By Fourier method this auxilary problem is reduced to equivalent integral equations. Then, using estimates of the Mittag-Leffler function and successive aproximation method, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel which will be used in study of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven.

In this article, we present new results for the computation of structured singular values of non-negative matrices subject to pure complex perturbations. We prove the equivalence of structured singular values and spectral radius of perturbed matrix (M∆). The presented new results on the equivalence of structured singular values, non-negative spectral radius and non-negative determinant of (M∆) is presented and analyzed. Furthermore, it has been shown that for a unit spectral radius of (M∆), both structured singular values and spectral radius are exactly equal. Finally, we present the exact equivalence between structured singular value and the largest singular value of (M∆).

This paper considers the inverse problem of determining the time-dependent coeffiicient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag-Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved.

We study a nonlocal problem for a differential Boussinesq-type equations in a multidimensional domain. Conditions for the existence and uniqueness of the solution are established, and a spectral decomposition of the solution is obtained.

We consider solutions to two boundary values problems for the Poisson equation on plane domains. We prove several estimates for integrals of solutions using geometric characteristics of domains.

Let τ be a faithful normal semifinite trace on a von Neumann algebra M. We investigate the block projection operator P_n (n ≥ 2) in the ^∗-algebra S(M, τ ) of all τ -measurable operators. We show that A ≤ nP_n(A) for any operator A ∈ S(M, τ )⁺. If an operator A ∈ S(M, τ )+ is invertible in S(M, τ ) then P_n(A) is invertible in S(M, τ ). Consider A = A^∗ ∈ S(M, τ ). Then (i)   if P_n(A) ≤ A (or if P_n(A) ≥ A) then P_n(A) = A; (ii) P_n(A) = A if and only if P_kA = AP_k for all k = 1, . . . , n; (iii) if  A,   n(A) are projections then n(A) = A. We obtain 4 corollaries. We also refined and reinforced one example from the paper “A. Bikchentaev, F. Sukochev, Inequalities for the block projection operators, J. Funct. Anal. 280 (7), article 108851, 18 p. (2021)”.

A natural generalization of Killing vector fields are conformally Killing vector fields which play an important role in the study of the group of conformal transformations of manifolds, Ricci flows on manifolds, and the theory of Ricci solitons. In this paper, we study conformally Killing vector fields on 2-symmetric indecomposable Lorentzian manifolds. It is established that the conformal factor of the conformal analogue of the Killing equation on them depends on the behavior of the Weyl tensor. In addition, in the case when the Weyl tensor is equal to zero, non-trivial examples of conformally Killing vector fields with a variable conformal factor are constructed using the Airy functions.

A functional differential equation with a discrete retarded argument and a constant concentrated delay is considered. The problem of the asymptotic stability of this equation is reduced to the problem of the location of the spectrum of the shift operator. Coefficient sufficient conditions for the asymptotic stability of this equation are obtained. The domain in the parameter space such that these conditions are necessary is obtained.