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Vol 61, No 3 (2025)
ЛЮДИ НАУКИ
ALEKSANDR BORISOVICh KURZhANSKIY
Differencial'nye uravneniya. 2025;61(3):291–292
291–292
ORDINARY DIFFERENTIAL EQUATIONS
ASYMPTOTICS OF EIGENVALUES AND EIGENFUNCTIONS OF THE STURM–LIOUVILLE OPERATOR WITH SINGULAR POTENTIAL ON A STAR GRAPH. II
Abstract
Spectral problems on a star-graph consisting of three edges with a Sturm–Liouville operator defined on each of them are investigated. The spectral properties of such operators have been studied, in particular, asymptotic formulas for eigenvalues and eigenfunctions of the operator with Dirichlet boundary conditions at free ends and continuity and Kirchhoff conditions at a common vertex have been obtained. The potential in the Sturm–Liouville problem is assumed to be singular, it is a derivative of a quadratically summable function in sense of distributions.
Differencial'nye uravneniya. 2025;61(3):293–304
293–304
ON PROPERTIES OF DIRAC OPERATOR WITH IRREGULAR BOUNDARY CONDITIONS
Abstract
The paper is concerned with the basis properties of root functions of the 2 × 2 Dirac operator with summable complex-valued potential and irregular boundary conditions. When certain conditions on the spectrum of the operator under consideration are satisfied, we prove that the system of root functions of this operator is incomplete in the space L2(0,π)⊕L2(0,π) but forms unconditional basis in the closure of its linear hull.
Differencial'nye uravneniya. 2025;61(3):305–315
305–315
ON THE CHANGE IN THE POWER OF THE SPECTRUM OF THE EXACT AND ABSOLUTE WANDERING EXPONENT DURING THE TRANSITION FROM A TWO-DIMENSIONAL NONLINEAR SYSTEM TO A SYSTEM OF ITS FIRST APPROXIMATION
Abstract
The sets of values (spectra) of the wandering exponents of solutions of differential systems are studied. Two-dimensional systems with nonlinearity of an arbitrarily specified higher order of smallness in the neighborhood of the origin are constructed, for which all solutions are infinitely extendable to the right and any of the spectra of their wandering exponents can coincide with both the segment [0, 1] and with any pre-specified non-empty subset of rational numbers of this segment, while the spectra of linear systems of their first approximation consist of only one element. Moreover, the spectra of the exponents of the original system coincide with the corresponding spectra of the wandering exponents of the narrowing of the constructed nonlinear two-dimensional systems to the direct product of any open neighborhood of the zero of the phase plane and the time semi-axis.
Differencial'nye uravneniya. 2025;61(3):316–329
316–329
INTEGRABLE DYNAMICAL SYSTEMS OF 9th ORDER WITH DISSIPATION
Abstract
New cases of integrable dynamical systems of the ninth order, homogeneous in terms of variables, are presented in which a system on a cotangent bundle to a four-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has a dissipation of different signs. The external field is introduced using some unimodular transformation and generalizes the previously considered fields. Complete sets of both the first integrals and invariant differential forms are given.
Differencial'nye uravneniya. 2025;61(3):330–353
330–353
PARTIAL DERIVATIVE EQUATIONS
354–365
CLASSIFICATION OF FIELD EQUATIONS FOR WEYL SPINORS AND ELKO SPINORS
Abstract
A class of field (relativistically invariant) equations is introduced for a wave function consisting of several Weyl spinors. The equations are such that each of these Weyl spinors satisfies the Klein–Gordon equation with the same mass. Subclasses of equations of Majorana type and Dirac-type are introduced. It is shown that the known equations for Elko spinors belong to the subclass of Dirac-type equations.
Differencial'nye uravneniya. 2025;61(3):366–373
366–373
CONTROL THEORY
ON SINGULARITIES OF A-ORBITAL FEEDBACK LINEARIZATION OF SINGLE-INPUT AFFINE CONTROL SYSTEMS
Abstract
For single-input affine control systems, we address the problem of A-orbital feedback linearization around singular points of the derived flag of the distribution associated with the control system. By a singular point of a derived flag we mean a point such that at least one of the elements of the derived flag in any neighborhood of this point is not a distribution of constant rank. We prove a local necessary and sufficient condition for A-orbital feedback equivalence of a single-input affine control system to a linear controllable system considered in a neighbourhood of the zero equilibrium point.
Differencial'nye uravneniya. 2025;61(3):374–393
374–393
FINITE STABILIZATION OF NOT FULLY CONTROLLED HYBRID LINEAR CONTINUOUS-DISCRETE SYSTEMS
Abstract
For hybrid linear autonomous continuous-discrete systems that do not have the property of complete controllability, an approach to designing two types of controllers that provide “incomplete finite stabilization” is proposed. The implementation of one of them — a controller of weak finite state stabilization — is based on knowledge of the values of the control system solution at discrete moments of time, multiples of the quantization step. The second type of controller — a weak finite stabilization controller by output — uses the observed output signal as feedback. The constructed regulators contain auxiliary variables described by additional equations with discrete time, and incomplete finite stabilization implies that for a closed system, finite functions will only be required for those components of the solution vector that are components of the solution vector of the initial (open) system. Criteria for the existence of the specified regulators and a method for their design are obtained.
Differencial'nye uravneniya. 2025;61(3):394–409
394–409
ON EXACT GLOBAL CONTROLLABILITY OF A SEMILINEAR EVOLUTIONARY EQUATION WITH NONSTATIONARY OPERATOR
Abstract
For a Cauchy problem associated with a controlled semilinear evolutionary equation with bounded nonstationary (id est depending on time) operator in a Hilbert space we obtain sufficient conditions for exact controllability to a given final state (and also to given intermediate states at intermediate time moments) on an arbitrarily fixed (without additional conditions) time interval. In fact, it is generalized an analogous result having been obtained by the author formerly for the case of a stationary operator. Like formerly, here we use the Minty–Browder’s theorem and also a chain technology of successive continuation of the solution to a controlled system to intermediate states. As example (of a specific interest) we consider a semilinear equation of the global electric circuit in the Earth atmosphere.
Differencial'nye uravneniya. 2025;61(3):410–428
410–428
BRIEF MESSAGES
429–432