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Vol 59, No 4 (2023)
Articles
On the Asymptotic Stability and Ultimate Boundedness of Solutions of a Class of Nonlinear Systems with Delay
Abstract
For a certain class of nonlinear systems of differential equations with constant delay, we study the conditions for the asymptotic stability of the zero solution and the ultimate boundedness of the solutions. To obtain such conditions, we propose special constructions of Lyapunov–Krasovskii full-type functionals. Estimates of the transient time are found, and an analysis of the influence of perturbations on the dynamics of systems is carried out. In addition, we study the case in which the systems have switching operation modes and determine conditions under which the asymptotic stability or ultimate boundedness is preserved for any admissible switching laws.
435-445
On the Study of Robust Exponential Stability of Continuous- and Discrete-Time Systems
Abstract
A technique for obtaining sufficient conditions for the robust exponential stability of a parametrically uncertain system is proposed. This technique is used to study both continuousand discrete-time parametrically uncertain systems. For a common Lyapunov function we take a positive definite quadratic form that is a Lyapunov function of the system for a specific parameter value and satisfies some constraints on the first derivative (first difference). The application of our technique is illustrated by specific examples.
446-455
Inverse Problem of Determining the Unknown Coefficient in the Beam Vibration Equation in an Infinite Domain
Abstract
For the equation of transverse vibrations of a homogeneous beam, we consider the direct initial value problem in an infinite domain and study the inverse problem of determining the time-dependent beam stiffness coefficient. The solution of the direct problem is given with the help of fundamental solutions, and the existence and uniqueness of this solution are proved. Stability estimates are obtained for the solution of the inverse problem. Existence and uniqueness theorems for the solution of the inverse problem are proved using Banach’s contraction mapping principle.
456-466
Spectral Properties of the Operator in the Problem of Oscillations in a Mixture of Viscous Compressible Fluids
Abstract
We study the problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids that fills a bounded domain in three-dimensional space with infinitely smooth boundary. We prove that the essential spectrum of the problem is a finite set of segments located on the real axis. The remaining spectrum consists of isolated eigenvalues of finite algebraic multiplicity and is located on the real axis, with the possible exception of finitely many complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution.
467-482
Uniqueness of the Solution of the Dirichlet Problem for the Poisson Equation with a Singular -Kipriyanov Operator
Abstract
Functions satisfying the Laplace equation for the Kipriyanov
-operator are said to be
-harmonic. The following properties of
-harmonic functions are presented and proved: the Green-type integral representation of
-functions, the spherical mean theorem, and the maximum principle. As a corollary, the uniqueness of the solution of the interior and exterior Dirichlet problems is proved.
483-493
On Some Properties of Solutions of Systems of Linear Difference Equations with Periodic Right-Hand Sides
Abstract
We consider homogeneous and inhomogeneous systems of linear difference equations with coefficients that are
-periodic functions of discrete time. For homogeneous systems, sufficient conditions for the existence of periodic and almost periodic solutions are obtained. For inhomogeneous systems, it is shown that a necessary and sufficient condition for the existence of an N-periodic solution is the existence of a bounded solution. Necessary and sufficient conditions for theN orthogonality of the fundamental matrix of the homogeneous system are established. Illustrative examples are given.
494-500
Initial–Boundary Value Problem for Flows of a Fluid with Memory in a 3D Network-Like Domain
Abstract
We consider an initial–boundary value problem for an integro-differential system that describes 3D flows of a non-Newtonian fluid with memory in a network-like domain. The problem statement uses the Dirichlet boundary conditions for the velocity and pressure fields as well as Kirchhoff-type transmission conditions at the internal nodes of the network. A theorem on the existence and uniqueness of a time-continuous weak solution is proved. In addition, an energy equality for this solution is derived.
501-511
A Special Version of the Collocation Method for One Class of Integro-Differential Equations
Abstract
A linear integro-differential equation with a singular differential operator in the main part is studied. To find its approximate solution in the space of generalized functions, a special version of the generalized collocation method is proposed and justified
512-519
On a Class of Control Problems with Mixed Constraints
Abstract
An optimal control problem with a nonregular mixed constraint linear in the control variable is studied. Necessary optimality conditions are proposed in the form of Pontryagin’s maximum principle for such a class of problems. The corresponding examples are considered.
520-530
Stabilization of a Differential-Difference System of Delay Type
Abstract
For a linear autonomous differential-difference system with commensurate delays, algorithms for constructing controllers that provide asymptotic, finite-time, or complete stabilization of this system are substantiated. A distinctive feature of the proposed approach is that it does not require a priori information about the location of the roots of the characteristic quasipolynomial of the original system. The results are illustrated with examples.
531-553
On the Stability of a Switched Affine System for a Class of Switching Signals
Abstract
We study the problem of stability of the zero equilibrium of a switched affine system closed by a linear static state feedback. The concept of feasible control for a given set of switching signals is introduced, and a constructive condition for checking this property for an arbitrary linear feedback is obtained. A sufficient condition for the stability of the zero equilibrium of a switched affine system closed by a feasible control is formulated.
554-562
563-566