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Том 54, № 8 (2018)
- Год: 2018
- Статей: 10
- URL: https://journal-vniispk.ru/0012-2661/issue/view/9352
Ordinary Differential Equations
Supremum of the Perron Exponent on the Solutions of a Linear System with Slowly Growing Coefficients is Metrically Typical
Аннотация
We prove that if the Lyapunov exponent of the norm of the coefficient matrix of a linear differential system is nonpositive, then the supremum of Perron exponents of the solutions issuing from any given affine subspace is attained and the set of initial vectors of solutions with the maximum Perron exponent has full Lebesgue measure in the subspace.
993-999
On a Version of the Hyperbolic Annulus Principle
Аннотация
A sufficiently general class of diffeomorphisms of the annulus (the direct product of a ball in \(\mathbb{R}^{k}\), k ≥ 2, by an m-dimensional torus) is studied. The so-called annulus principle, i.e., a set of sufficient conditions under which the diffeomorphisms of the class under study have a mixing hyperbolic attractor, is obtained.
1000-1025
Stability Conditions for Solutions of Multidimensional Completely Integrable Differential Equations
Аннотация
New sufficient tests are given for the stability and asymptotic stability of the zero solution of a nonautonomous completely integrable equation on an arbitrary salient convex closed cone and on a finitely generated cone. The class of Lyapunov functions suitable for studying the asymptotic behavior of solutions of nonautonomous completely integrable equations is significantly extended by substantially weakening the sign negativeness condition, traditional in the Lyapunov second method, for the derivative of the Lyapunov function at the interior points of the cone.
1026-1031
Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval
Аннотация
For the system of root functions of an operator defined by the differential operation −u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L2(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L2 and W2−1, respectively, and may have singularities at the endpoints of G such that q(x) = qR(x) +q′S(x) and the functions qS(x), p(x), q2S (x)w(x), p2(x)w(x), and qR(x)w(x) are integrable on the whole interval G, where w(x) = x(1 − x).
1032-1048
Partial Differential Equations
Complex Cauchy Problem in a Scale of Analytic Functions with Power-Law Singularities
Аннотация
We consider the Cauchy problem for systems of complex linear partial differential equations and obtain necessary and sufficient conditions for this problem to be well posed in a scale of Banach spaces of analytic functions having power-law singularities as the independent variable tends to the lateral surface of a cone.
1049-1056
Homogenization Method in the Problem of Long Wave Propagation from a Localized Source in a Basin over an Uneven Bottom
Аннотация
In the framework of the linearized shallow water equations, the homogenization method for wave type equations with rapidly oscillating coefficients that generally cannot be represented as periodic functions of the fast variables is applied to the Cauchy problem for the wave equation describing the evolution of the free surface elevation for long waves propagating in a basin over an uneven bottom. Under certain conditions on the function describing the basin depth, we prove that the solution of the homogenized equation asymptotically approximates the solution of the original equation. Model homogenized wave equations are constructed for several examples of one-dimensional sections of the real ocean bottom profile, and their numerical and asymptotic solutions are compared with numerical solutions of the original equations.
1057-1072
Total Preservation of the Solvability of the Semilinear Global Electric Circuit Equation
Аннотация
Conditions for the total (over the whole set of admissible controls occurring in the higher coefficient and the right-hand side) preservation of global solvability and uniqueness of the solution are obtained for an initial–boundary value problem related to a control semilinear differential equation of a global electric circuit.
1073-1082
Control Theory
1083-1101
Algebraic Approach to the Stabilization of a Differential System of Retarded Type
Аннотация
For a spectrally controllable linear autonomous systems with commensurable delays, we construct state feedbacks ensuring the complete damping of the original system (finite stabilization) as well as the complete damping of the original system and the asymptotic stability of the closed-loop system (complete stabilization). The spectral reduction and asymptotic stabilization problems are considered as auxiliary problems. The argument is constructive, and the results are illustrated by an example.
1102-1114
Digital Stabilizer Design for a Switched Linear Control Delay System
Аннотация
The problem of designing a digital controller stabilizing a continuous-time switched linear control delay system is studied. The approach to stabilization successively includes the construction of a continuous-time–discrete-time closed-loop system with a digital controller, the transition to its discrete-time model, and the construction of a discrete-time controller by simultaneous stabilization methods.
1115-1124