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Том 53, № 5 (2017)
- Год: 2017
- Статей: 14
- URL: https://journal-vniispk.ru/0012-2661/issue/view/9311
Ordinary Differential Equations
Positive invertibility of matrices and stability of Itô delay differential equations
Аннотация
We study the global exponential p-stability (1 ≤ p < ∞) of systems of Itô nonlinear delay differential equations of a special form using the theory of positively invertible matrices. To this end, we apply a method developed by N.V. Azbelev and his students for the stability analysis of deterministic functional-differential equations. We obtain sufficient conditions for the global exponential 2p-stability (1 ≤ p < ∞) of systems of Itô nonlinear delay differential equations in terms of the positive invertibility of a matrix constructed from the original system. We verify these conditions for specific equations.
571-582
Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential
Аннотация
For the self-adjoint Schrödinger operator ℒ defined on ℝ by the differential operation −(d/dx)2 + q(x) with a distribution potential q(x) uniformly locally belonging to the space W2−1, we describe classes of functions whose spectral expansions corresponding to the operator ℒ absolutely and uniformly converge on the entire line ℝ. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.
583-594
595-606
Perturbation of Fredholm eigenvalues of linear operators
Аннотация
We use the reduction method, which allows one to reduce the study of perturbations of multiple eigenvalues to perturbations of simple eigenvalues, to analyze the general perturbation problem for Fredholm points of the discrete spectrum of linear operator functions analytically depending on the spectral parameter. The same method is used to study a perturbation of multiple Fredholm points of the discrete Schmidt spectrum (s-numbers) of a linear operator. We present an example of a problem on a perturbation of the domain of the Sturm–Liouville problem for a second-order differential operator.
607-616
Partial Differential Equations
Fundamental solution of the elasticity theory equations in displacements for a transversely isotropic medium
Аннотация
We consider a linear fourth-order elliptic partial differential equation describing the displacements of a transversely isotropic linearly elastic medium. We find the symmetries of this equation and of the inhomogeneous equation with the delta function on the right-hand side. Based on the symmetries of the inhomogeneous equation, we construct an invariant fundamental solution in elementary functions.
617-622
Boundary value problem for a second-order elliptic equation in the exterior of an ellipse
Аннотация
We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.
623-629
630-645
Harnack inequality for the solutions of the p-Laplacian with a partially Muckenhoupt weight
Аннотация
We consider a class of quasilinear elliptic second-order equations of divergence structure admitting uniform degeneration in the domain. We prove that the classical Harnack inequality fails and establish a Harnack inequality corresponding to the equation in question.
646-657
658-664
665-676
Solvability of the Dirichlet problem for a mixed-type equation of the second kind
Аннотация
We obtain sufficient conditions for the solvability of the Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain. The solution is represented by a convergent series constructed from the problem data. Some cases of nonuniqueness of the solution are described.
677-685
Control Theory
Reduction of systems to a form with relative degree using dynamic output transformation
Аннотация
A form with the extraction of the zero dynamics is the most convenient canonical form of a linear time-independent multivariable control system. Only systems with vector relative degree can be reduced to such a form. There exist control systems that, together with any system obtained from them by a time-independent change of outputs, have no relative degree. To ensure the relative degree conditions, we suggest to use an invertible dynamic change of measured outputs of the system, which allows one to solve the problem on the reduction of a linear time-independent MIMO-system to a form with relative degree in the most general case.
686-700
Short Communications
On multidimensional difference operators and equations
Аннотация
We study the solvability of multidimensional difference equations in Sobolev–Slobodetskii spaces. In the simplest model case, we describe the solvability picture for such equations. In the general case, we present conditions for the Fredholm property and a theorem on the zero index.
701-703
Spectral problem for a triple differentiation operator with asymmetric weight
Аннотация
We study the spectrum of a class of two-point boundary value problems for an ordinary differential operator of triple differentiation with weight. We show that the spectrum of a problem with asymmetric weight and with periodic boundary conditions fills the entire complex plane. We present an example of a problem with asymmetric weight to which one cannot assign a given spectrum by changing only one of the boundary conditions.
704-707