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Vol 52, No 1 (2016)
- Year: 2016
- Articles: 12
- URL: https://journals.rcsi.science/0012-2661/issue/view/9210
Ordinary Differential Equations
Methods for studying the asymptotic properties of systems with two linear delays
Abstract
We study methods for the instability analysis of neutral-type systems with two delays linearly depending on the argument. In the case of instability of the neutral part of a system with constant coefficients, we suggest a stabilization algorithm. In addition, by using the Laplace transform, we obtain sufficient conditions for the instability of the solution of a given system.
1-17
Quadratic functionals and nondegeneracy of boundary value problems on a geometric graph
Abstract
Quadratic functionals defined on the space of functions differentiable on a geometric graph are considered. Analogs of the Lagrange and Dubois–Raymond lemmas are proved. Necessary extremum conditions for these quadratic functionals are obtained. A boundary value problem with conditions posed locally at the vertices of a geometric graph is shown to be selfadjoint if and only if it is generated by a quadratic functional. A subclass of quadratic energy functionals is singled out. The space of solutions of the homogeneous boundary value problem generated by a quadratic energy functional is described, and nondegeneracy criteria for such boundary value problems are derived.
18-27
Asymptotic expansion of the solution of a matrix differential-difference equation of the general form
Abstract
We obtain the asymptotic expansion of the solution of an inhomogeneous matrix differential-difference equation that belongs to the retarded or neutral type. The case of unbounded delays is considered. We take into account the influence of roots of the characteristic equation. An integral estimate of the remainder with a semimultiplicative weight is obtained depending on the semimultiplicative moment of the free matrix term in the equation.
28-38
Partial Differential Equations
Criterion for the solvability of the cauchy problem for an abstract Euler–Poisson–Darboux equation
Abstract
We consider the Cauchy problem for an abstract Euler–Poisson–Darboux equation in a Banach space and prove a necessary and a sufficient condition for the solvability of this problem. The conditions are stated in terms of an estimate for the norm of a fractional power of the resolvent and its derivatives. The properties of solutions are established, and examples are given.
39-57
Repeated quantization method and its applications to the construction of asymptotics of solutions of equations with degeneration
Abstract
We study the asymptotics of solutions of homogeneous nth-order differential equations with a cusp degeneration for the case in which the principal symbol has multiple roots. We describe a new method for constructing the asymptotics, which we call the repeated quantization method. Examples of application of the method are given.
58-75
Dynamics of solutions of the Cauchy problem for semilinear parabolic stochastic partial differential equations with power-law singularities
Abstract
We prove a comparison theorem for bounded solutions of the Cauchy problem for stochastic partial differential equations of the parabolic type with linear leading part. The drift and diffusion coefficients have locally bounded derivatives with respect to the state variable. We use this comparison theorem to study the dynamics of solutions of an equation with an absorber and an equation with a source.
76-83
84-91
Control Theory
Finite spectrum assignment and complete damping of a differential system of the neutral type by a single controller
Abstract
For a linear autonomous system of the neutral type with commensurable delays, we construct a dynamic feedback that ensures a complete damping of the original system and an asymptotically stable finite spectrum of the closed-loop system. The results are illustrated by examples.
92-110
On a majorant–minorant criterion for the total preservation of global solvability of distributed controlled systems
Abstract
For distributed controlled systems that can be represented by a functional-operator equation of the Hammerstein type with an additional term on the right-hand side in the form of a linear operator (an extended equation of the Hammerstein type), we prove a criterion for the total (over the entire set of admissible controls) preservation of global solvability. In this connection, we develop an earlier-proved majorant–minorant criterion for the total preservation of global solvability. As an example, we study a controlled semilinear integro-differential equation describing radiation transport with Compton scattering diagram. The choice of this example is explained, in particular, by the fact that the preceding version of the majorant–minorant criterion is not applicable to it.
111-121
Short Communications
On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator
Abstract
We study a nonlocal problem for a fractional partial differential equation with the Dzhrbashyan–Nersesyan fractional differentiation operator. By separation of variables, we prove a theorem on the existence and uniqueness of a regular solution of this problem.
122-127
128-132
Mixed problem for a system of fractional partial differential equations
Abstract
We study a mixed boundary value problem in the general setting for a system of Riemann–Liouville fractional partial differential equations with constant matrix coefficients. By using a system of Volterra integral equations of the second kind, we reduce the problem to a special case for which the solution was earlier constructed in terms of the Green matrix. Existence and uniqueness theorems are proved for the problem in question.
133-138