Vol 23, No 122 (2018)

The report provides semidiscretization diagram for semilinear differential inclusions of fractional order.

In the present paper the method of integral guiding potentials is applied to study the problem of the asymptotic behavior of solutions for a differential inclusion with a causal multioperator. At first we consider the case when the multioperator is closed and convex-valued. Then the case of a non-convex-valued and lower semicontinuous right-hand part is considered.

Basic properties of the system of eigenfunctions for even order functional differential equation under special boundary conditions are obtained. Equivalence of a serie of classical affirmations is established. Among them are the Vallee-Poussin affirmation and positivity of the corresponding quadratic functional.

The paper addresses the test integral Volterra equations of the first kind in integral models of developing systems. Earlier, models consisted of two and three age groups of elements. Here the authors generalized the results to the case of an arbitrary number of groups. Along with the theoretical results, numerical calculations are given for the test example. Calculations illustrate the validity of theoretical estimates.

We consider a linear autonomous neutral functional differential equation. We obtain formulas relating the fundamental solution and the Cauchy function for this equation. On the basis of the formulas the asymptotic behavior of solutions of the equation is studied.

The concept of orderly covers extend to mappings acting from an ordered space X into space Y with a reflexive binary relation. An assertion is obtained about the existence of a solution x∈X of the equation Υx, x = y, where y∈Y ; the mapping Υ :X 2 → Y one by one from the arguments is a covering, and on the other - antitone. An example of a concrete an equation satisfying the assumptions of the proved assertion, to which are not applicable known results, since Y is not an ordered space.

A numerical method for constructing an unknown effect in a nonlinear system of ODEs is considered.

In the present paper we consider a new inequality of Carity type and prove a theorem on a fixed point. Further, relying on the theorem obtained, we study maps (generalized contractions) that compress relative to some function of two vector arguments. This function does not need to be a metric or even continuous.

In the paper, an estimate from above of the fractional Riemann-Liouville derivative of an order α∈0;1 of the composition of two functions is proved for the case when the inner function is assumed only to be represented by the fractional Riemann-Liouville integral of a measurable essentially bounded function. The necessity of such an estimate arises in control problems of dynamical systems described by differential equations with fractional derivatives.

In this paper, we consider the problem of maximizing a sufficiently general functional defined on solutions of the controlled nonlinear Goursat-Darboux system. The right-hand side of the differential equation is a Caratheodory function. We study singular controls in the sense of the pointwise maximum principle, i.e. the controls for which this principle degenerates. We consider strong degeneration of the pointwise maximum principle (this principle is the necessary first-order optimality conditions by using of needle-shaped variation of a control) when the maximum principle degenerates together with second-order optimality conditions. Sufficient conditions for the strong degeneration of maximum principle and necessary conditions for the optimality of corresponding singular controls are given. These conditions generalize the conditions that are known for the case of the terminal quality functional and the case of the smoother right side of the equation.

In this paper we consider the assertion on the estimate of the closeness of the solution of a perturbed inclusion to a preassigned continuous function, and the proof of this assertion is given.

We propose a method for constructing attainability sets for controllable systems with integral constraints on the control and trajectory of the system, which is based on the use of the Pontryagin maximum principle for characterizing the boundary points of the attainable set.