ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Boundedness of solutions of some class of linear systems

### Author(s):

Boris Filippovich Ivanov

Head of the Department of Higher Mathematics
Saint Petersburg State University of Industrial Technologies and Design
Higher School of Technology and Energy,
Str. Ivan Chernykh, 4, 198095 Saint Petersburg, Russia.
PhD in physics and mathematics, Associate Professor

ivanov-bf@yandex.ru

### Abstract:

We consider a linear nonhomogeneous system of differential equations. In this system, the matrix-coefficient in the linear part is the sum of an absolutely summable matrix and a matrix summable with the degree which greater one but not exceeding two. It is also assumed that there exists a neighborhood of zero in which the Fourier transform of each element of the second matrix equals zero. The nonhomogeneity is assumed to be summable with any degree which greater than one or essentially bounded, has a bounded integral, and the sum of the reciprocal of the summability of the second matrix and the nonhomogeneity is less than one. In previous works we introduced the concepts of the set of resonant points, resonant and non-resonant conditions for functions summable with some degree or essentially bounded. This paper proposes a criterion of boundedness of solutions in the form of conditions on the resonant points of the coefficients of the system. It was proved that if for a number of coefficients of the system non-resonant conditions (which for finite resonance points have the form of arithmetic relations between their coordinates, i.e resonant frequencies) are fulfilled, then each solution is bounded. In addition, it has been established that in the resonance case it is always possible to choose such an arbitrarily small perturbations of the coefficients of the system (in the sense of the norms of the corresponding spaces) that the resonance sets of the system coefficients will not increase, the resonance condition will be satisfied but the perturbed system will have unbounded solutions.

### Keywords

• boundedness of solutions of linear systems
• resonance

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