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 Vladikavkaz. Mat. Zh., 2020, Volume 22, Number 1, Pages 49–65 (Mi vmj714)

Stability of impulse systems of two linear Ito differential equations with delay

a Dagestan State University, 43 a M. Gadzhiyeva St., Makhachkala 367000, Russia
b Dagestan Federal Research Center of the Russian Academy of Sciences, 45 M. Gadzhiyeva St., Makhachkala 367032, Russia

Abstract: The problems $2p$-stability $(1 \le p < \infty)$ of systems of two linear Ito differential equations with delay and impulse impacts on one component of solutions are studied on the base of the theory of positively reversible matrices. Ideas and methods developed by N. V. Azbelev and his followers to study the stability problems of deterministic functional-differential equations are applied for this purpose. Sufficient conditions for the $2p$-stability and exponential $2p$-stability of systems of two linear Ito differential equations with delay and impulse impacts on one component of solutions are given in terms of positive reversibility of the matrices constructed from the parameters of the original systems. The validity of these conditions is checked for specific equations. Sufficient conditions for exponential moment stability of a system of two deterministic linear differential equations with constant delay and coefficients with pulse influences on one component of solutions are received in terms of parameters of this system. It is shown that in this case from the general statements it is possible to receive new results for the studied system.

Key words: Ito's equations, stability of solutions, impulse impacts, positive invertibility of a matrix.

DOI: https://doi.org/10.23671/VNC.2020.1.57571

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UDC: 517.929.4+519.21
MSC: 34K20, 34K50

Citation: R. I. Kadiev, “Stability of impulse systems of two linear Ito differential equations with delay”, Vladikavkaz. Mat. Zh., 22:1 (2020), 49–65

Citation in format AMSBIB
\Bibitem{Kad20} \by R.~I.~Kadiev \paper Stability of impulse systems of two linear Ito differential equations with delay \jour Vladikavkaz. Mat. Zh. \yr 2020 \vol 22 \issue 1 \pages 49--65 \mathnet{http://mi.mathnet.ru/vmj714} \crossref{https://doi.org/10.23671/VNC.2020.1.57571}