Study Features of Asymptotic Properties and Calming of Some Systems with Linear Delay

This paper proposes methods for obtaining sufficient conditions for asymptotic stability and instability for systems of differential equations containing linear delay. By changing the argument, such systems are reduced to systems with constant delay, which, however, contain an exponential factor on the right-hand side, i.e., are not Lipschitz continuous. Based on these conditions, several systems of linear differential equations are studied, and for one of them, an algorithm for stabilization over an infinite time interval is proposed. It is shown that similar sufficient conditions are also valid for singularly perturbed systems with constant delay and a small parameter at the derivative. Based on the obtained asymptotic stability conditions, an algorithm for stabilizing certain systems with linear delay is proposed. This algorithm can also be used for calming the singular systems under study. Sufficient conditions for asymptotic stability are obtained for systems of differential equations containing linear delay. Systems of this type are encountered in problems in mechanics, physics, biology, economics, and queuing theory. Based on the stability and instability conditions obtained by the authors, the asymptotic behavior of certain systems of linear differential equations is studied, with one of them stabilized over an infinite time interval.

Keywords

instability; asymptotic stability; exponential estimate; first approximation; stabilization.

References

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