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This study uses a delayed matrix exponential to illustrate the solution of both linear and nonlinear delay problems using permutable matrices. We outline several prerequisites that are necessary to ensure the exponential stability of the trivial solution of delay systems. The conditions that are necessary and sufficient for the linear system to possess relative controllability are established by introducing a delayed Grammian matrix‐like criterion. A sufficient condition is obtained by Krasnoselskii's fixed point theorem for the relative controllability of nonlinear systems. Numerical examples provided at the end of the paper demonstrate the validity of our theoretical findings. [ABSTRACT FROM AUTHOR]

Abstract We study the results of existence and continuous dependence on neutral impulsive stochastic differential equations with Poisson jumps. We have also created some conditions confirming exponential stability.

This paper studies the exponential stability for random impulsive non-linear singular differential systems. We established some new sufficient conditions for the proposed singular differential system by using the Lyapunov function method with random impulsive time points. Further, to validate the theoretical results' effectiveness, we finally gave two numerical examples that study with graphical illustration and an additional example involving matrices with complex entries, proving the results to be true in that case as well. [ABSTRACT FROM AUTHOR]

The exponential stability of hybrid event‐triggered singular time‐delay (STD) systems is established in this paper. To ensure their exponential stability, we give sufficient conditions for linear matrix inequalities (LMIs) for STD systems with average impulsive gain. Furthermore, zeno behavior is avoided by carefully designing the settings for event‐triggering conditions. A few numerical examples demonstrate the usefulness of the suggested findings. [ABSTRACT FROM AUTHOR]