Abstract
The main objective of this paper is to further investigate the exponential stability of a class of impulsive delay differential equations. Several new criteria for the exponential stability are analytically established based on Razumikhin techniques. Some sufficient conditions, under which a class of linear impulsive delay differential equations are exponentially stable, are also given. An Euler method is applied to this kind of equations and it is shown that the exponential stability is preserved by the numerical process.
Highlights
Impulsive differential equations arise widely in the study of medicine, biology, economics, engineering, and so forth
There is a little work done on exponential stability for impulsive differential delay equations (IDDEs) by the Lyapunov-Razumikhin method
There are a few papers on numerical methods of impulsive differential equations
Summary
Impulsive differential equations arise widely in the study of medicine, biology, economics, engineering, and so forth. There are a few papers on numerical methods of impulsive differential equations. In [20, 21], the authors studied the stability of Runge-Kutta methods for linear impulsive ordinary differential equations. In [4], Ding et al studied the convergence property of an Euler method for IDDEs. In [18], asymptotic stability of numerical solutions and exact solutions of a class of linear IDDEs was studied by the property of DDEs without impulsive perturbations. The results obtained are applied to a class of linear IDDEs. In the last section, we prove that the Euler method for the linear IDDEs preserves the analytic exponential stability
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