By utilizing the following identity for the exponential integral function given in ([43], Eq. (06.34.06.0007.01), when z→0, we get { {E_{1}}}!left z right) propto -ln z) + O(1).
Some global exponential stability criteria are derived for the error system by utilizing the stability analysis of impulsive differential equations and differential inequalities and, moreover, the exponential convergence rate can be specified.
Caraballo and Liu, in [2], have considered the exponential stability in -moment and almost surely exponential stability for mild solution to system (3.24) by utilizing the Gronwall inequality.
By utilizing the stability theory for impulsive functional differential equations, several impulsive control laws are derived to guarantee the exponential synchronization of neural networks with time-varying and distributed delays.
When and, the global exponential stability criteria for (2.2) have been established in [18] by utilizing the Lyapunov functional method.
Specifically, by utilizing the π-equivalent saturation model, we design an observer/controller strategy that achieves semi-global exponential rotor position tracking and only requires stator current, rotor velocity, and rotor position measurements.
By utilizing the Lyapunov stability and impulsive control theory on delayed dynamical networks, some sufficient conditions of exponential synchronization for a switching complex dynamical network are presented.
For instance, in [1], by utilizing the Lyapunov-Krasovkii functional and combining with the linear matrix inequalities (LMIs) approach, the authors analyzed the global exponential stability of neutral-type impulsive NNs.
By utilizing the stability theory of dynamical systems and some analysis techniques such as the Barbalat lemma, several sufficient criteria are obtained to ensure global exponential synchronization for the controlled complex networks of networks.
