Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods

Abstract

a b s t r a c tIn this paper, we consider the input-to-state stability (ISS) of impulsive control systems with and without time delays. We prove that, if the time-delay system possesses an exponential Lyapunov-Razumikhin function or an exponential Lyapunov-Krasovskii functional, then the system is uniformly ISS provided that the average dwell-time condition is satisfied. Then, we consider large-scale networks of impulsive systems with and without time delays and prove that the whole network is uniformly ISS under the small-gain and the average dwell-time condition. Moreover, these theorems provide us with tools to construct a Lyapunov function (for time-delay systems, a Lyapunov-Krasovskii functional or a Lyapunov-Razumikhin function) and the corresponding gains of the whole system, using the Lyapunov functions of the subsystems and the internal gains, which are linear and satisfy the small-gain condition. We illustrate the application of the main results on examples.

Constructions of ISS-Lyapunov functions for interconnected impulsive systems

“…Also we prove, that if all subsystems possess exponential ISS Lyapunov functions, and the gains are power functions, then the exponential ISS Lyapunov function for the whole system can be constructed. This generalizes [4,Theorem 4.2], where this result for linear gains has been proved.…”

Stability of uncertain impulsive complex-variable chaotic systems with time-varying delays

“…Therefore, the impulsive control has been widely used to stabilize and synchronize chaotic systems [1][2][3][4][5][6][7][8][9][10][11][12], and the references cited therein.…”

Exponential stability via event-triggered impulsive control for continuous-time dynamical systems

“…Thus, if the comparison system (19) is GUAS, then we have lim k→∞ z(k) = 0. Then by (22) and (12), we get for all t ∈ (t k ,t k+1 ],…”

“…In the case of delayed ETIC, the CDS becomes an impulsive system with delayed impulses. Here, we use the comparison method instead of the method of Lyapunov-Krasovskii function (LKF) (see [17][18][19][20]) to derive GUES criteria for this impulsive system with delayed impulses. Since the CDS is general and its vector ¿eld may be uncertain or known incompletely, this brings dif¿culty in using the LKF approach.…”