## Exponential stability of nonlinear time-delay systems with delayed impulse effects.(English)Zbl 1233.93080

Summary: The problem of exponential stability for nonlinear time-delay systems with delayed impulses is addressed in this paper. Lyapunov-based sufficient conditions for exponential stability are derived, respectively, for two kinds of delayed impulses (i.e., destabilizing delayed impulses and stabilizing delayed impulses). It is shown that if a nonlinear impulsive time-delay system without impulse input delays is exponentially stable, then under some conditions, its stability is robust with respect to small impulse input delays. Moreover, it is also shown that for a stable nonlinear impulsive time-delay system, if the magnitude of the delayed impulses is sufficiently small, then under some conditions, the delayed impulses do not destroy the stability irrespective of the sizes of the impulse input delays. The efficiency of the proposed results is illustrated by three numerical examples.

### MSC:

 93D20 Asymptotic stability in control theory 93C10 Nonlinear systems in control theory 93D09 Robust stability 93C15 Control/observation systems governed by ordinary differential equations

### Keywords:

impulsive systems; delayed impulses; exponential stability
Full Text:

### References:

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