Incremental stability and contraction via impulsive control for continuous-time dynamical systems

Abstract This paper investigates the incremental stability and contraction via impulsive control for continuous-time dynamical systems (CDS). By using the methods of Lyapunov-like function and average dwell-time (ADT), criteria are established for incremental stability including incremental global asymptotic stability ( δ GAS) and incremental global exponential stability ( δ GES). The notion of rate of control is proposed and the ADT condition is shown to be equivalent to the condition via the inferior limit of rate of control. And it is shown that the δ GES via impulsive control is robust with respect to the time-delays and data dropout. The results are then applied to time-varying Lipschitz-type CDS which can achieve δ GES via a linear impulsive control. Moreover, the notion of global contracting via impulsive control is proposed for CDS. The global contracting via impulsive control is proved to be equivalent to the δ GES via impulsive control. Some less conservative criteria are derived for global contracting/ δ GES via impulsive control. And the equivalence is used to derive a byproduct of δ GES and global contracting for CDS with destabilizing impulses. Finally, three examples with numerical simulations are presented for illustrations.