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Adaptive Finite Time Stability of Delayed Systems With Applications to Network Synchronization
The finite time stability (FnTSta) theory of delayed systems has not been set
up until now. In this paper, we propose a two-phases-method (2PM), to achieve
this object. In the first phase, we prove that the time for norm of system
error evolving from initial values to 1 is finite; then in the second phase, we
prove that the time for norm of the error evolving from 1 to 0 is also finite,
thus FnTSta is obtained. Considering the cost and complexity of controller, we
use only two simple terms to realize this aim. For the proposed 2PM, time
delays can be the same or asynchronous, bounded or unbounded, etc; the norm can
be -norm, -norm, -norm, which show that 2PM is powerful and has a
wide scope of applications. Furthermore, we also prove the adaptive finite time
stability (AFnTSta) theory of delayed systems using 2PM. As an application of
the obtained FnTSta theory, we consider the finite time outer synchronization
(FnTOSyn) for complex networks with asynchronous unbounded time delays,
corresponding criteria are also obtained. Finally, two numerical examples are
given to demonstrate the validity of our theories