Stability and robust stability for systems with a time-varying delay

To concern the stability and robust stability criteria for systems with time-varying delays, this note uses not only the time-varying-delayed state x(t-h(t))x(t-h(t)) but also the delay-upper-bounded state x(t-h¯) to exploit all possible information for the relationship among a current state x(t)x(t), an exactly delayed state x(t-h(t))x(t-h(t)), a marginally delayed state x(t-h¯), and the derivative of the state x˙(t), when constructing Lyapunov–Krasovskii functionals and some appropriate integral inequalities, originally suggested by Park (1999. A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Transactions on Automatic Control, 44(4), 876–877). Two fundamental criteria are provided for the cases where no bound of delay derivative is assumed and where an upper bound of delay derivative is assumed. Examples show the resulting criteria outperform all existing ones in the literature.