Time delayed system stability theory in the sense of non-Lyapunov delay independent and delay dependent approach: New results

The article provides sufficient conditions for both practical and finite time stability of linear continuous time delay systems described as ẋ(t) = A₀X(t) + A₁X(t - τ). Considering a finite time stability concept, the new delay independent conditions have been derived using the approach based on the Lyapunov-like functions. These functions do not need to have the properties of positivity in the whole state space and negative derivatives along the system trajectories. When the practical stability has been analyzed the above mentioned approach was combined and supported by the classical Lyapunov technique to guarantee the attractivity property of the system behavior.