Asymptotic behavior analysis of a coupled time-varying system: application to adaptive systems

Asymptotic behavior of a partial state of a coupled ordinary and/or partial differential equation is investigated. It is specifically shown that if a signal x(t) is a solution to a dynamic system existing for all t⩾0 in an abstract Banach space and pth (p⩾1) power integrable, then x(t)→0 as t→∞. The system is allowed to be nonautonomous and assumes the existence of a Lyapunov function. Since the derivative of the Lyapunov function is negative semidefinite, stability or uniform stability in the sense of Lyapunov would be concluded. However, this paper further asserts that the partial state which remains in the time derivative of the Lyapunov function converges to zero asymptotically