We address Lyapunov differential equation construction and new conditions for integral Mittag-Leffler asymptotic stability of linear fractional-order time-varying (LFOTV) systems associated with different orders and random initial time put on the real axis. The first ingredient establishes new sufficient conditions dealing with linkage to the coefficient matrix and an elementary integral promising a final value at \(\infty \) and gives new notions of integral asymptotic stability of zero solutions to such systems. The second ingredient enables a novel platform to construct a time-varying quadratic Lyapunov function by solving a Lyapunov differential equation (LDE) for such fractional-order systems. The third ingredient introduces a generalized concept of integral Mittag-Leffler asymptotic stability to LFOTV systems. The fourth ingredient tackles the issue of random initial-time initialization, which has been considered in a control problem that deals with the stabilization of linear fractional-order time-varying systems. For computational demonstration, fractional-order Mathieu’s system has served as an example to show the novel significance of the theoretical result controlling state responses.
