New insight to multi-order Mittag-Leffler stability and Lyapunov theorems for random initialization time fractional order systems

We consider fractional order systems associated with different orders and random initialization time placed on a real number line. We introduce a new concept of multi-order Mittag-Leffler stability and formulate new proofs to Lyapunov stability theorems for random initialization time fractional order systems. The new theorems give way to measuring fractional derivatives of scalar Lyapunov functions and enable a pathway to estimate decay associated with trajectories of such systems. A few examples that deal with applications of interest have been discussed.