Stability Analysis of Distributed Order Fractional Differential Equations

and Applied Analysis 3 fractional order to distributed order fractional. In Section 4, we introduce the distributed order fractional evolution systems C doD α t x t A C doD β t x t Bu t , x 0 x0, 0 < β < α ≤ 1, 1.5 where u t is control vector, and generalize the results obtained in Section 3 for this case. Finally, the conclusions are given in the last section. 2. Elementary Definitions and Theorems In this section, we consider the main definitions and properties of fractional derivative operators of single and distribute order and the Mittag-Leffler function. Also, we recall two important theorems in inverse of the Laplace transform. 2.1. Fractional Derivative of Single and Distributed Order The fractional derivative of single order of f t in the Caputo sense is defined as 16, 27 C soD α t f t 1 Γ m − α ∫ t 0 f m τ t − τ α−m 1 dτ, 2.1 for m − 1 < α ≤ m, m ∈ N, t > 0. The Caputo’s definition has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their integer order which is the case in most physical processes. Fortunately, the Laplace transform of the Caputo fractional derivative satisfies L { C soD α t f t } sαL{f t } − m−1 ∑ k 0 f k 0 sα−1−k, 2.2 wherem−1 < α ≤ m and s is the Laplace variable. Now, we generalize the above definition in the fractional derivative of distributed order in the Caputo sense with respect to order-density function b α ≥ 0 as follows: C doD α t f t ∫m m−1 b α CdoD α t f t dα, 2.3 and the Laplace transform of the Caputo fractional derivative of distributed order satisfies L { C doD α t f t } ∫m m−1 b α [ sF s − m−1 ∑ k 0 sα−1−kf k 0 ] dα