*2.2. The FDE/FDS Initial Value Problem*

Let us consider the elementary FDE

$$D\_t^n(\mathbf{x}(t)) = f(\mathbf{x}(t), u(t)) \, 0 < n < 1,\tag{6}$$

where *n* is the fractional order and *x*(*t*) = *x*(0) at *t* = 0.

Contrary to the integer-order case, several approaches are derived from the fractional derivative definitions of *D<sup>n</sup> <sup>t</sup>* (*x*(*t*)). The main popular ones are the Caputo and Riemann–Liouville derivatives [3].

Practically, Equation (6) is integrated with the Caputo derivative definition, since its initial condition is considered equal to *x*(0).

Then, in order to prove the existence and the uniqueness of the solution *x*(*t*) of (6), Picard's method [3–5] is frequently used.

In the linear multidimensional commensurate order case, Equation (6) becomes:

$$\begin{cases} D\_t^n(\underline{x}(t)) = A\underline{x}(t) + Bu(t) \\ y(t) = \mathbb{C}x(t) \end{cases} \quad 0 < n < 1,\tag{7}$$

where *x*(*t*) -*RN*, and *A* and *B* are matrices of appropriate dimensions.

The general solution of (7), expressed in terms of the Mittag–Leffler matrix function [53]

$$\Phi(t) = E\_{n,1}(At^n), \ \Phi(t) \in R^{N \times N}$$

is

$$\underline{\mathbf{x}}(t) = \Phi(t)\underline{\mathbf{x}}(0) + \int\_{0}^{t} \Phi(t-\tau)B\tilde{u}(\tau)d\tau \quad \text{with} \quad \tilde{u}(\tau) = D^{1-n}(u(\tau)). \tag{8}$$

As mentioned in the introduction, the main objective of this paper is to revisit the integration of the FDE/FDS initial-value problem, using the infinite-state approach, which is directly related to the integer order ODE case and does not need any derivative definition, as it is exhibited in Section 4.
