**1. Introduction**

The integration of Fractional Differential Equations (FDE) and Systems (FDS) is considered to be a well-founded and approved topic for most fractional calculus researchers. Therefore, the title of the paper appears as an ingenuous and unrealistic objective to revisit an established mathematical result. Nevertheless, our purpose is to provide an objective analysis of this fundamental problem and to formulate a satisfactory solution to fractionalorder initial-value problems.

In fact, the initial-value problem, or Cauchy problem, is obviously trivial in the integer order case [1,2]. On the other hand, the solution of the fractional-order case appears as a generalization of the integer-order one. However, due to the multiplicity of fractional-order derivative definitions, researchers have considered it necessary to adapt the classical approach by referring to a particular derivative and its corresponding initial conditions [3–5]. Practically, most of the time, the Caputo derivative [3,6] is used because its "initial conditions" can be physically interpreted. Many critics have already addressed this choice, based on initialization considerations [7–15]. In those papers, the authors emphasize the inability of the Caputo derivative technique to solve the initialization problem, but, contrary to the history function technique [9,10,16–21] and the infinite-state approach [13,22–24], they do not provide a solution to this problem. Recently, some solutions based on new fractional derivatives (see, for example, [25–28]), which are in fact local derivatives [29,30], have been proposed. Practically, the direct consequence of these multiple choices is that different theoretical free responses are possible for the same FDE/FDS problem, which is of course physically inconsistent.

Our objective in this paper is to prove (in fact to recall) theoretically, using an elementary initial value problem, that the solution predicted by the Caputo derivative approach

**Citation:** Maamri, N.; Trigeassou, J.-C. A Plea for the Integration of Fractional Differential Systems: The Initial Value Problem. *Fractal Fract.* **2022**, *6*, 550. https://doi.org/10.3390/ fractalfract6100550

Academic Editors: Thach Ngoc Dinh, Shyam Kamal, Rajesh Kumar Pandey and Norbert Herencsar

Received: 14 July 2022 Accepted: 19 September 2022 Published: 28 September 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

leads to a false free response. Then, we treat the same example with the frequencydistributed model of the fractional integrator [31–34]. We demonstrate that using a distributed initial condition, in fact that of the fractional integrator, provides the good solution to the considered problem. The conclusion of this analysis is that any fractional-order initial-value problem has to be treated such as in the integer-order case, using essentially the fractional integrator and its distributed initial state or its initialization function. Then, this technique is applied to the modeling of FDE/FDS and the formulation of their transients in the linear case. Two expressions are derived, one using the classic Mittag–Leffler function and a new one based on the definition of a distributed exponential function.

The theory developed in the paper is not a new one, since the first paper [35] related to the fractional integrator was published in 1999. Since that original publication, this research has been applied to the modeling and identification of real-world diffusive processes: electrochemical [36], thermal [37], and rotor skin effect, see chapter 5, volume 1 of [34]. The modeling of fractional systems based on the fractional integrator, known as the infinitestate approach, has been presented in several articles, see, for example, [22,33,38], with a particular focus on system initialization [23,24]. Moreover, it has been applied to the stability analysis of linear and nonlinear systems with a distributed formulation of the Lyapunov function [39]. The theory of the infinite state approach and its applications to various domains of control theory are presented in a two-volume monograph [34]. However, in spite of its contributions to initialization and Lyapunov system stability, this theory is ignored or considered as an exotic contribution to fractional calculus, although it has been adopted by researchers for initialization purposes [21,40–42] and Lyapunov stability analysis [43–48]. Moreover, although the pseudo initial conditions of the Caputo derivative are frequently criticized [8,9,16,20], mainly for their use in system initialization [7,14,15,21], they are still used because they provide apparently simple solutions. Consequently, there is an important challenge to provide a general and satisfactory solution to the initialization problem, using the same approach as in the integer-order case, where the initial conditions are those of the fractional integrator.

Thus, this paper intends to treat the FDE initial-value problem with a new and theoretical presentation of the infinite state approach, demonstrating that we do not have to refer to any fractional derivative and, on the contrary, can focus on the Riemann–Liouville integral and its distributed initial conditions. It is important to note that the authors have privileged a theoretical formalism contrary to their previous publications, where numerical simulations were abundantly used. So, the reader can refer to these previous papers to find numerical illustrations related to initialization. A restricted version of the paper has already been published in a recent conference [49].

The paper is composed of six sections and a conclusion. Section 1 is the introduction. Sections 2–4 present the materials and methods related to initial-value problems and the infinite state approach. In Section 5, an elementary counter-example permits us to invalidate the usual Caputo derivative initial value approach. In Section 6, the frequency-distributed integrator model is used to solve the previous counter-example and to formulate a new approach to the FDE initial-value problem. This methodology is used in Section 6 to express the dynamics of the general FDS initial-value problem.
