Stabilization of Multi Fractional Order Differential Equation with Delay Time and Feedback Control

: The purpose of this article is to introduce the original results which devoted with the nonlinear control system problems involves of nonlinear differential equations of fractional orders. Thus, this system is described with a mixed of ordinary derivatives in the first and second order that, are unstable before feedback gain. More precisely, we investigate and analysis the nonlinear control system in related to feedback gain matrix. In addition, we prove that the considered system is locally asymptotically stabilizable via certain conditions. Then, this work reinforce through some application examples that programmed for illustrating and showing the stabilizability of the current systems with high efficiency and accuracy.


Introduction
The field of control and systems is currently one of the most important topics that play a good role in simplifying some systems.Thus, the control is involved in nonlinear systems and the interpretation of complex phenomena, which is of great benefit in modernizing human civilization day after day [1].
Fractional calculus contributes to many important aspects such as science, engineering and physical applications.We mention some of its applications with fractional optimal control problems (FOCPs) that are subject to dynamic constraints with the objective function problems, in, bioscience [2], economic [3], and so on .
The stability of a nonlinear Langevin system of Mittag-Leffler (ML)-type fractional derivative affected by time-varying delays and differential feedback control stability has been studied by Zhao in ref. [4].Then, Li and et al. are studying of the global stability problem for feedback control systems of impulsive fractional differential equations on networks [5].Another direction studied by Qasim and et for some classes with composition FOCPs as in [6].The stabilization and destabilization of fractional oscillators via a delayed feedback control has been considered by Čermák et al. in [7].
The Mittag-Leffler stabilization of fractional-order nonlinear systems with unknown control coefficients is verified and examined by Wang in [8].[19], [20] The main objective of this work is to study non-linear systems with multiple fractional orders between zero and one with an ordinary derivative for control systems.The systems that are unstable were examined, then a feedback describes gain matrix the presence of control.So, after that we investigate and demonstrate the local stabilizabiliy with complete accuracy for nonlinear systems.
This outline of paper is organized as follows: Section 2, present some basic preliminaries concept and some auxiliary definitions.In section 3, we obtain the rigorous new results for the multiplying (fractional-one order ordinary) differential nonlinear feedback control system with some applications.Finally, we provide the results that have discovered which focused on the stabilizability problem of non-linear feedback control systems.

Preliminaries
In this section, we will present some important definitions and characterizations which play a good role to achieve the stabilizability concept of the considered system.
And the Mittag-Leffler function with two parameters: , where  ∈  and ,  > 0.

Definition 2.2 [10]:
The Gamma function is defined by the integral formula With the property of Gamma function ( + 1) = ().
In particular, when And

Definition 2.8 [13]:
The power function and the constant function of the Caputo's derivative, is: ii

When
→ ∞, ‖()‖ → 0 For  >  3 {( − ) −1 }.Consequently, the system (1) is asymptotically locally stable.∎ Example 3.2: Consider the following differential nonlinear without feedback control system: This nonlinear control system consists of nonlinear differential equations of fractional orders with a mixed of ordinary derivatives of the first order that are unstable before feedback gain matrix.Thus, we examine this nonlinear control system after applying feedback gain matrix and prove the asymptotic local stabilizability of the system by using the conditions in the theorem (3.1).] The nonlinear control system (25) with feedback takes the following form:

Conclusions
A new outcomes has been explored in nonlinear dynamical systems for double fractional with ordinary order in this paper related to some necessary conditions.Then, the stabilizability of nonlinear systems with control was obtained by feedback gain for the nonlinear systems class.So that, the precise results are obtained in locally case according to the conditions of theorem 3.1, which have demonstrated their accuracy in some applications.Also, as the work technology has been programmed and reinforced with the illustrative examples that have shown the efficiency of the stabilizability of the considered systems.Finally, may be interested to extend the obtained results in this work to the case of regional observer problem in distributed parameter systems as in [17][18].