Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations☆
Introduction
It is well-known that fractional differential equation is an alternative mathematical model corresponding to integer differential equation, which provides an excellent tool for the description of memory and hereditary properties of various materials and processes [1], [2], [3], [4], [5], [6]. Numerous research papers on basic theory analysis for fractional differential equations as well as control problems are investigated by many researchers, see for example, [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] and the references cited therein.
Finite time stability (FTS) is concerned with the behavior of systems over a finite intervals, which is not like Lyapunov stability, asymptotic stability, and exponential stability that are concerned with the behavior of systems within an infinite time interval. Recently, FTS of fractional differential equations has recently received a lot of attention and now constitutes a significant branch of automatic control. It is remarkable that Lazarević [31] initially investigate the finite time stability of linear fractional delay differential equations wicontrols. Some sufficient conditions to guarantee the FTS are established. Thereafter, FTS of fractional differential equations has been studied in [32], [33], [34], [35], [36], [37], [38] with the help of linear matrix inequality and Gronwall’s integral inequality.
Recently, there is a quick development on seeking the explicit formula of solution to delay differential/discrete equations via the concept of continuous/discrete delayed exponential matrix [39], [40]. One of the most advantages of continuous/discrete delayed exponential matrix is used to transfer the classical idea to represent the solution of linear ODEs into linear delay differential/discrete equations. For more continued contributions, one can refer to existence and stability of solutions to several classes of delay differential/discrete equations [41], [42], [43], [44], [45], [46], [47], [48], [49], [50] and some relative controllability problems [51], [52], [53], [54].
Very recently, Li and Wang [55] initially study the following linear homogenous fractional delay differential equations: where is the Caputo derivative of order α ∈ (0, 1) and of lower limit zero, denotes constant matrix, for a fixed τ is a fixed delay time. We introduced the notation of delayed Mittag-Leffler type matrix (see [55, Definition 2.3] or Definition 2.5) for (1), which is an extension of one parameter Mittag-Leffler matrix function . Then, the solution of (1) can be given by (see [55, Theorem 3.2]) Moreover, sufficient conditions to guarantee finite time stability of (1) via are also presented.
Motivated by Li and Wang [55], the first objective of this paper is to seeking representation of solutions of the following linear nonhomogeneous fractional delay differential equations: For this purpose, we introduce the notation of delayed two parameters Mittag-Leffler type matrix (see Definition 2.4), which is an extension of classical two parameters Mittag-Leffler matrix function α > 0, .
Thereafter, we study existence and FTS of the following nonlinear fractional delay differential equations: Existence and FTS results for solutions of (4) are given by using delayed one parameter and two parameter Mittag-Leffler type matrixes and singular Gronwall inequality.
The rest of the paper is organized as follows. In Section 2, we recall some necessary notations, definitions and give some necessary estimation on delayed one parameter and two parameter Mittag-Leffler type matrixes. In Section 3, we give the explicit formula of solutions to linear nonhomogeneous fractional delay differential equations. In Section 4, sufficient conditions ensuring the existence and uniqueness of solutions is presented by using classical contraction mapping principle. In Section 5, sufficient conditions on FTS of nonlinear delay equations are established. An example is given to illustrate our theoretical results in final section.
Section snippets
Preliminaries
Throughout the paper, we denote and which are the Euclidean vector norm and matrix norm, respectively; yi and aij are the elements of the vector y and the matrix A, respectively. Denote by the Banach space of vector-value continuous function from endowed with the norm for a norm ‖ · ‖ on . We introduce a space . In addition, we note .
We recall definitions of
Representation of solutions to (3)
In this section, we seek the explicit formula of solutions to linear nonhomogeneous fractional delay differential equations by adopting the classical ideas to find solution of linear fractional ODEs.
In [55, Theorem 3.2], the formula of solutions to linear homogeneous fractional delay differential equations (1) is given in (2). To achieve our aim, we need to find a special solution to (3). The next theorem will show that how to find this special solution.
Theorem 3.1 A solution of (3)
Existence of solutions to (4)
Definition 4.1 A function is called the solution of (4) if it satisfies the following integral equation
We introduce the following assumptions:
[H1] and there exists a such that for any x ∈ J and all .
[H2] where k ∈ Λ is a fixed number.
Denote . Now we are ready to present the following existence and uniqueness result.
Theorem 4.2 Assume that [
Finite time stability results for (4)
In this section, we continue to study FTS of (4). We give the basic definition of FTS as follows.
Definition 5.1 (see [32]) Let y be a solution of (4). We say (4) is finite time stable with respect to {0, J, τ, δ, η} if and only if ‖φ‖C < δ implies ‖y(x)‖ < η, ∀ x ∈ J, where is the initial time of observation, δ, η are real positive numbers and δ < η.
We impose the following assumptions:
[H3] There exists a such that ‖f(x, y)‖ ≤ ω(x), for x ∈ J and .
[H4] There exists a
An example
In this section, we give an example to demonstrate the validity of our theoretical results.
Let and . Consider where
A solution of (16) can be expressed in the following form:
Obviously, for
Acknowlegments
The authors thanks the referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper. The authors thank the help from the editor too.
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