Practical stability for Riemann–Liouville delay fractional differential equations

In this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several sufficient conditions for practical stability in time are obtained using Lyapunov functions and the modified Razumikhin technique. Two types of derivatives of Lyapunov functions are used. Some examples are given to illustrate the introduced definitions and results.

and economics. In the case of zero initial conditions the RL, Grünwald-Letnikov (GL) and Caputo fractional derivatives coincide [25]. For this reason, some authors either study Caputo derivatives, or use RL derivatives but avoid the problem of initial values of fractional derivatives by treating only the case of zero initial conditions. This leads to the consideration of mathematical correct problems, but without taking into account the physical nature of the described process. As mentioned sometimes, such as in the case of the impulse response, nonzero initial conditions appear (see, for example, [13]).
In connection with the main idea of stability properties we will consider in this paper nonzero initial conditions for RL fractional equations and we will define in an appropriate way practical stability properties which are slightly different than those for Caputo fractional differential equations. Note stability properties of delay differential equations can be considered by an application of the Lyapunov-Krasovskii method by functionals or by the Razumikhin method by Lyapunov functions. Both of the above mentioned methods are applied for the stability study of Caputo fractional delay differential equations in the literature (see, for example, [2,3,7,8,10], respectively).
In the case of delay fractional differential equations with the RL fractional derivative, following the idea of initial conditions in ordinary delay differential equations and the above-mentioned idea concerning the initial condition without any delay for RL fractional differential equations we will set up initial conditions in an appropriate way. Note any solution of the defined initial conditions with RL fractional derivatives is not continuous at zero (the initial point) which is the same as in the case without any delay. Delay RL fractional differential equations are set up and studied in [19] but the initial condition does not correspond to the idea of the case of delay differential equations with ordinary derivatives (the lower bound of the RL fractional derivative coincides with the left end side of the initial interval). The existence of the solution of RL fractional differential equations was studied in [1] but the initial condition as well as the integral presentation does not correspond to the RL fractional derivative.
Asymptotic stability for RL fractional differential equations with delays was studied in [9,20,22] but only the autonomous case is considered. Also a Lyapunov functional and its integer order derivative is applied. This functional is similar to the one used in the theory of differential equations with ordinary derivative and delay. On one hand the application of the ordinary derivative of the Lyapunov functional is not similar to the fractional derivatives used in the equation and on the other hand it leads to some restrictions on both the delay and the right-hand side parts of the equation ( [21]). Additionally, in [9] the initial condition is not adequately associated with the RL fractional derivative. RL fractional equations with delays were studied recently in [4,5] but there are unclear parts in the statement of the problem (the lower limit of the RL derivative is different than the initial time point) as well as in the initial condition (the RL fractional integral has no meaning, compare with [24] at the initial time). The Razumikhin method is applied to RL fractional differential equations in [10] but the initial condition is not connected with the RL fractional derivative.
There are many papers in the literature which study various types of stability of solutions of differential equations via Lyapunov functions. One type of stability, useful in real world problems, is the so called practical stability (see the book [18] for the basic definitions and applications to differential equations with ordinary derivatives). In this paper, a modified practical stability is defined and we call it practical stability in time. This stability is studied by Lyapunov functions and the modified Razumikhin technique. In connection with the application of Lyapunov functions to fractional equations it is necessary to define in an appropriate way the derivative of Lyapunov function among the studied fractional differential equations. Two different types of derivatives of Lyapunov functions among the studied fractional differential equations are applied. Several sufficient conditions for practical stability in time are obtained by the application of these derivatives. Some examples illustrating the definitions and the results are provided.

Notes on fractional calculus
We will give the main definition of fractional derivatives used in the literature (see, for example, [11,12,24]). We will give these definitions for scalar functions. Throughout the paper we will assume q ∈ (0, 1).
-The Grünwald-Letnikov fractional derivative is given by and the Grünwald-Letnikov fractional Dini derivative by and t h denotes the integer part of the fraction t h . [11,24]). Note the fractional derivatives for scalar functions could be easily generalized to the vector case by taking fractional derivatives with the same fractional order for all components.
First we note a known result from the literature (the space C 1−q will be defined below).

Statement of the problem and basic definitions
Let τ > 0 be a given number. Define the classes Similarly, we define Let ρ > 0 be a given number. Consider the following set Consider the following system of nonlinear Riemann-Liouville fractional delay differential equations (RLFrDDE) of fractional order q ∈ (0, 1) : with initial conditions Remark 3.1 According to Proposition 2.3 the second equation in the initial conditions (2) could be replaced by the equality 0 I We give a definition for various types of practical stability of the zero solution of RLFrDDE (1). In our further considerations below we will assume the existence of the solution of the IVP for RLFrDDE (1), (2) and we will denote it by x(t; φ) ∈ C 1−q ([0, ∞), R n ) (some existence results for RL fractional differential equations were obtained in [6,15,17]).
Note practical stability for differential equations with ordinary derivatives is defined in [18]. Now, using the classical ideas of practical stability and taking into account the meaning of RL fractional derivative, the required initial condition at the initial time, we will introduce a modified practical stability for RLFrDDE (1).

Definition 3.2 Let positive constants λ,
The practical stability in time is closer to boundedness in time than the stability in time. We will illustrate it on an example without delay because it is easier to obtain the exact solution.

Example 3.3 Consider the RL fractional differential equations
where u ∈ R. The solution of (3) is The zero solution is not stable because (1) is practically stable in time with respect to (λ, A), then it is practically stable in time with respect to (λ, B) with B > A, but we are interested in the smallest possible value of A.

Lyapunov functions and their derivatives among nonlinear RL delay fractional differential equations
One approach to study practical stability in time of RLFrDDE (1) is based on using Lyapunov functions. The first step is to define a Lyapunov function. The second step is to define its derivative among the fractional equation.
We will use the class of Lyapunov functions.
Definition 4.1 Let J ∈ R + be a given interval, and ⊂ R n be a given set. We will say that the function V (t, x) ∈ C(J × , R + ) belongs to the class (J, ) if it is locally Lipschitz with respect to its second argument.
In connection with the RL fractional derivative it is necessary to use appropriate derivative of Lyapunov functions among the studied equation. We will use two types of derivatives of Lyapunov functions from the class (J, ) to study the practical stability properties of RL fractional differential equations (1): where We will illustrate the above-defined derivatives on some simple examples of Lyapunov functions.
be a solution of the IVP for RLFrDDE (1), (2). Then the RL fractional derivative of V (t, x) is given by Note it is difficult to obtain the RL fractional derivative in the general case for any solution of (1).
Case 2. Dini fractional derivative. Let φ ∈ C 0 with n = 1 and t > 0. Then applying (5) we obtain Comparing with the RL fractional derivative of V , the Dini fractional derivative is easier to obtain.

Example 4.3 (multi-dimensional case) Consider the Lyapunov function
. . , φ n ) and t > 0. Then similar to Example 2 the Dini fractional derivative is

Main results
We will study practical stability properties of the RL delay fractional nonlinear system (1). We will use the above defined fractional derivatives of Lyapunov functions.

RL fractional derivative of Lyapunov functions
Theorem 5.1 Let the following conditions be satisfied: holds where a ∈ K; (ii) there exists a positive number λ > 0 such that for any function y holds with b ∈ K; (iii) for any initial function φ ∈ S λ and the corresponding solution Then the RLFrDDE (1) is practically stable in time w.r.t. (λ, a −1 (b(λ))).
According to condition 1(iii)

Lemma 5.2
Assume the following conditions are satisfied: holds with b ∈ K; (ii) there exists a number C > 0 such that for any point t > 0 such that (t + s) 1 holds.

Theorem 5.3
Let the following conditions be satisfied: holds where a ∈ K; (ii) there exists a number λ ≥ 0 such that for any function y holds with b ∈ K; (iii) there exists a number C > 0 such that for any initial function φ ∈ S λ and the corresponding solution holds.

Dini fractional derivative of Lyapunov functions
We will use Dini fractional derivative defined by (5) to obtain some sufficient conditions for practical stability in time of the RLFrDDE(1).

Theorem 5.4
Let the following conditions be satisfied: 1. There exist a function V ∈ ([0, ∞), R n ) such that: (i) there exists a number T > 0 such that the inequality holds where a ∈ K; (ii) there exists a positive number λ > 0 such that for any function y ∈ holds with b ∈ K; (iii) for any initial function φ ∈ S λ and the corresponding solution Then the RLFrDDE (1) is practically stable in time w.r. t. (λ, a −1 (b(λ))).
The proof of Theorem 5.4 is similar to the one of Theorem 5.1 where Remark 2.1 is applied and inequality (20) for t = ξ is applied instead of (11).
When the Dini fractional derivative defined by (5) is used then the comparison result is: (5), i.e.

2(ii)* . there exists a number C > 0 such that for any point t
Then the inequality V (t, x * (t)) ≤ C + λ t 1−q for t ∈ (0, T ) holds. Proof A part of the proof of Lemma 5.5 is similar to the one of Lemma 5.2 and we will emphasize only on the differences which are connected with the application of the Dini fractional derivative and condition 2(ii)*.

Discussions about the type of Lyapunov function
Note the main condition for Lyapunov functions in all Theorems is connected with the application of RL fractional derivative and the corresponding initial condition, i.e. the condition about lim t→0+ t 1−q V (t, x(t)). Let . Let x(t) be a solution of (1), (2) with the initial function ||φ|| 0 < λ. Then lim t→0+ t 1−q V (t, x(t)) = n i=1 lim t→0+ t 1−q |x i (t)| < nλ (q) , i.e. condition 1(ii) of Theorems 1, 2, 3 is fulfilled with b(s) = n (q) s.
i and x(t) be a solution of (1), (2) with the initial function ||φ|| 0 < λ. Then the following holds, i.e. condition 1(ii) of Theorems 5.1, 5.3, 5.4 is not satisfied for the quadratic Lyapunov function. Note the quadratic Lyapunov function is one of the most applicable functions in studying stability properties of differential equations with ordinary derivatives or Caputo derivatives. But it is not the same for RL fractional derivatives.
The above examples and discussions show that one useful Lyapunov function is V (t, x) = t 1−q m(t) n i=1 x 2 i .

Applications
Example 6.1 Consider the nonlinear RL fractional differential equation with a variable delay where u ∈ R.
Consider V (t, x) = t 1−q x 2 . Then there exists a number T = 1 > 0 such that t 1−q ≥ 1 for t ≥ T and thus the condition 1(i) of Theorem 5.4 is satisfied with a(s) = s 2 .