Scalar linear impulsive Riemann-Liouville fractional differential equations with constant delay-explicit solutions and finite time stability

1 Introduction

Fractional differential equations have been applied as more adequate models of real-world problems in engineering, physics, finance, etc. ([1,2]). One of the main qualitative problems is connected with finite time stability (FTS). FTS of linear fractional delay differential equations with controls was studied with the help of an inequality of Gronwall type in [3,4,5]. For other related contribution, one can refer to [6,7,8].

The question about Riemann-Liouville (RL) fractional differential equations is still at the initial stage of investigations (see, e.g., [9,10,11]). Li and Wang introduced the concept of a delayed Mittag-Leffler-type matrix function, and then they presented the finite-time stability results by virtue of a delayed Mittag-Leffler-type matrix in [12,13,14]. They study the case when the lower limit of the RL fractional derivative coincides with the left side end of the initial interval. It is not only different than the idea of the initial value problem (IVP) for delay equations but also it requires strong conditions for the initial function.

In this article, we study IVPs of systems of RL fractional differential equations with a constant delay and impulses at the fixed initially given points 0=t0<t1<t2<…<tN<tN+1=T. We study the following two cases: when the lower limit of the fractional derivative is fixed on the whole interval of consideration, i.e.,

and the case when the lower limit of the fractional derivative is changed at any point of impulse, i.e.,

where A,B are constants, τ>0 is a constant delay and T<∞.

Similar to the case of the ordinary derivative, the differential equation is given to the right of the initial time interval. It requires the lower bound of the RL fractional derivative to coincide with the right side end of the initial interval (usually this point is zero). Note that in this case any solution of an IVP with RL fractional derivatives is not continuous at the initial point. That is why RL fractional delay differential equations are convenient for the modeling process with impulsive types of initial conditions. This type of process can be found in physics, chemistry, engineering, biology, and economics. To determine the law of the initial impulsive reaction we need to add to the usual initial condition (e.g., x(t)=ϕ(t) on the initial interval [−τ,0], τ>0 is the delay) a fractional condition. This conclusion is based on the results obtained in [1] and [8] concerning the physical interpretation of the RL fractional derivatives and initial conditions which include derivatives of the same kind. Based on the above, we set up appropriate IVPs for RL linear fractional differential equations with a lower limit of the RL derivative equal to the right side point of the initial interval, i.e., we study the initial conditions of the type

Similarly, the impulsive conditions are given by

where Dk,k=1,2,…,N are constants.

In this article, we study scalar linear RL fractional differential equations with a constant delay and impulses. The main contributions of the study are as follows:

1. Two types of fractional equations are studied:

1. the equation in which the lower bound of the fractional derivative is fixed at the initial time point;

2. the equation in which the lower limit of the fractional derivative is changing at each point of impulse.

1. The impulsive conditions are set up in both the aforementioned cases. It is connected with the presence of RL fractional derivative and the delay.

2. Explicit formulas of the solutions are obtained in both the aforementioned cases.

3. The obtained explicit formulas for the solutions are applied to study the FTS.

The rest of this article is organized as follows. In Section 2, we outline some basic notations and results from fractional calculus. In Section 3, the main two types of the interpretation of the presence in impulses in RL fractional differential equations are presented. In Section 4, explicit solutions of IVPs (1), (3) and (4) as well as of IVPs (2), (3) and (4) are given. In Section 5, the formulas for the exact solutions are applied to the study of the FTS of both IVPs.

2 Preliminary notes on fractional derivatives and equations

Let 0≤t0<T<∞. In this article, we will use the following definitions for fractional derivatives and integrals:

1. RL fractional integral of order q∈(0, 1) [15,16]

   Itqt0m(t)=1Γ(q)∫t0tm(s)(t−s)1−qds,t∈[t0,T],

   where Γ(⋅) is the Gamma function when the integral exists.

   This is called by some authors the left RL fractional integral of order q (because we integrate to t from the left).

   Note sometimes the notation Dt−qt0m(t)=Itqt0m(t) is used.

2. RL fractional derivative of order q∈(0, 1) [15,16]

where m(t) is measurable on [t0,T].

This is also called the left RL fractional derivative. We will call the point t0 a lower limit of the RL fractional derivative.

We will give fractional integrals and RL fractional derivatives of some elementary functions which will be used later.

The definitions of the initial condition for fractional differential equations with RL-derivatives are based on the following result.

Let Ep,q(z)=∑j=0∞zjΓ(jp+q) be the Mittag-Leffler function with two parameters (see, e.g., [16]). In the case of a scalar linear RL fractional differential equation, we have the following result.

3 Interpretations of the impulses in the RL fractional equations

Let an increasing sequence of non-negative points {ti}i=0N+1 be given with t0=0,T=tN+1.

The interpretation of the impulse in differential equations at a point τ is that there is an instantaneous jump of the solution x(t), which is determined by the value x(τ+0)=limε→0+x(τ+ε) depending significantly on the value of the solution x(τ−0)=limε→0+x(τ−ε) before the jump. The presence of the RL fractional derivative in the differential equation and the inequality Ibqam(b)≠Icqam(c)+Ibqcm(b) with a<c<b<∞ (which is not true for the ordinary case q=1) leads to two basic interpretations of the solution (see, e.g., [10,18]):

1. Fixed lower limit of the RL fractional derivative – in this case the lower limit of the fractional derivative is kept equal to the initial time t0 on the whole interval of consideration. At points of impulses the amount of jump is taken into account.

2. Changed lower limit of the RL fractional derivative at each time of impulses – this is based on the fact that the value of the solution is changed at each impulsive point and it is determined by the differential equation on each interval between two consecutive impulsive points. In this case, the impulsive time is considered as an initial time of the fractional differential equation. Then, the lower limit of the RL fractional derivative, being equal to the initial time, is changed at each impulsive time.

For some explanations about the presence of impulses in the fractional differential equation without any delays and Caputo fractional derivative we refer to [18,19]. In the case of the RL fractional derivative, impulses and no delays, a discussion about the interpretation of the solutions is given in [10].

Define the set PL1loc([0,T],ℝ)={u:[0,T]→ℝ:u∈L1loc((tk,tk+1],ℝ),k=0,1,…,N} with u(tk)=u(tk−0)=limε→0+u(tk−ε),u(tk+0)=limε→0+u(tk+ε).

4 Explicit formulas for the solutions

In this section, we will study the case F(t,x)≡F(t).

5 Finite time stability

In this section, we use the obtained above exact solutions to study the FTS of the IVP for linear RL fractional differential equations with a constant delay and impulses.

In our further considerations, we will assume that F(t,0)≡0, i.e., the linear RL fractional differential equations with zero initial condition g(t)≡0, will have a zero solution.

Note that because of the singularities of tq−1 at 0 and (t−tk)q−1 at tk we could prove the FTS on intervals which do not contain the initial time 0 as well as the impulsive points tk.