Stability analysis of impulsive fractional differential systems with delay

$Page 1: Stability analysis of impulsive fractional differential systems with delay$
Stability analysis of impulsive

fractional differential systems

with delay

By Qi Wang, Dicheng Lu, Yuyun Fang

Presentation by Mostafa Shokrian Zeini

Important Questions:

- What is an impulsive differential equation? And what are its applications?

- Why is the Gronwall inequality developed for? What is the application of

the generalized Gronwall inequality?

- What is the main approach for the stability analysis of delayed impulsive

fractional differential systems?

Impulsive Differential Equations

BUT

• Differential equations have been used in modeling the dynamicsof changing processes.

SO

• The dynamics of many evolving processes are subject to abruptchanges, such as shocks, harvesting and natural disasters.

THUS

• These phenomena involve short-term perturbations fromcontinuous and smooth dynamics.

AS A CONSEQUENCE

• In models involving such perturbations, it is natural to assumethese perturbations act in the form of “impulses”.

Impulsive Differential Equations

IN

• Impulsive differential equations have been developed inmodeling impulsive problems

physics, population dynamics, ecology, biological systems,

biotechnology, industrial robotics, pharmacokinetics, optimal control, etc.

Gronwall Inequality and its Generalized Form

Integral inequalities play an important role in thequalitative analysis of the solutions to differential andintegral equations.

The Gronwall (Gronwall–Bellman–Raid) inequalityprovides explicit bounds on solutions of a class oflinear integral inequalities.


If

𝑥 𝑡 ≤ ℎ 𝑡 +

𝑡0

𝑡

𝑘 𝑠 𝑥 𝑠 𝑑𝑠 , 𝑡 ∈ 𝑡0, 𝑇 ,

where all the functions involved are continuous on 𝑡0, 𝑇 , 𝑇≤ +∞, and 𝑘(𝑡) ≥ 0, then 𝑥 𝑡 satisfies

𝑥 𝑡 ≤ ℎ 𝑡 +

𝑡0

𝑡

ℎ(𝑠)𝑘 𝑠 exp[

𝑠

𝑡

𝑘 𝑢 𝑑𝑢]𝑑𝑠 , 𝑡 ∈ 𝑡0, 𝑇 .

The Standard GronwallInequality


If

𝑥 𝑡 ≤ ℎ 𝑡 +

𝑡0

𝑡

𝑘 𝑠 𝑥 𝑠 𝑑𝑠 , 𝑡 ∈ 𝑡0, 𝑇 ,

and in addition, ℎ 𝑡 is nondecreasing, then

𝑥 𝑡 ≤ ℎ 𝑡 + exp

𝑡0

𝑡

𝑘 𝑠 𝑑𝑠 , 𝑡 ∈ 𝑡0, 𝑇 .

The Standard GronwallInequality


sometimes we need a different form, to discuss the weaklysingular Volterra integral equations encountered infractional differential equations.

we present a slight generalization of the Gronwallinequality which can be used in a fractional differentialequation.

However

S

o


Suppose 𝑥 𝑡 and 𝑎 𝑡 are nonnegative and locally

integrable on 0 ≤ 𝑡 < 𝑇 (some 𝑇 ≤ +∞), and 𝑔(𝑡) is anonnegative, nondecreasing continuous function definedon 0 ≤ 𝑡 < 𝑇, 𝑔 𝑡 ≤ 𝑀 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, and 𝛼 > 0 with

𝑥 𝑡 ≤ 𝑎 𝑡 + 𝑔(𝑡)

0

𝑡

(𝑡 − 𝑠)𝛼−1𝑥 𝑠 𝑑𝑠

on this interval. Then

𝑥 𝑡 ≤ 𝑎 𝑡 + 𝑔(𝑡)

0

𝑡

[

𝑛=1

∞(𝑔(𝑡)𝛤(𝛼))𝑛

𝛤(𝑛𝛼)(𝑡 − 𝑠)𝑛𝛼−1𝑎(𝑠)]𝑑𝑠

The Generalized

GronwallInequality

Impulsive Fractional Differential Systems

Non-

autonomous

autonomous

System 1

System 2

Stability Analysis: Definitions and Theorems

Definition

Non-autonomous Impulsive Fractional Differential Systems


Theorem 1


1st Approach


applying the .

a solution of system 1 in the form of the equivalent Volterraintegral equation

the property of the fractional order

0 < 𝛼 < 1



substituting 𝐷𝛼𝑥(𝑡) by the right side of the equation of system 1

knowing that

applying the . on system 1



by using

and

therefore


Some Preliminaries by using the Generalized

Gronwall Inequality

Under the hypothesis of the Generalized GronwallInequality theorem, let 𝑎(𝑡) be a nondecreasingcontinuous function defined on 0 ≤ 𝑡 < 𝑇, then we have

𝑥 𝑡 ≤ 𝑎 𝑡 𝐸𝛼(𝑔 𝑡 𝛤 𝛼 𝑡𝛼)

where 𝐸𝛼 is the Mittag-Leffler function defined by

𝐸𝛼 𝑧 = 𝑘=0∞ 𝑧𝑘 𝛤 𝑘𝛼 + 1 .

Corollary


According to the definition

𝜓 𝐶 < 𝛿

Let 𝑎 𝑡 = 𝜓𝑥 𝐶 1 +𝜎𝑚𝑎𝑥01𝑡

𝛼

𝛤(𝛼+1)+ 0<𝑡𝑘<𝑡 𝜎𝑚𝑎𝑥(𝐶𝑘) 𝑥(𝑡𝑘)

+𝛼𝑢𝜎𝑚𝑎𝑥(𝐵0)𝑡

𝛼

𝛤(𝛼+1)

𝑎 𝑡 is a nondecreasing function



Therefore by the condition (*), we have

by using the corollary



Theorem 2


2nd Approach


By the condition that 0<𝑡𝑘<𝑡𝜎𝑚𝑎𝑥 𝐶𝑘 < 1

Similar to the proof of Theorem 1



by using the definition and the corollary

Let 𝑎 𝑡 =𝜓𝑥 𝐶 1+

𝜎𝑚𝑎𝑥01𝑡𝛼

𝛤(𝛼+1)+𝛼𝑢𝜎𝑚𝑎𝑥(𝐵0)𝑡

𝛼

𝛤(𝛼+1)

1− 0<𝑡𝑘<𝑡𝜎𝑚𝑎𝑥(𝐶𝑘) 𝑥(𝑡𝑘)

𝑎 𝑡 is a nondecreasing function



Therefore by the condition (**), we have



Theorem 3


3rd Approach

Some Preliminaries by using the Generalized

Gronwall Inequality

Let 𝑢 ∈ 𝑃𝐶(𝐽, 𝑅) satisfy the following inequality

𝑢 𝑡 ≤ 𝐶1 𝑡 + 𝐶2

0

𝑡

𝑡 − 𝑠 𝑞−1 𝑢 𝑠 𝑑𝑠 +

0<𝑡𝑘<𝑡

𝜃𝑘 𝑢 𝑡𝑘

where 𝐶1 is nonnegative continuous and nondecreasing on 𝐽,and 𝐶2, 𝜃𝑘 ≥ 0 are constants. Then

𝑢 𝑡 ≤ 𝐶1 𝑡 1 + 𝜃𝐸𝛽 𝐶2𝛤 𝛽 𝑡𝛽𝑘𝐸𝛽 𝐶2𝛤 𝛽 𝑡

𝛽

where 𝑡 ∈ 𝑡𝑘 , 𝑡𝑘+1 𝑎𝑛𝑑 𝜃 = max 𝜃𝑘: 𝑘 = 1,2, … ,𝑚 .

Lemma


Let 𝐶1 𝑡 = 𝜓𝑥 𝐶 1 +𝜎𝑚𝑎𝑥01𝑡

𝛼

𝛤(𝛼+1)+𝛼𝑢𝜎𝑚𝑎𝑥(𝐵0)𝑡

𝛼

𝛤(𝛼+1), and 𝐶2

=𝜎𝑚𝑎𝑥01

𝛤(𝛼), and 𝐶 = max{𝜎𝑚𝑎𝑥 𝐶𝑘 , 𝑘 = 1,2, … ,𝑚}

𝐶1 𝑡 is a nondecreasing function and 𝐶2, 𝐶 ≥ 0

Similar to the proof of Theorem 1



Therefore by the condition (***), we have

by using the definition and the lemma



Theorem 4

Autonomous Impulsive Fractional Differential Systems


Theorem 5



Theorem 6


References

1. Q. Wang, D. Lu, Y. Fang, “Stability analysis of impulsive fractional

differential systems with delayˮ, 2015, Applied Mathematics Letters,

40, pp. 1-6.

2. H. Ye, J. Gao, Y. Ding, “A generalized Gronwall inequality and its

application to a fractional differential equationˮ, 2007, J. Math. Anal.

Appl., 328, pp. 963-968.

3. M. Benchohra, J. Henderson, S. Ntouyas, “Impulsive Differential

Equations and Inclusionsˮ, 2006, Contemporary Mathematics and Its

Applications, volume 2, Hindawi Publishing Corporation, NY.

4. M.P. Lazarević, Aleksandar M. Spasić, “Finite-time stability analysis

of fractional order time-delay systems: Gronwall’s approachˮ, 2009,

Math. Comput. Modelling, 49, pp. 475-481.

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Stability analysis of impulsive fractional differential systems with delay