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 dynamics
of changing processes.
SO
• The dynamics of many evolving processes are subject to abrupt
changes, such as shocks, harvesting and natural disasters.
THUS
• These phenomena involve short-term perturbations from
continuous and smooth dynamics.
AS A
CONSEQUENCE
• In models involving such perturbations, it is natural to assume
these perturbations act in the form of “impulses”.


Impulsive Differential Equations
IN
• Impulsive differential equations have been developed in
modeling 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 the
qualitative analysis of the solutions to differential and
integral equations.
The Gronwall (Gronwall–Bellman–Raid) inequality
provides explicit bounds on solutions of a class of
linear 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
Gronwall
Inequality


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
Gronwall
Inequality


sometimes we need a different form, to discuss the weakly
singular Volterra integral equations encountered in
fractional differential equations.
we present a slight generalization of the Gronwall
inequality which can be used in a fractional differential
equation.
However
S
o


Suppose 𝑥 𝑡 and 𝑎 𝑡 are nonnegative and locally
integrable on 0 ≤ 𝑡 < 𝑇 (some 𝑇 ≤ +∞), and 𝑔(𝑡) is a
nonnegative, nondecreasing continuous function defined
on 0 ≤ 𝑡 < 𝑇, 𝑔 𝑡 ≤ 𝑀 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, and 𝛼 > 0 with
𝑥 𝑡 ≤ 𝑎 𝑡 + 𝑔(𝑡)
0
𝑡
(𝑡 − 𝑠) 𝛼−1
𝑥 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−1 𝑎(𝑠)]𝑑𝑠
The
Generalized
Gronwall
Inequality


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 Volterra
integral equation
the property of the fractional order
0 < 𝛼 < 1


substituting 𝐷 𝛼 𝑥(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 Gronwall
Inequality theorem, let 𝑎(𝑡) be a nondecreasing
continuous function defined on 0 ≤ 𝑡 < 𝑇, then we have
𝑥 𝑡 ≤ 𝑎 𝑡 𝐸 𝛼(𝑔 𝑡 𝛤 𝛼 𝑡 𝛼)
where 𝐸 𝛼 is the Mittag-Leffler function defined by
𝐸 𝛼 𝑧 = 𝑘=0
∞
𝑧 𝑘 𝛤 𝑘𝛼 + 1 .
Corollary


According to the definition
𝜓 𝐶 < 𝛿
Let 𝑎 𝑡 = 𝜓 𝑥 𝐶 1 +
𝜎 𝑚𝑎𝑥01 𝑡 𝛼
𝛤(𝛼+1)
+ 0<𝑡 𝑘<𝑡 𝜎 𝑚𝑎𝑥(𝐶 𝑘) 𝑥(𝑡 𝑘)
+
𝛼 𝑢 𝜎 𝑚𝑎𝑥(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<𝑡 𝑘<𝑡 𝜎 𝑚𝑎𝑥 𝐶 𝑘 < 1
Similar to the proof of Theorem 1


by using the definition and the corollary
Let 𝑎 𝑡 =
𝜓 𝑥 𝐶 1+
𝛤(𝛼+1)
+
𝛤(𝛼+1)
1− 0<𝑡 𝑘<𝑡 𝜎 𝑚𝑎𝑥(𝐶 𝑘) 𝑥(𝑡 𝑘)
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<𝑡 𝑘<𝑡
𝜃 𝑘 𝑢 𝑡 𝑘
where 𝐶1 is nonnegative continuous and nondecreasing on 𝐽,
and 𝐶2, 𝜃 𝑘 ≥ 0 are constants. Then
𝑢 𝑡 ≤ 𝐶1 𝑡 1 + 𝜃𝐸𝛽 𝐶2 𝛤 𝛽 𝑡 𝛽
𝑘
𝐸𝛽 𝐶2 𝛤 𝛽 𝑡 𝛽
where 𝑡 ∈ 𝑡 𝑘, 𝑡 𝑘+1 𝑎𝑛𝑑 𝜃 = max 𝜃 𝑘: 𝑘 = 1,2, … , 𝑚 .
Lemma


Let 𝐶1 𝑡 = 𝜓 𝑥 𝐶 1 +
𝛤(𝛼+1)
+
𝛤(𝛼+1)
, and 𝐶2
=
𝜎 𝑚𝑎𝑥01
𝛤(𝛼)
, and 𝐶 = max{𝜎 𝑚𝑎𝑥 𝐶 𝑘 , 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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