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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.
Gronwall Inequality and its Generalized Form
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
Gronwall Inequality and its Generalized Form
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
Gronwall Inequality and its Generalized Form
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
Gronwall Inequality and its Generalized Form
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
Stability Analysis: Definitions and Theorems
Theorem 1
Non-autonomous Impulsive Fractional Differential Systems
1st Approach
Stability Analysis: Definitions and Theorems
applying the .
a solution of system 1 in the form of the equivalent Volterraintegral equation
the property of the fractional order
0 < 𝛼 < 1
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
substituting 𝐷𝛼𝑥(𝑡) by the right side of the equation of system 1
knowing that
applying the . on system 1
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
by using
and
therefore
Non-autonomous Impulsive Fractional Differential Systems
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
Stability Analysis: Definitions and Theorems
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
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
Therefore by the condition (*), we have
by using the corollary
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
Theorem 2
Non-autonomous Impulsive Fractional Differential Systems
2nd Approach
Stability Analysis: Definitions and Theorems
By the condition that 0<𝑡𝑘<𝑡𝜎𝑚𝑎𝑥 𝐶𝑘 < 1
Similar to the proof of Theorem 1
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
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
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
Therefore by the condition (**), we have
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
Theorem 3
Non-autonomous Impulsive Fractional Differential Systems
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
Stability Analysis: Definitions and Theorems
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
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
Therefore by the condition (***), we have
by using the definition and the lemma
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
Theorem 4
Autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
Theorem 5
Autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
Theorem 6
Autonomous Impulsive Fractional Differential Systems
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.