stability analysis of impulsive fractional differential systems with delay
TRANSCRIPT
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
โข Differential equations have been used in modeling the dynamicsof changing processes.
SO
โข The dynamics of many evolving processes are subject to abruptchanges, such as shocks, harvesting and natural disasters.
THUS
โข These phenomena involve short-term perturbations fromcontinuous and smooth dynamics.
AS A CONSEQUENCE
โข In models involving such perturbations, it is natural to assumethese perturbations act in the form of โimpulsesโ.
Impulsive Differential Equations
IN
โข 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 (GronwallโBellmanโRaid) inequalityprovides explicit bounds on solutions of a class oflinear integral inequalities.
Gronwall Inequality and its Generalized Form
If
๐ฅ ๐ก โค โ ๐ก +
๐ก0
๐ก
๐ ๐ ๐ฅ ๐ ๐๐ , ๐ก โ ๐ก0, ๐ ,
where all the functions involved are continuous on ๐ก0, ๐ , ๐โค +โ, and ๐(๐ก) โฅ 0, then ๐ฅ ๐ก satisfies
๐ฅ ๐ก โค โ ๐ก +
๐ก0
๐ก
โ(๐ )๐ ๐ exp[
๐
๐ก
๐ ๐ข ๐๐ข]๐๐ , ๐ก โ ๐ก0, ๐ .
The Standard GronwallInequality
Gronwall Inequality and its Generalized Form
If
๐ฅ ๐ก โค โ ๐ก +
๐ก0
๐ก
๐ ๐ ๐ฅ ๐ ๐๐ , ๐ก โ ๐ก0, ๐ ,
and in addition, โ ๐ก is nondecreasing, then
๐ฅ ๐ก โค โ ๐ก + exp
๐ก0
๐ก
๐ ๐ ๐๐ , ๐ก โ ๐ก0, ๐ .
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 ๐ฅ ๐ก and ๐ ๐ก are nonnegative and locally
integrable on 0 โค ๐ก < ๐ (some ๐ โค +โ), and ๐(๐ก) is anonnegative, nondecreasing continuous function definedon 0 โค ๐ก < ๐, ๐ ๐ก โค ๐ = ๐๐๐๐ ๐ก๐๐๐ก, and ๐ผ > 0 with
๐ฅ ๐ก โค ๐ ๐ก + ๐(๐ก)
0
๐ก
(๐ก โ ๐ )๐ผโ1๐ฅ ๐ ๐๐
on this interval. Then
๐ฅ ๐ก โค ๐ ๐ก + ๐(๐ก)
0
๐ก
[
๐=1
โ(๐(๐ก)๐ค(๐ผ))๐
๐ค(๐๐ผ)(๐ก โ ๐ )๐๐ผโ1๐(๐ )]๐๐
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 < ๐ผ < 1
Non-autonomous Impulsive Fractional Differential Systems
Stability Analysis: Definitions and Theorems
substituting ๐ท๐ผ๐ฅ(๐ก) 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 ๐(๐ก) be a nondecreasingcontinuous function defined on 0 โค ๐ก < ๐, then we have
๐ฅ ๐ก โค ๐ ๐ก ๐ธ๐ผ(๐ ๐ก ๐ค ๐ผ ๐ก๐ผ)
where ๐ธ๐ผ is the Mittag-Leffler function defined by
๐ธ๐ผ ๐ง = ๐=0โ ๐ง๐ ๐ค ๐๐ผ + 1 .
Corollary
Stability Analysis: Definitions and Theorems
According to the definition
๐ ๐ถ < ๐ฟ
Let ๐ ๐ก = ๐๐ฅ ๐ถ 1 +๐๐๐๐ฅ01๐ก
๐ผ
๐ค(๐ผ+1)+ 0<๐ก๐<๐ก ๐๐๐๐ฅ(๐ถ๐) ๐ฅ(๐ก๐)
+๐ผ๐ข๐๐๐๐ฅ(๐ต0)๐ก
๐ผ
๐ค(๐ผ+1)
๐ ๐ก 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<๐ก๐<๐ก๐๐๐๐ฅ ๐ถ๐ < 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 ๐ ๐ก =๐๐ฅ ๐ถ 1+
๐๐๐๐ฅ01๐ก๐ผ
๐ค(๐ผ+1)+๐ผ๐ข๐๐๐๐ฅ(๐ต0)๐ก
๐ผ
๐ค(๐ผ+1)
1โ 0<๐ก๐<๐ก๐๐๐๐ฅ(๐ถ๐) ๐ฅ(๐ก๐)
๐ ๐ก 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 ๐ข โ ๐๐ถ(๐ฝ, ๐ ) satisfy the following inequality
๐ข ๐ก โค ๐ถ1 ๐ก + ๐ถ2
0
๐ก
๐ก โ ๐ ๐โ1 ๐ข ๐ ๐๐ +
0<๐ก๐<๐ก
๐๐ ๐ข ๐ก๐
where ๐ถ1 is nonnegative continuous and nondecreasing on ๐ฝ,and ๐ถ2, ๐๐ โฅ 0 are constants. Then
๐ข ๐ก โค ๐ถ1 ๐ก 1 + ๐๐ธ๐ฝ ๐ถ2๐ค ๐ฝ ๐ก๐ฝ๐๐ธ๐ฝ ๐ถ2๐ค ๐ฝ ๐ก
๐ฝ
where ๐ก โ ๐ก๐ , ๐ก๐+1 ๐๐๐ ๐ = max ๐๐: ๐ = 1,2, โฆ ,๐ .
Lemma
Stability Analysis: Definitions and Theorems
Let ๐ถ1 ๐ก = ๐๐ฅ ๐ถ 1 +๐๐๐๐ฅ01๐ก
๐ผ
๐ค(๐ผ+1)+๐ผ๐ข๐๐๐๐ฅ(๐ต0)๐ก
๐ผ
๐ค(๐ผ+1), and ๐ถ2
=๐๐๐๐ฅ01
๐ค(๐ผ), and ๐ถ = max{๐๐๐๐ฅ ๐ถ๐ , ๐ = 1,2, โฆ ,๐}
๐ถ1 ๐ก is a nondecreasing function and ๐ถ2, ๐ถ โฅ 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, โStability analysis of impulsive fractional
differential systems with delayหฎ, 2015, Applied Mathematics Letters,
40, pp. 1-6.
2. H. Ye, J. Gao, Y. Ding, โA generalized Gronwall inequality and its
application to a fractional differential equationหฎ, 2007, J. Math. Anal.
Appl., 328, pp. 963-968.
3. M. Benchohra, J. Henderson, S. Ntouyas, โImpulsive Differential
Equations and Inclusionsหฎ, 2006, Contemporary Mathematics and Its
Applications, volume 2, Hindawi Publishing Corporation, NY.
4. M.P. Lazareviฤ, Aleksandar M. Spasiฤ, โFinite-time stability analysis
of fractional order time-delay systems: Gronwallโs approachหฎ, 2009,
Math. Comput. Modelling, 49, pp. 475-481.