Ahmed M.Elshenhab,Xingtao Wang,Fatemah Mofarreh and Omar Bazighifan
1School of Mathematics,Harbin Institute of Technology,Harbin,150001,China
2Department of Mathematics,Faculty of Science,Mansoura University,Mansoura,35516,Egypt
3Mathematical Science Department,Faculty of Science,Princess Nourah Bint Abdulrahman University,Riyadh,11546,Saudi Arabia
4Section of Mathematics,International Telematic University Uninettuno,Roma,00186,Italy
ABSTRACT We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their solutions.As an application,we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayed matrix functions.The obtained results are new,and they extend and improve some existing ones.Finally,an example is presented to illustrate the validity of our theoretical results.
KEYWORDS Representation of solutions;conformable fractional derivative;conformable delayed matrix function;conformable fractional delay differential equations;finite time stability
In recent years,particularly in 2014,Khalil et al.[1]introduced a new definition of the fractional derivative called the conformable fractional derivative that extends the classical limit definition of the derivative of a function.The conformable fractional derivative has main advantages compared with other previous definitions.It can,for example,be used to solve the differential equations and systems exactly and numerically easily and efficiently,it satisfies the product rule and quotient rule,it has results similar to known theorems in classical calculus,and applications for conformable differential equations in a variety of fields have been extensively studied,see [2–10] and the references therein.On the other hand,in 2003,Khusainov et al.[11]represented the solutions of linear delay differential equations by constructing a new concept of a delayed exponential matrix function.In 2008,Khusainov et al.[12] adopted this approach to represent the solutions of an oscillating system with pure delay by establishing a delayed matrix sine and a delayed matrix cosine.This pioneering research yielded plenty of novel results on the representation of solutions,which are applied in the stability analysis and control problems of time-delay systems;see for example[13–28]and the references therein.Thereafter,in 2021,Xiao et al.[29]obtained the exact solutions of linear conformable fractional delay differential equations of orderα∈(0,1]by constructing a new conformable delayed exponential matrix function.
However,to the best of our knowledge,no study exists dealing with the representation and stability of solutions of conformable fractional delay differential systems of orderα∈(1,2].
Motivated by these papers,we consider the explicit formula of solutions of linear conformable fractional differential equations with pure delay
by constructing new conformable delayed matrix functions.Moreover,the representation of solutions of Eq.(1) is used to obtain a finite time stability result onW= [0,L],L >0,whereis called the conformable fractional derivative of orderα∈(1,2] with lower index zero,y(x)∈Rn,ψ∈C2([-τ,0],Rn),B∈Rn×nis a constant nonzero matrix andf∈C([0,∞),Rn)is a given function.
The paper is organized as follows: In Section 2,we present some basic definitions concerning conformable fractional derivative and finite time stability,and construct new conformable delayed matrix functions and derive their properties for use when we discuss the representation of solutions and finite time stability.In Section 3,by using the new conformable delayed matrix functions,we give the explicit formula of solutions of Eq.(1).In Section 4,as an application,we derive a finite time stability result using the representation of solutions.Finally,we give an example to illustrate the main results.
Throughout the paper,we denote the vector norm and matrix norm,respectively,as‖y‖=and‖B‖ =yandbare the elements of the vectoryand the matrixB,respectively.iijDenoteC(W,Rn)the Banach space of vector-value continuous function fromW→Rnendowed with the norm ‖y‖C= maxx∈W‖y(x)‖ for a norm ‖·‖ on Rn.We introduce a spaceC1(W,Rn)={y∈C(W,Rn):y′∈C(W,Rn)}.Furthermore,we see‖ψ‖C=maxυ∈[-τ,0]‖ψ(υ)‖.
We recall some basic definitions of conformable fractional derivative,fractional exponential function,and finite time stability.
Definition 2.1.([2,Definition 2.2]).Letf:[a,∞)→Rnbe a differentiable function atx.Then the conformable fractional derivative forfof orderα=(1,2]is given by
if the limit exists.
Remark 2.1.As a consequence of Definition 2.1,we can show that
whereα=(1,2],andfis 2-differentiable atx >a.
Definition 2.2.([2]).We define the fractional exponential function as follows:
Definition 2.3.([30]).The system in Eq.(1)is finite time stable with respect to{0,W,τ,δ,β},δ <βif and only ifη <δimplies‖y(x)‖<βfor allx∈W,whereη=maxandδ,βare real positive numbers.
Next,we construct new conformable delayed matrix functions that are the fundamental solution matrices of Eq.(1).
Definition 2.4.The conformable delayed matrix functionsHτ,α(Bxα)andMτ,α(Bxα)are defined as
respectively,wherem=0,1,2,...,Iis then×nidentity matrix andΘis then×nnull matrix.
Lemma 2.1.The following rule is true:
Proof.First,whenx∈(-∞,-τ),we obtainHτ,α(Bxα)=Hτ,α(B(x-τ)α)=Θ,and we can see that Lemma 2.1 holds.Following that,set(m-1)τ≤x <mτ,m=0,1,2,...,we get
Applying Remark 2.1,we get
This completes the proof.
In the same way that we proved Lemma 2.1,we can derive the next result.
Lemma 2.2.The following rule is true:
To conclude this section,we provide a norm estimation of the conformable delayed matrix functions,which is used while discussing finite time stability.
Lemma 2.3.For anyx∈[(m-1)τ,mτ],m=0,1,2,...,we have
Proof.Taking the norm of Eq.(2),we get
This completes the proof.
Lemma 2.4.For anyx∈[(m-1)τ,mτ],m=0,1,2,...,we have
This completes the proof.
In this section,we give the exact solutions of Eq.(1)via the conformable delayed matrix functions and the method of variation of constants.To do this,we consider the homogeneous system of linear conformable fractional delay differential equations
and the linear inhomogeneous conformable fractional delay system
Theorem 3.1.The solutiony(x)of Eq.(4)has the representation
Proof.We seek for a solution of Eq.(4)in the form
or
wherec1andc2are unknown constants vectors on Rn,andr(x)is an unkown twice continuously differentible vector function.From Lemmas 2.1 and 2.2,we deduce thatHτ,α(Bxα)andMτ,α(Bxα)are solutions of Eq.(4).We notice that Eq.(6)is a solution of Eq.(4)due to the linearity of solutions for arbitraryc1,c2andr(x).Now we find the constantsc1andc2,and the vector functionr(x)so that the initial conditionsy(x)≡ψ(x),y(x)≡ψ′(x)for -τ≤x≤0,are satisfied.That is,the following relations hold for-τ≤x≤0:
and
Consider Eq.(8).If-τ≤x <0,then
and
which implies that
and
Substituting Eq.(11)into Eq.(10),we get
Differentiating Eq.(12)with respect tox,we have
As a result,we find that the equalities obtained Eqs.(12)and(13)are true if
Substituting Eq.(14)into Eq.(7),we obtain Eq.(6).This finishes the proof.
Theorem 3.2.The particular solutiony0(x)of Eq.(5)has the representation
Proof.We try to find a particular solutiony0(x)of Eq.(5)in the form
by applying the method of variation of constants,whereξ(υ),0<υ≤x,is an unknown function.Taking the conformable derivative of Eq.(16),we get
Substituting Eqs.(16)and(17)into Eq.(5),and noting that
We havex2-αξ(x)=f (x).Substitutingξ(x)=xα-2f (x)into Eq.(16),we obtain Eq.(15).This completes the proof.
Corollary 3.1.The solutiony(x)of Eq.(1)can be represented as
Remark 3.1.Letα=2 in Eq.(1).Then Corollary 3.1 coincides with Corollary 1 in[13].
Remark 3.2.Letα= 2,B=B2in Eq.(1)such that the matrixBis a nonsingularn×nmatrix.Then
where cosτ(Bx)and sinτ(Bx)are called the delayed matrix of cosine and sine type,respectively,defined in[12].Therefore,Corollary 3.1 coincides with Theorems 1 and 2 in[12].
In this section,we establish some sufficient conditions for the finite time stability results of Eq.(1)by using a norm estimation of the conformable delayed matrix functions and the formula of general solutions of Eq.(1).
Theorem 4.1.The system Eq.(1)is finite time stable with respect to{0,W,τ,δ,β},δ <βif
Proof.By using Definition 2.3,and Theorems 3.1 and 3.2,we haveη <δand
Note thatMτ,α(Bxα)=Θifx∈(-∞,-τ).For-τ≤υ≤0,we get
Thus
Therefore,from Lemma 2.4,we have
for-τ≤υ≤0,x∈W,and sinceEαis increasing function whenx≥υ.From Eq.(21),we get
From Lemma 2.4,we have
From Eqs.(20),(22)and(23),we get
for allx∈W.Combining Eq.(19)with Eq.(24),we obtain‖y(x)‖<βfor allx∈W.This completes the proof.
Corollary 4.1.Letα=2 in Eq.(1).Then the system
is finite time stable with respect to{0,W,τ,δ,β},δ <βif
Remark 4.1.Letα= 2,B=B2in Eq.(1)such that the matrixBis a nonsingularn×nmatrix.Then the representation of solution Eq.(18)coincides with the conclusion of Theorems 1 and 2 in[12],which leads to the same of the finite time stability results in[27].
Consider the conformable delay differential equations
where
From Theorems 3.1 and 3.2,for all 0 ≤x≤1,and through a basic calculation,we can obtain
which implies that
and
where
and
Thus the explicit solutions of Eq.(25)are
where 0 ≤x≤0.5,which implies that
and
where 0.5 ≤x≤1,which implies that
By calculating we obtainη= max= 0.3,‖B‖ = 2,‖f‖C= 3,Eα=4.0104,= 1.4871,then we setδ= 0.31>0.3 =η.Fig.1 shows the statey(x)and the norm‖y(x)‖of Eq.(25).Now Theorem 4.1 implies that‖y(x)‖ ≤5.930254,we just takeβ=5.9303,which implies that‖y(x)‖<βand Eq.(25)is finite time stable.
Figure 1:The state y(x)and||y(x)||of Eq.(25)
In this work,using new conformable delayed matrix functions,we derived explicit solutions of linear conformable fractional delay systems of orderα∈(1,2],which extend and improve the corresponding and existing ones in[12,13]in the case ofα=2 without any restrictions on the matrix coefficient of the linear part,by removing the condition thatBis a nonsingular matrix and replacing the matrix coefficient of the linear partB2in[12]by an arbitrary,not necessarily squared,matrix.In addition,using the formula of general solutions and a norm estimation of the conformable delayed matrix functions,we established some sufficient conditions for the finite time stability results,which extend and improve the existing ones in[27]in the case ofα= 2.Ultimately,an illustrative example was given to show the validity of the proposed results.
Following the topic of this paper,we outline some possible next research directions.The first direction will include applying the results of this paper on control problems for conformable fractional delay systems of orderα∈(1,2].The second direction is to consider the explicit solutions of linear conformable fractional delay systems of the form
which lead to new results on stability and control problems.Depending on these results and delayed arguments,we will try to prove a generalized Lyapunov-type inequality for the conformable and sequential conformable boundary value problems
and
which leads to new results on the conformable Sturm-Liouville eigenvalue problem.
Acknowledgement:The authors would like to thank Princess Nourah bint Abdulrahman University Researchers Supporting Project No.(PNURSP2022R27),Princess Nourah bint Abdulrahman University,Riyadh,Saudi Arabia.
Funding Statement:Princess Nourah bint Abdulrahman University Researchers Supporting Project No.(PNURSP2022R27),Princess Nourah bint Abdulrahman University,Riyadh,Saudi Arabia.
Conflicts of Interest:The authors declare that they have no conflicts of interest to report regarding the present study.
Computer Modeling In Engineering&Sciences2023年2期