# Partial Stability of Conformable Stochastic Systems

Authors: Abdellatif Ben Makhlouf; Lassaad Mchiri
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, we investigate the almost sure exponential stability (ASES) and the exponential stability in the p-th moment (ESPM) with respect to part of the variable of conformable stochastic systems (CSSs) by using the Lyapunov methods and the stochastic calculus techniques. In the last section, we illustrate our results with a theoretical example.


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## Body

## 1. Introduction

Many deterministic phenomena were described by the differential equations. Taking into account the random phenomena, formally one must take into account the “stochastic perturbed term,” which transforms the equations into stochastic differential equations (SDEs).The Caputo derivative and Riemann–Liouville derivative do not verify the chain and Leibniz rules, which sometimes prevent us from applying these derivatives to some physical problem with the standard Newton derivative. Recently, in [1], the authors invented the new fractional derivative called “conformable integral and fractional derivatives.” This well-known derivative satisfies the chain and Leibniz rules.In fact, according to the special characteristics of the fractional derivative (which is different from the property of the classical derivative), the compatibility of the stochastic integral and fractional integral encounters many difficulties. Fractional integrals contain singular integral terms, which makes it difficult to calculate its numerical solution and estimate its asymptotic solution. Taking into account all the considerations, we present conformable stochastic differential equations.In particular, the asymptotic behaviour and the stability theory of the solution of ordinary stochastic systems, conformable fractional-order stochastic systems, and deterministic systems were investigated by many researchers (see [2–6]).In the last decades, it has established its effectiveness as an essential tool in many real-world applications, such as physics [7], biology [4, 8, 9], and control theory [2, 3, 5, 10–14]. In fact, Neirameh in [4] proposed a new method to find exact solutions of the general biological population model. By using the trial solution algorithm and the complete discrimination system, Odabasi in [9] has proved the existence of the exact solutions of the biological population model. Using the fixed-point method, Das in [2] showed the controllability and the existence of mild solution for a class of nonlinear systems. On the other hand, Younus et al. in [14] investigated the observability of linear time-invariant control systems.In the last years, the stability analysis of conformable systems was investigated by many researchers. In [2, 3, 12], the authors studied the stability of the solution of conformable deterministic systems. In the literature, there is a few work about the stability of the solution of conformable stochastic systems (see [5, 6]). On the other hand, in many real-world phenomena, such stability is sometimes too difficult to be satisfied. Consequently, the concept of partial stability (see [15–20]) has been presented, and the Lyapunov technique, as an important tool, has been utilized to study the stability with respect to part of the variable in various real-world important phenomena. To the best of our knowledge, there is no existing paper on the stability with respect to part of the variables of CSSs. In [17, 19], the authors studied the partial asymptotic stability in the probability of ordinary stochastic differential equations by using the Lyapunov methods. In our paper, we will extend the results in [17, 19] to the case of nonlinear conformable systems and we will investigate the ASES and the ESPM with respect to part of the variable.In our paper, we investigate the partial ESPM and the partial ASES. In this sense, our results present a complete generalization of the work [17].The main contributions of our paper are as follows:(i)
Study the partial ASES and the partial ESPM of CSSs under some new criteria(ii)
Different from the previous results in [17], taking into account the influence of the conformable fractional derivative in the partial stability theory make our results more generalThis paper is organized as follows. In Section2, we recall some basic notions. In Section 3, we investigate the partial ESPM and the partial ASES of CSSs. In Section 4, we study the exponential instability in the p-th moment with respect to all variables of CSSs. In Section 5, we illustrate our theory with an example.


## 2. Preliminaries

In this section, some basic results are presented.The complete probability space is denoted byΩ,F,Fδδ≥0,P, where Fδδ≥0 is a filtration satisfying the usual conditions. Wδ is an m-dimensional Brownian motion defined on Ω,F,Fδδ≥0,P. The set of ℝn-valued Fδ-adapted processes ψδt≤δ≤s is denoted by ℒ2t,s,ℝn such that ∫tsψδ2dδ<∞ a.s. Let C−γ,0;ℝn denote the set of functions ψ from −γ,0 to ℝn that are right-continuous and have limits on the left. C−γ,0;ℝn is equipped with the norm ψ=sup−γ≤δ≤0ψδ and m=mTm for any m∈ℝn. The set of all F0-measurable bounded C−γ,0;ℝn-valued random variables ν=νκ:−γ≤κ≤0 are denoted by CF0b˜−γ,0;ℝn. Let LFδ2−γ,0;ℝn, δ≥0, denote the set of all Fδ-measurable, C−γ,0;ℝn-valued random variables ϖ=ϖκ:−γ≤κ≤0 satisfies sup−γ≤κ≤0Eϖκ2<∞.Definition 1.
[21]. The conformable derivative (CD) of a function G:Θ,∞⟶ℝ is defined by(1)TΘαGδ=limd⟶0Gδ+dδ−Θ1−α−Gδd,for all δ>Θ, α∈0,1, while(2)TΘαGΘ=limd⟶0TΘαGδ.Definition 2.
[21]. The conformable integral (CI) of a function G:Θ,∞⟶ℝ is defined by(3)IΘαGδ=∫Θδξ−Θα−1Gξdξ,for all δ>Θ, α∈0,1.Lemma 1.
[21]. LetG∈CΘ,∞,ℝand0<α<1. Then,∀δ>Θ, we obtain(4)TΘαIΘαGδ=Gδ.Lemma 2.
[22]. Letα∈0,1, υ1, υ2, ς, ν∈ℝ, and the functionsG1,G2:Θ,+∞⟶ℝsuch thatTΘαGiδexists onΘ,+∞fori=1,2. Then,(i)
TΘαυ1G1+υ2G2=υ1TΘαG1+υ2TΘαG2(ii)
TΘαν=0(iii)
TΘαG1G2=G1TΘαG2+G2TΘαG1Remark 1.
LetG:Θ,∞⟶ℝn such that TΘαGδ exists on Θ,∞. Then, TΘαGTGδ exists on Θ,∞ and(5)TΘαGTGδ=2GδTTΘαGδ,∀δ>a.Definition 3.
[3]. The conformable exponential function is defined by(6)Ξαν,δ=expνδαα,where α∈0,1 and ν∈ℝ.Remark 2.
Since the conformable derivative has no spatial characteristic for the fractional derivative, we can directly explicit the solution of the system, evaluate the numerical solution, and approximate the error of the asymptotic solution.


## 3. Main Results

Consider the following CSS onδ0,∞, for any χ∈1/2,1:(7)Tδ0χy=hδ0δ,y+ϕδ0δ,ydWδdδ,yδ0=y0,where the initial condition y0=y01,y02T∈ℝd×ℝk, with d+k=n, which is independent of W⋅, y=y1,y2T∈ℝd×ℝk, hδ0⋅,⋅:δ0,∞×ℝn⟶ℝn, δ0>0, and ϕδ0⋅,⋅:δ0,∞×ℝn⟶ℝn are continuous. Suppose that hδ0 and ϕδ0 are smooth enough, i.e., satisfying the Lipschitz and the growth condition to ensure the existence and the uniqueness of a global solution of system (7) (for more details, see Theorem 4.3 in [6]).Definition 4.
We call that anℝn-valued stochastic process y⋅ is a solution of (7) if yδ is continuous, Fδ-adapted, and(8)yδ=y0+∫δ0δhδ0l,yll−δ0χ−1dl+∫δ0δϕδ0l,yll−δ0χ−1dWl,δ∈δ0,∞.Definition 5.
Letp>0. System (7) is said to be y1-ASES, if there exist positive constants c1 and c2 satisfying, for any yδ0∈ℝn,(9)y1δp≤c1yδ0pΞχ−c2,δ−δ0,∀δ≥δ0≥0.Definition 6.
[16]. Let p>0. System (7) is said to be y1-ESPM, if there exist positive constants ξ1 and ξ2 satisfying, for any yδ0∈ℝn,(10)Ey1δp≤ξ1yδ0pΞχ−ξ2,δ−δ0,∀δ≥δ0≥0.Remark 3.
Note that, in Definition7, when p=2, system (7) is said to be y1-exponentially stable in mean square (y1-ESMS).
LetWδ,y∈C1,2δ0,+∞×ℝn,ℝ+. By Itô formula (see [6]), one has the following: for δ>δ0,(11)Tδ0χWδ,y=ℒχδ0Wδ,y+Wyδ,yϕδ0δ,ydWδdδ,(12)dWδ,y=ℒχδ0Wδ,yδ−δ0χ−1dδ+Wyδ,yϕδ0δ,yδ−δ0χ−1dWδ,where(13)ℒχδ0Wδ,y=Wδδ,yδ−δ01−χ+Wyδ,yhδ0δ,y+12traceϕδ0δ,yδWyyδ,yϕδ0δ,yδ−δ0χ−1,(14)Wδδ,y=∂Wδ,y∂δδ,y,Wyδ,y=∂Wδ,y∂y1δ,y,…,∂Wδ,y∂ynδ,y,Wyyδ,y=∂2Wδ,y∂yi∂yjδ,yn×n.Theorem 1.
Letpbe a positive constant. Suppose that there exist positive constantsα1, α2, andα3. Assume that there existWδ,y∈C1,2δ0,+∞×ℝn,ℝ+such that(i)
α1y1p≤Wδ,y≤α2y1p,∀δ,y∈δ0,+∞×ℝn(ii)
ℒχδ0Wδ,yδ≤−α3y1δp,∀δ,y∈δ0,+∞×ℝn
Then, system (7) isy1-ESPM.Proof.
Lety0≠0 be fixed and denoted by yδ the solution of system (7). For each y0≤d, the stopping time is denoted by σd such that(15)σd=infδ≥δ0;yδ≥d.
It is easy to see thatσd⟶∞ as d⟶∞. By Itô’s formula, we obtain for δ≥δ0(16)Ξχα3α2,δ∧σd−δ0Wδ∧σd,yδ∧σd=Wδ0,yδ0+∫δ0δ∧σdΞχα3α2,l−δ0l−δ0χ−1α3α2Wl,yl+ℒχδ0Wl,yldl+∫δ0δ∧σdΞχα3α2,l−δ0l−δ0χ−1Wyl,ylϕδ0l,yldWl.
Using thatMδ=∫δ0δΞχα3/α2,l−δ0l−δ0χ−1Wyl,ylϕδ0l,yldWl is a local martingale with Mδ0=0 (see [23]) and taking the expectation on both sides of (16), we obtain(17)EΞχα3α2,δ∧σd−δ0Wδ∧σd,yδ∧σd=Wδ0,yδ0+E∫δ0δ∧σdΞχα3α2,l−δ0l−δ0χ−1α3α2Wl,yl+ℒχδ0Wl,yldl.
By conditionsi and ii, one has(18)α1EΞχα3α2,δ∧σd−δ0y1δ∧σdp≤EΞχα3α2,δ∧σd−δ0Wδ∧σd,yδ∧σd≤α2yδ0p.
Lettingd⟶∞, then(19)α1EΞχα3α2,δ−δ0y1δp≤α2yδ0p.
Hence, we can derive that(20)Ey1δp≤α2α1Ξχ−α3α2,δ−δ0yδ0p.
Therefore, system (7) is y1-ESPM. □Remark 4.
It is necessary to give the relation between equation (20) and the y1-ASES to make our result more interesting.Theorem 2.
Letp>2/2χ−1. Suppose that (20) holds and there is a positive constantMsuch that for allδ,y∈δ0,∞×ℝn(21)Ehδ0δ,y1p+Eϕδ0δ,y1p≤MEy1δp.Then, system (7) is y1-ASES.Proof.
Lety0≠0 be fixed. It follows from (20) that there exists positive constants ξ1 and ξ2 such that(22)Ey1δp≤ξ1yδ0pΞχ−ξ2,δ−δ0,∀δ≥δ0≥0.
From ([23], p. 178), for any t1, t2, t3≥0, we have(23)t1+t2+t3p≤3t1∨t2∨t3p≤3pt1p∨t2p∨t3p≤3pt1p+t2p+t3p.
Letd=1,2,3,… By system (7), we can derive that, for δ0+d−1≤δ≤δ0+d,(24)y1δ=y1δ0+d−1+∫δ0+d−1δhδ0l,y1ll−δ0χ−1dl+∫δ0+d−1δϕδ0l,y1ll−δ0χ−1dWl.
Then,(25)yδ≤y1δ0+d−1+∫δ0+d−1δhδ0l,y1ll−δ0χ−1dl+∫δ0+d−1δϕδ0l,y1ll−δ0χ−1dWl.
Hence,(26)y1δp≤3py1δ0+d−1p+3p∫δ0+d−1δhδ0l,y1ll−δ0χ−1dlp+3p∫δ0+d−1δϕδ0l,y1ll−δ0χ−1dWlp.
Therefore,(27)supδ0+d−1≤δ≤δ0+dy1δp≤3py1δ0+d−1p+3pE∫δ0+d−1δ0+dhδ0l,y1ll−δ0χ−1dlp+3psupδ0+d−1≤δ≤δ0+d∫δ0+d−1δϕδ0l,y1ll−δ0χ−1dWlp.Taking the expectation of (27), we obtain(28)Esupδ0+d−1≤δ≤δ0+dy1δp≤3pEy1δ0+d−1p+3pE∫δ0+d−1δ0+dhδ0l,y1ll−δ0χ−1dlp+3pEsupδ0+d−1≤δ≤δ0+d∫δ0+d−1δϕδ0l,y1ll−δ0χ−1dWlp.
Using (22), we have(29)Ey1δ0+d−1p≤ξ1yδ0pΞχ−ξ2,d−1.
Define(30)I1=E∫δ0+d−1δ0+dhδ0l,y1ll−δ0χ−1dlp.(31)I2=Esupδ0+d−1≤δ≤δ0+d∫δ0+d−1δϕδ0l,y1ll−δ0χ−1dWlp.
In view of (21) and (22) and the Hölder inequality,(32)I1≤∫δ0+d−1δ0+dl−δ0pχ−1p−1dlp−1∫δ0+d−1δ0+dEhδ0l,y1lpdl≤L1p−1∫δ0+d−1δ0+dEhδ0l,y1lpdl≤L1p−1M∫δ0+d−1δ0+dEy1lpdl≤L1p−1Mξ1yδ0p∫δ0+d−1δ0+dΞχ−ξ2,δ−δ0dl≤L1p−1Mξ1yδ0pΞχ−ξ2,d−1,where L1=p−1/χp−1dχp−1/p−1−d−1χp−1/p−1.
On the other hand, by (21) and (22), the Burkholder–Davis–Gundy inequality and Hölder inequality, we derive that(33)I2≤ΘpE∫δ0+d−1δ0+dϕδ0l,y1l2l−δ02χ−1dlp/2≤Θp∫δ0+d−1δ0+dl−δ02pχ−1p−2dlp−2/2E∫δ0+d−1δ0+dϕδ0l,y1lpdl≤ΘpL2p−2/2E∫δ0+d−1δ0+dϕδ0l,y1lpdl≤ΘpL2p−2/2Mξ1yδ0pΞχ−ξ2,d−1,where L2=p−2/p2χ−1−2dp2χ−1−2/p−2−d−1p2χ−1−2/p−2 and Θp>0 which only depends on p. Using (29)–(33) and in view of (28),(34)Esupδ0+d−1≤δ≤δ0+dy1δp≤3pξ1yδ0pΞχ−ξ2,d−11+ML1p−1+ΘpL2p−2/2.
Letυ∈0,ξ2ε be arbitrary, then Chebyshev theorem yields that(35)ℙsupδ0+d−1≤δ≤δ0+dy1δp≤ξ1yδ0pΞχ−ξ2−υ,d≤Ξχξ2−υ,dξ1yδ0pEsupδ0+d−1≤δ≤δ0+dy1δp≤3pΞχξ2−υ,dΞχ−ξ2,d−11+ML1p−1+ΘpL2p−2/2.
SetVd=Ξχξ2−υ,dΞχ−ξ2,d−1. It is obvious to see that Vd∼Ξχ−υ,d.
By the Borel–Cantelli lemma, there existsd0=d0ω˜ such that for all ω˜∈Ω˜ and d≥d0, we obtain(36)supδ0+d−1≤δ≤δ0+dy1δp≤ξ1yδ0pΞχ−ξ2−υ,d,a.s.which implies that, for δ∈δ0+d−1,δ0+d and d∈ℕ∗,(37)y1δp≤ξ1yδ0pΞχ−ξ2−υ,d,a.s.
Lettingυ⟶0, we deduce that, for any δ∈δ0+d−1,δ0+d and d∈ℕ∗,(38)y1δp≤ξ1yδ0pΞχ−ξ2,δ−δ0,a.s.,which completes the proof.Remark 5.
We can see here the importance of Theorem2 to make the relation between y1-ESPM and y1-ASES, which means that under condition (21) the y1-ESPM implies y1-ASES.


## 4. Exponential Instability in thep-th Moment

This section is devoted to presenting a sufficient condition that guarantees the exponential instability of system (7) in the p-th moment. The interest of this section is to provide the importance to prove the instability of system (7) in the p-th moment with respect to all variables before showing the partial stability in our example in Section 5.Definition 7.
Letp>0. The solution yδ=y1δ,y2δ of system (7) is called exponentially unstable in the p-th moment, if there exist positive numbers ξ1 and ξ2 satisfying, for any yδ0∈ℝn,(39)Eyδp≥ξ1yδ0pΞχ−ξ2,δ−δ0,∀δ≥δ0≥0.
Ifp=2, system (7) is exponentially unstable in mean square.Remark 6.
In order to show the importance of Theorem1, we cite the following theorem.Theorem 3.
Letpbe a positive constant. Suppose that there exist positive constantsα1, α2, andα3. Assume that there existWδ,y∈C1,2δ0,+∞×ℝn,ℝ+such that(i)
α1yp≤Wδ,y≤α2yp,∀δ,y∈δ0,+∞×ℝn(ii)
ℒχδ0Wδ,yδ≥−α3yδp,∀δ,y∈δ0,+∞×ℝn
Then, system (7) is exponentially unstable in the p-th moment.Proof.
Proceeding as the proof of Theorem1, we have(40)EΞχα3α1,δ∧σd−δ0Wδ∧σd,yδ∧σd=Wδ0,yδ0+E∫δ0δ∧σdΞχα3α1,l−δ0l−δ0χ−1α3α1Wl,yl+ℒχδ0Wl,yldl.
By conditionsi and ii, one has(41)α2EΞχα3α1,δ∧σd−δ0yδ∧σdp≥EΞχα3α1,δ∧σd−δ0Wδ∧σd,yδ∧σd≥α1yδ0p.
Lettingd⟶∞, thus(42)α2EΞχα3α1,δ−δ0yδp≥α1yδ0p.
Therefore, we can obtain(43)Eyδp≥α1α2Ξχ−α3α1,δ−δ0yδ0p.
Finally, system (7) is exponentially unstable in the p-th moment.Remark 7.
The instability analysis of Theorem3 is based on the Lyapunov function Wδ,y. It is worth pointing out that the instability Theorem 3 can also handle the cases of conformable stochastic functional differential equations and conformable neutral stochastic functional differential equations.


## 5. Example

Consider the following CSS:(44)Tδ0χy1δ=h1δ,y+ϕ1δ,ydW1δdδ,Tδ0χy2δ=h2δ,y+ϕ2δ,ydW2δdδ,where χ∈1/2,1, yδ=y1δ,y2δ∈ℝ2, yδ0=y0=y01,y02, and Wiδi∈1,2 are one-dimensional Brownian motions. Let(45)h1δ,y=−2y1δ1+y22δ,h2δ,y=y2δ,ϕ1δ,y=y1δ21+y22δδ−δ01−χ2,ϕ2δ,y=2y1δ1+y22δ2δ−δ01−χ2.We will show that system (44) is exponentially unstable in mean square with respect to all variables.LetWδ,y=y12+y22.Thus,(46)y2≤Wδ,y≤2y2,(47)ℒχδ0Wδ,yδ=2y12δ+y22δ≥−2yδ2.Consequently, by Theorem3, system (44) is exponentially unstable in mean square.Now, setW1δ,y=y12.Then,(48)y12≤Wδ,y≤2y12,(49)ℒχδ0Wδ,yδ=−2y12δ1+y22δ≤−2y12δ.Then,α1=1, α2=α3=2, and p=2. Therefore, all assumptions of Theorem 1 are satisfied. Therefore, system (44) is y1-ESMS.


## 6. Conclusion

This paper investigated conformable stochastic systems which include the conformable fractional derivative and the Brownian motion.Using the Lyapunov method and some classical stochastic calculus techniques like Itô’s formula, we prove the ASES and the ESPM with respect to part of the variables of conformable stochastic systems. Finally, an example has been presented to show the interest of our results. Also, there are some pending questions to be solved in future work, for example, the partial stabilization of an unstable conformable neutral stochastic functional differential equations.



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