Partial Stability of Conformable Stochastic Systems

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.

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 general

This paper is organized as follows. In Section 2, 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 by , where is a filtration satisfying the usual conditions. is an -dimensional Brownian motion defined on . The set of -valued -adapted processes is denoted by such that a.s. Let denote the set of functions from to that are right-continuous and have limits on the left. is equipped with the norm and for any . The set of all -measurable bounded -valued random variables are denoted by . Let , , denote the set of all -measurable, -valued random variables satisfies .

Definition 1. [21]. The conformable derivative (CD) of a function is defined byfor all , , while

Definition 2. [21]. The conformable integral (CI) of a function is defined byfor all , .

Lemma 1. [21]. Letand. Then,, we obtain

Lemma 2. [22]. Let, , , , , and the functionssuch thatexists onfor. Then,(i)(ii)(iii)

Remark 1. Let such that exists on . Then, exists on and

Definition 3. [3]. The conformable exponential function is defined bywhere and .

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 on , for any :where the initial condition , with , which is independent of , , , , and are continuous. Suppose that and 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 an -valued stochastic process is a solution of (7) if is continuous, -adapted, and

Definition 5. Let . System (7) is said to be -ASES, if there exist positive constants and satisfying, for any ,

Definition 6. [16]. Let . System (7) is said to be -ESPM, if there exist positive constants and satisfying, for any ,

Remark 3. Note that, in Definition 7, when , system (7) is said to be -exponentially stable in mean square (-ESMS).
Let . By Itô formula (see [6]), one has the following: for ,where

Theorem 1. Letbe a positive constant. Suppose that there exist positive constants, , and. Assume that there existsuch that(i)(ii)Then, system (7) is-ESPM.

Proof. Let be fixed and denoted by the solution of system (7). For each , the stopping time is denoted by such thatIt is easy to see that as . By Itô’s formula, we obtain for Using that is a local martingale with (see [23]) and taking the expectation on both sides of (16), we obtainBy conditions and , one hasLetting , thenHence, we can derive thatTherefore, system (7) is -ESPM. □

Remark 4. It is necessary to give the relation between equation (20) and the -ASES to make our result more interesting.

Theorem 2. Let. Suppose that (20) holds and there is a positive constantsuch that for all

Then, system (7) is -ASES.

Proof. Let be fixed. It follows from (20) that there exists positive constants and such thatFrom ([23], p. 178), for any , , , we haveLet By system (7), we can derive that, for ,Then,Hence,Therefore,Taking the expectation of (27), we obtainUsing (22), we haveDefineIn view of (21) and (22) and the Hölder inequality,where .
On the other hand, by (21) and (22), the Burkholder–Davis–Gundy inequality and Hölder inequality, we derive thatwhere and which only depends on . Using (29)–(33) and in view of (28),Let be arbitrary, then Chebyshev theorem yields thatSet . It is obvious to see that .
By the Borel–Cantelli lemma, there exists such that for all and , we obtainwhich implies that, for and ,Letting , we deduce that, for any and ,which completes the proof.

Remark 5. We can see here the importance of Theorem 2 to make the relation between -ESPM and -ASES, which means that under condition (21) the -ESPM implies -ASES.

4. Exponential Instability in the p-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. Let . The solution of system (7) is called exponentially unstable in the p-th moment, if there exist positive numbers and satisfying, for any ,If , system (7) is exponentially unstable in mean square.

Remark 6. In order to show the importance of Theorem 1, we cite the following theorem.

Theorem 3. Letbe a positive constant. Suppose that there exist positive constants, , and. Assume that there existsuch that(i)(ii)Then, system (7) is exponentially unstable in the p-th moment.

Proof. Proceeding as the proof of Theorem 1, we haveBy conditions and , one hasLetting , thusTherefore, we can obtainFinally, system (7) is exponentially unstable in the p-th moment.

Remark 7. The instability analysis of Theorem 3 is based on the Lyapunov function . 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:where , , , and are one-dimensional Brownian motions. Let

We will show that system (44) is exponentially unstable in mean square with respect to all variables.

Let .

Thus,

Consequently, by Theorem 3, system (44) is exponentially unstable in mean square.

Now, set .

Then,

Then, , , and . Therefore, all assumptions of Theorem 1 are satisfied. Therefore, system (44) is -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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was funded by the Deanship of Scientific Research at Jouf University under Grant No. DSR-2021-03-0328.