New stability theorems of uncertain differential equations with time-dependent delay

Abstract: Stability in measure and stability in mean for uncertain differential equations with timedependent delay have been investigated, which are not applicable for all situations, for the sake of completeness, this paper mainly gives the concepts of stability almost surely, in p-th moment and in distribution, and proves the sufficient conditions for uncertain differential equations with timedependent delay being stable almost surely, in p-th moment and in distribution, respectively. In addition, the relationships among stability in measure, stability in p-th moment, and stability in distribution for the uncertain differential equation with time-dependent delay are also discussed.


Introduction
More than half a century ago, when the Itô's [1] landmark work on stochastic differential equations (Itô, 1951) comes out, the stochastic differential equations (SDEs), as a new branch of mathematics, have aroused great interest in academic circles. In recent decades, SDEs have been widely applied in many fields such as physical, biological, engineering, medical, social sciences, economics, finance and other models hidden in the observed data. One of the best important works in this research field is to discuss the stability of such systems. For example, in 2017, Zhu and Zhang [2] considered the pth moment exponential stability criterion instead of the mean square exponential stability criterion and discrete-time state observations depend on time delays. In 2018, Zhu [3] studied the pth moment exponential stability problem for a class of stochastic delay differential equations driven by Lévy processes. In 2019, Zhu [4] investigated the stabilization problem of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control by introducing the notation of input-to-state practical stability and an event-triggered strategy. In 2020, Ma et al. [5] focused on the the mean square and almost sure practical exponential stability for a class of stochastic age-dependent capital system with Lévy noise. Zhu and Huang [6] investigated the pth moment exponential stability problem for a class of stochastic time-varying delay nonlinear systems driven by G-Brownian motion.
It is well known that all results about SDEs are based on an axiomatic probability theory, and large amounts of sample data is needed to obtain the frequency of their random disturbances. Furthermore, their distribution functions can be obtained. However, in reality, people seem to lack data or the size of sample data applied in practice are less in some cases, such as the emerging infectious disease model, the new stock model and so on. Although sometimes we have a lot of available sample data, the frequency obtained by sample data is, unfortunately, not close enough to the distribution function obtained in some practical problems, and we need to invite some domain experts to evaluate the belief degree that each event may happen in these situations. Human uncertainty with respect to belief degrees can play a crucial role in addressing the issue of indeterminate phenomenon. For describing human uncertainty, a considerable amount of literatures about uncertainty theory (for example, Liu [7,8]) have been published on Springer-Verlag. Recently, investigators have examined the effects of uncertainty theory on programming model and application [9], risk and reliability analysis [10], set theory and control theory [11], logic logic for modeling human language [12] and currency model [13] and the references therein.
In the framework of uncertainty theory, there has been an increasing amount of literature on uncertain differential equations [14][15][16][17]. In particular, uncertain process and Liu process for such perturbed systems were investigated extensively in [14,15]. Existence and uniqueness theorem for uncertain differential equations were derived in [16,17]. A broader perspective has been adopted by Liu [18] who argued that an analytic method can solve uncertain differential equations. Similarly, Yao [19], Liu [20], Wang [21] found that the nonlinear uncertain differential equation, semi-linear uncertain differential equation, the general uncertain differential equation have their analytic solution. More importantly, numerous studies have attempted to explain solution of an uncertain differential equation (for example, Yao and Chen, [22]; Yao, [23]). Using these theoretical approaches, researchers have been able to develop the applications of uncertain differential equation successfully. A great deal of previous research into applications has focused on financial systems [24,30], game theory [25], heat conduction [26], uncertain wave equation [27][28][29], uncertain differential equations with jump [31,32] and the references therein.
Since 2009, after the concept of stability [15] for an uncertain differential equation showed firstly, there have been more interest in the stabilities of uncertain differential equation. Being stable in measure [33], being stable in p-th moment [34], stability in mean [35], exponential stability [36], almost sure stability [37] and being stable in inverse distribution [38] were successively investigated. Researchers often use uncertain delay differential equations to describe such physical conditions that are related to both the current state and the past state (for example, Barbacioru [39]; Ge et al. [40]; Wang et al. [41]). Following that, Wang et al. [42,43] put forward the stability in measure, stability in mean and stability in p-th moment for uncertain delay differential equations, and proved the corresponding stability theorems. Jia et al. [44] discussed the stability in distribution. The above uncertain delay differential equations can be seen as constant delay. Recently, Wang et al. [45] proposed another class of uncertain differential equations with time-dependent delay including stability in measure, stability in mean, which are not applicable for all situations. For completeness, our goal is to provide the almost sure stability, stability in p-th moment and stability in distribution of the uncertain differential equations with time-dependent delay as a supplement. The equation is introduced as follows: where C k is a Liu process with respect to time k, and h and q are two continuous functions, Z 0 is the initial value, δ : [0, +∞) → [0, τ] is a delay function. Its equivalent integral form is as follows: (1. 2) The rest of this article is organized as follows. In Section 2, we define almost sure stability, p-th moment stability and stability in distribution for uncertain differential equations with time-dependent delay, In Section 3, firstly, we derive a sufficient condition of almost sure stability and a corollary for uncertain differential equations with time-dependent delay. Secondly, the p-th moment stability theorem and corollary of uncertain differential equations with time-dependent delay are presented. Thirdly, we prove a sufficient condition of being stable in distribution for uncertain differential equations with time-dependent delay. Fourthly, we prove some relationships about stabilities for the uncertain differential equation with time-dependent delay. Finally, we make a brief conclusion in Section 4.

Some important concepts
In this section, we will define some concepts of stability. Definition 1. The uncertain differential equation with time-dependent delay (1.1) is said to be stable almost surely if for any two solutions Z k andẐ k with different initial states,respectively. We have where M is uncertain measure and Γ is a nonempty set defined in [8].
where E is uncertain expected value defined in [8].
Definition 3. Let Z k andẐ k be two solutions of uncertain differential equation with time-dependent delay (1.1) with different initial states. Assume the uncertainty distributions of Z k andẐ k are Φ k (z) and Ψ k (z), respectively. Then the uncertain differential equation with time-dependent delay (1.1) is said to be stable in distribution if ∀z at which Φ and Ψ are continuous.

Main results
We will give a sufficient condition for (1.1) being stable almost surely.

Stable almost surely
Theorem 1. Assume uncertain differential equation with time-dependent delay (1.1) has an unique solution for each given initial state. Then it is stable almost surely if the coefficients h(k, z,ẑ) and q(k, z,ẑ) satisfy where N k is a bounded function satisfying +∞ 0 N k dk < +∞.

3)
Then for a Lipschitz continuous sample C k (γ), it holds that
According to Gronwall inequality [45], then we have Thus we have Thus, by Theorem 2 in [33], we have Since +∞ 0 N r dr < +∞.
We have is stable almost surely according to Definition 1. This completes the proof.
Corollary 1. Supposing that u ik , v ik , and η ik (i = 1, 2) are real-valued functions, then the linear uncertain differential equation with time-dependent delay is almost surely stable if u ik , v ik , and η ik (i = 1, 2) are bounded, and satisfy According to Theorem 4.1 in [45], we have that linear uncertain differential equation with timedependent delay (3.6) with initial states has a unique solution. Since By using Theorem 1, the linear uncertain differential equation with time-dependent delay (3.6) is almost surely stable. Example 1. Consider an uncertain differential equation with time-dependent delay and According to Theorem 4.1 in [45], we have that linear uncertain differential equation with timedependent delay (3.7) with initial states has a unique solution. Since According to Corollary 1, linear uncertain differential equation with time-dependent delay (3.7) is stable almost surely.
By (3.7), we get Because ∆C k is a normal uncertain variable with expected value 0 and variance ∆k 2 , the distribution function of ∆C k is Φ( We may get a sample pointẑ k of ∆C k from −π ln( 1 α − 1),α ∈ (0, 1), where α is belief degree. So ∆Z k can be given by the following equation.
Thus, Z k can be simulated by the following ordinary differential equation with time-dependent delay.

Stability in p-th moment
This part mainly investigates the stability in p-th moment for uncertain differential equation with time-dependent delay (1.1).
Theorem 2. Assume that uncertain differential equation with time-dependent delay (1.1) has an unique solution for each given initial state. Then it is stable in p-th moment if the coefficients h(k, z,ẑ) and q(k, z,ẑ) satisfy where N k and G k are two bounded functions satisfying .

10)
Then for a Lipschitz continuous sample C k (γ), we have

11)
and where H(γ) is the Lipschitz constant of C k (γ).
According to Gronwall inequality [45], then we have Thus we have where H is a nonnegative uncertain variable. Taking expected value on both sides of expression (3.13), we have . Thus, It follows from the definition of uncertain expected value, Definition 6 in [15] and Theorem 2 [33] in that . Let π √ 3κp = a, so a > 1. Then, Thus, we can get lim Corollary 2. Supposing that u ik , v ik , and η ik (i = 1,2) are real-valued functions, then the linear uncertain differential equation with time-dependent delay is almost surely stable if u ik , v ik , and η ik (i = 1, 2) are bounded, Proof Take h(k, z,ẑ) = u 1k z + v 1kẑ + η 1k and q(k, z,ẑ) = u 2k z + v 2kẑ + η 2k . Let Q denote a common upper bound of |u ik |, |v ik |, and |η ik | (i = 1,2). The inequalities According to Theorem 4.1 in [45], we have that linear uncertain differential equation with timedependent delay (3.15) with initial states has a unique solution. Since . By using Theorem 2, uncertain differential equation with time-dependent delay (3.15) is stable in p-th moment.
Example 2. Consider an uncertain differential equation with time-dependent delay It follows from Theorem 4.1 in [45] that uncertain differential equation with time-dependent delay has a unique solution with given initial states. In addition, real-valued functions exp(−k) is bounded on the interval [0, +∞), and +∞ 0 exp(−k)dk = 1 < +∞.
According to Corollary 2, uncertain differential equation with time-dependent delay (3.16) is stable in p-th moment. Similar to Example 1, we take a = 3 and give the Figure 2, which illustrates the stability furthermore.  Remark 1. When p = 1, stability in p-th moment is just stability in mean.

Stability in distribution
In this section, we investigate the stability in distribution for uncertain differential equation with time-dependent delay, and prove a sufficient condition for an uncertain differential equation with timedependent delay being stable in distribution.
Theorem 3. Assume the uncertain differential equation with time-dependent delay (1.1) has an unique solution for each given initial state. Then it is stable in distribution if the coefficients h(k, z,ẑ) and q(k, z,ẑ) satisfy where N k is a bounded function satisfying +∞ 0 N k dk < +∞.

18)
and Then for a Lipschitz continuous sample C k (γ), we have

20)
and (3.21) By Theorem 4.1 in [38], the inverse uncertainty distributions Φ −1 k (α) and Ψ −1 k (α) of Z k andẐ k satisfy the ordinary differential equation with time-dependent delay It follows that By the Gronwall's inequality, we have there exists a real number Q > 0 such that for any k ≥ 0. Thus, for any given > 0, we set δ = /Q such that So, the uncertain differential equation with time-dependent delay (1.1) is stable in distribution.

Comparison of stability
Theorem 4. If the uncertain differential equation with time-dependent delay (1.1) is stable in p-th moment, then it is stable in measure.
Proof Suppose that Z k andẐ k are two solutions of the Eq (1.1) with different initial states ϕ(k) and ψ(k)(k ∈ [−τ, 0]), respectively. Then, it follows from the definition of stability in p-th moment that (3.26) By Markov inequality, for any given real number > 0, we have Therefore, from Definition 5.1 in [45], p-th moment stability implies the stability in measure.
Theorem 5. For any two real numbers p 1 and p 2 (0 < p 1 < p 2 < +∞), if the uncertain differential equation with time-dependent delay (1.1) is stable in p 2 -th moment, then it is stable in p 1 -th moment.
Proof Suppose that Z k andẐ k are two solutions of the Eq (1.1) with different initial states ϕ(k) and ψ(k)(k ∈ [−τ, 0]), respectively. Then, it follows from the definition of stability in p 2 -th moment that According to Holder's inequality, we have Thus, stability in p 2 -th moment implies stability in p 1 -th moment when p 1 < p 2 .
Theorem 6. If the uncertain differential equation with time-dependent delay (1.1) is stable in measure, then it is stable in distribution.
Proof Suppose that Z k andẐ k are two solutions of the Eq (1.1) with different initial states ϕ(k) and ψ(k)(k ∈ [−τ, 0]), respectively. According to Definition 5.1 in [45], we have for any given number > 0. Suppose z is a given real number. On the one hand, for anyẑ > z, we can get By the monotonicity theorem and subadditivity axiom in uncertainty theory [8], it holds that

Thus we have
We can get M sup On the other hand, for anyz < z, it is obvious that By the monotonicity theorem and subadditivity axiom in uncertainty theory [7], it holds that Thus we have We can get M sup According to Definition 3, the uncertain differential equation with time-dependent delay (1.1) is stable in distribution.
Theorem 7. If the uncertain differential equation with time-dependent delay (1.1) is stable in p-th moment (0 < p < +∞), then it is stable in distribution.
Proof If the uncertain differential equation with time-dependent delay (1.1) is stable in p-th moment (0 < p < +∞), by Theorem 4, the uncertain differential equation with time-dependent delay (1.1) is stable in measure. And by Theorem 6, we obtain the uncertain differential equation with timedependent delay (1.1) is stable in distribution.
Remark 2. Theorem 4, Theorem 5, Theorem 6, and Theorem 7 give the sufficient condition but not the necessary condition for the comparison of stability.

Conclusions
The main goal of the current study is to propose new stabilities called almost sure stability, stability in p-th moment and stability in distribution for uncertain differential equations with time-dependent delay. Meanwhile, the sufficient conditions of these theorems are also provided. Some examples and figures are given to illustrate the stability furthermore. The applications about uncertain differential equations with time-dependent delay will be the focus of our future research.