Pth moment attractivity analysis for first order uncertain differential systems

Abstract

It is generally considered that attractivity is a concept that describes the overall characteristics of a system. This paper aims to study Pth moment attractivity for one order uncertain differential systems. According to the theory of uncertain differential systems, the concept of Pth moment attractivity is given. Moreover, the Pth moment attractivity of a class of nonlinear uncertain differential systems is studied and the judgment conditions of linear uncertain differential systems are derived.

Keywords

Pth moment attractivity uncertain differential systems concept judgment conditions

1 Introduction

Differential system can accurately describe objective phenomena. It is a very important method to solve mathematical and life problems. Lyapunov proposed the motion stability theory, which is used to solve the trend problem that the initial perturbation of the solution of the equation does not affect the solution of the original equation, and has been widely used in engineering technology, astronomy, and physics.

The motion of the state of a system can be described by differential equations. Due to the complexity of the world, the system is often disturbed by external factors. In these cases, stochastic or fuzzy differential system may be used to deal with this situation. So research on these systems arised. For example, see the literatures [1–6] and so on.

There is a lot of uncertainty in all its forms, for instance “about 1000m”, “cold”, “high”, etc. These are what we call human uncertainty. For handling human uncertainty, the theory with uncertainty was founded [7] and refined [8] by Liu. Since stability for uncertain differential systems(u-d-ss) was introduced by Liu [9], the study of stability has received much attention. Some investigations on the stability of u-d-ss may be found in literature such as stability for u-d-ss [10–12], multi-dimensional u-d-s [13–15], u-d-s with jumps [16–19], uncertain delay differential systems [20, 21].

For a stable system, if a perturbation from the initial state is small enough, it will lead to a corresponding deviation (disturbed motion from the equilibrium state of a system), which is also small enough. Furthermore, if all disturbed motion goes back to the original equilibrium state when time tends to infinity, this stable system is asymptotically stable. The difference between stability and asymptotically stability lies in whether the system is attractive. Some attractivity concepts of u-d-ss was introduced in [22–25]. And the corresponding judging conditions for attractivity are given.

With the widespread use of u-d-ss, there is a need to study their attractivity. In this paper, we will introduce the Pth moment attractivity concept of solutions for u-d-ss. Then the corresponding conditions of attractivity will be deduced. The rest of this article is structured as below. The basic theories and theorems supporting this paper will be introduced in Section 2. In Section 3, we will present the concept of Pth moment attractivity of solutions. In Section 4, the Pth moment attractivity of a class of nonlinear u-d-ss will be studied and the judgment conditions of linear u-d-ss will be derived.

2 Preliminary

In order to facilitate, we will review some useful concepts and theorems established by Professor Liu in the theory of uncertainty. Suppose that the triplet (Γ, $L$ , $M$ ) is an uncertainty space. If ζ is a measurable function from (Γ, $L$ , $M$ ) to the set of real numbers, it is an uncertain variable, so to speak. In other words, for every Borel set of real numbers, {ζ ∈ B} = {γ ∈ Γ ∣ ζ (γ) ∈ B} is an event.

Definition 2.1. ([9]) In any partition of closed interval [A₀, A₁], arbitrarily insert (n+1) points, and set point for A₀ = t₁ < t₂ ⋯ < t_n+1 = A₁, the mesh can be noted as following

$Δ = max_{1 \leq j \leq n} | t_{j + 1} - t_{j} | .$

If the following limit exists a.s. and is finite.

$lim_{Δ \to 0} \sum_{j = 1}^{n} Z_{t_{j}} \cdot (C_{t_{j + 1}} - C_{t_{j}}) .$

Then we call this limit value the integral of Z_t, i.e,

$\int_{A_{0}}^{A_{1}} Z_{t} d C_{t} = lim_{Δ \to 0} \sum_{j = 1}^{n} Z_{t_{j}} \cdot (C_{t_{j + 1}} - C_{t_{j}}) .$

Such a case, Z_t is considered integrable.

Definition 2.2. ([9]) With respect to t, supposed that g_t is an integrable function. Then, at each time s

$\int_{0}^{s} g_{t} d C_{t}$

is a normal uncertainty variable with

$\int_{0}^{s} g_{t} d C_{t} \sim N (0, \int_{0}^{s} | g_{t} | d t) .$

Definition 2.3. ([26]) An uncertain differential system is as follows

$d Z_{t} = G_{1} (t, Z_{t}) d t + G_{2} (t, Z_{t}) d C_{t}$ (1)

And the solution Z_t of (1) is an uncertainty process which satisfies $Z_{t} = Z_{0} + \int_{0}^{t} G_{1} (s, Z_{s}) d s + \int_{0}^{t} G_{2} (s, Z_{s}) d C_{s}$ .

Definition 2.4. ([22]) If for any two solutions Z_1t and Z_2t whose initial values are Z₁₀ and Z₂₀, respectively. $d Z_{t} = G_{1} (t, Z_{t}) d t + G_{2} (t, Z_{t}) d C_{t}$ is considered

attractive in mean if here exists $σ \in R^{+}$ which satisfies when |Z₁₀ - Z₂₀| < σ, it is obtained

$lim_{t \to + \infty} E [| Z_{1 t} - Z_{2 t} |] = 0 .$

attractive in measure if for any given $ϵ \in R^{+}$ , here exists $σ \in R^{+}$ which satisfies when |Z₁₀ - Z₂₀| < σ, it is obtained

$lim_{t \to + \infty} M {| Z_{1 t} - Z_{2 t} | > ϵ} = 0 .$

Theorem 2.1 ([7]) If $H$ defined on [0, ∞) is a nonnegative even function and monotonically increasing, then for any given $t \in R^{+}$ , one can be obtained.

$M {| ς | \geq t} \leq \frac{E [H (ς)]}{H (t)} .$

where ς is an uncertain variable.

Theorem 2.2 ([27]) If Z_t is an integrable uncertainty process on [m₁, m₂] about t, it can be obtained.

$| \int_{m_{1}}^{m_{2}} Z_{t} (γ) d C_{t} (γ) | \leq Lip (γ) \int_{m_{2}}^{m_{2}} | Z_{t} (γ) | d t,$

in which where K (γ) is the Lipschitz constant of the sample path C_t (γ).

Theorem 2.3 ([28]) If $G_{1}$ is a binary function,

$d Z_{t} = G_{1} (t, Z_{t}) d t + σ_{t} Z_{t} d C_{t}$ (2)

has one solution

$Z_{t} = Y_{t}^{- 1} X_{t}$

where

$Y_{t} = exp (- \int_{0}^{t} σ_{s} d s)$

Moreover, X_t satisfies the system

$d X_{t} = Y_{t} G_{1} (t, Y_{t}^{- 1} X_{t}) d t$

with initial value X₀ = Z₀.

Theorem 2.4 ([11]) If C _t is a canonical process, it can be obtained

$lim_{x \to + \infty} M {γ \in Γ ∣ K (γ) \leq x} = 1 .$

in which, for each γ, K (γ) is a Lipschitz constant of C_t (γ).

3 Concept of Pth moment attractivity

We will introduce Pth moment attractivity concept for

$d Z_{t} = G_{1} (t, Z_{t}) d t + G_{2} (t, Z_{t}) d C_{t}, t \geq t_{0} .$ (3)

Definition 3.1. If Z_1t and Z_2t are any two solutions of ((3)) with different initial values Z₁₀ and Z₂₀, respectively, the system ((3)) is considered attractive in Pth moment if here exists σ > 0 which satisfies when |Z₁₀ - Z₂₀| < σ, it follows that

$lim_{t \to + \infty} E [| Z_{1 t} - Z_{2 t} |^{P}] = 0 .$

Especially, if P = 1, it is attractive in mean.

Example 3.1. Analyse a system with the following form

$d Z_{t} = - \frac{1}{t + 1} Z_{t} d t + \frac{1}{4 t^{2} + 1} Z_{t} d C_{t} .$

Then we can easily get

$\begin{matrix} Z_{1 t} - Z_{2 t} = (Z_{10} - Z_{20}) \\ exp (\int_{0}^{t} - \frac{1}{s + 1} d s + \int_{0}^{t} \frac{1}{4 s^{2} + 1} d C_{s}) . \end{matrix}$

It follows that

$\begin{matrix} E [| Z_{1 t} - Z_{2 t} |^{2}] = | Z_{10} - Z_{20} |^{2} exp \\ (2 \int_{0}^{t} - \frac{1}{s + 1} d s) E \\ [exp (2 \int_{0}^{t} \frac{1}{4 s^{2} + 1} d C_{s})] . \end{matrix}$

Since $\int_{0}^{+ \infty} | \frac{1}{4 s^{2} + 1} | d s < \frac{π}{2 \sqrt{3}}$ and $\int_{0}^{t} \frac{1}{4 s^{2} + 1} d C_{s} \sim N (0, \int_{0}^{t} | \frac{1}{4 s^{2} + 1} | d s)$ , we have

$\begin{matrix} E [exp (2 \int_{0}^{t} \frac{1}{4 s^{2} + 1} d C_{s})] \\ = \int_{0}^{1} exp (\frac{2 \sqrt{3} \int_{0}^{t} | \frac{1}{4 s^{2} + 1} | d s}{π} ln \frac{α}{1 - α}) d α \\ = \int_{0}^{1} {(\frac{α}{1 - α})}^{\frac{2 \sqrt{3} \int_{0}^{t} \frac{1}{4 s^{2} + 1} d s}{π}} d α \\ = \int_{0}^{1} {(\frac{1 - α}{α})}^{\frac{2 \sqrt{3} \int_{0}^{t} \frac{1}{4 s^{2} + 1} d s}{π}} d α \\ < \int_{0}^{1} {(\frac{1}{α})}^{\frac{2 \sqrt{3} \int_{0}^{+ \infty} \frac{1}{4 s^{2} + 1} d s}{π}} d α \\ < + \infty . \end{matrix}$

we immediately obtain

$lim_{t \to + \infty} E [| Z_{1 t} - Z_{2 t} |^{2}] = 0 .$

Thus $d Y_{t} = - \frac{1}{t + 1} Z_{t} d t + \frac{1}{4 t^{2} + 1} Z_{t} d C_{t}$ is attractive in 2th moment.

Example 3.2. We analyse a u-d-ss with the following form

${dZ}_{t} = μ d t + σ d C_{t} .$

For two different initial values Z₁₀ and Z₂₀, we suppose that Z_1t and Z_2t are any two solutions with them, respectively. Then

$Z_{1 t} = Z_{10} + μ t + σ C_{t},$

$Z_{2 t} = Z_{20} + μ t + σ C_{t} .$

Since

$\begin{matrix} lim_{t \to + \infty} E [| Z_{1 t} - Z_{2 t} |^{P}] \\ = lim_{t \to + \infty} | Z_{10} - Z_{20} |^{P} = | Z_{10} - Z_{20} |^{P} \neq 0 . \end{matrix}$ (4)

Then the u-d-s dZ_t = μ dt + σ dC_t is not attractive in Pth moment.

4 Attractivity results

This section is divided into two parts. In part 1, we deduce that Pth moment attractivity implies attractivity in measure, and consider the judgment conditions of Pth moment attractivity for a special class of nonlinear u-d-s. In part 2, we present the sufficient and necessary of Pth moment attractivity for linear u-d-ss.

4.1 Attractivity for nonlinear u-d-s

Theorem 4.1 If the u-d-s ((3)) is attractive in Pth moment, then it is attractive in measure.

Proof. According to the definition of attractivity in Pth moment, here exists $σ \in R^{+}$ , if |Z₁₀ - Z₂₀| < σ, it is easy to get

$lim_{t \to + \infty} E [| Z_{1 t} - Z_{2 t} |^{P}] = 0 .$

For any given $ϵ \in R^{+}$ , from Markov inequality, we have

$lim_{t \to + \infty} M {| Z_{1 t} - Z_{2 t} | > ϵ} \leq \frac{{lim}_{t \to + \infty} E [| Z_{1 t} - Z_{2 t} |^{P}]}{ϵ^{P}} = 0 .$

Thus attractivity in Pth moment implies attractivity in measure.

Theorem 4.2 For any two positive integer P₀ and P₁, and P₁ > P₀ > 0, then u-d-s ((3)) is attractive in P₀th moment if it is attractive in P₁th moment.

Proof. Following the definition of attractivity in P₁th moment, there exists σ > 0 satisfying when |Z₁₀ - Z₂₀| < σ, we can get

$lim_{t \to + \infty} E [| Z_{1 t} - Z_{2 t} |^{P_{1}}] = 0,$

From $h \ddot{o} {lder}^{,} s$ inequality, we can get

$\begin{matrix} E [| Z_{1 t} - Z_{2 t} |^{P_{0}}] & = E [| | Z_{1 t} - Z_{2 t} |^{P_{0}} \cdot 1 |] \\ \leq \sqrt[P_{1} / P_{0}]{E [| Z_{1 t} - Z_{2 t} |^{P_{0} P_{1} / P_{0}}]} \\ \sqrt[P_{1} / (P_{1} - P_{0})]{E [1^{P_{1} / (P_{1} - P_{0})}]} \\ = \sqrt[P_{1} / P_{0}]{E [| Z_{1 t} - Z_{2 t} |^{P_{1}}]} . \end{matrix}$

Thus attractivity in P₁th moment implies attractivity in P₀th moment for u-d-s ((3)) with P₁ > P₀ > 0. Then the theorem has been proved.

Theorem 4.3. $\forall x_{1}, x_{2} \in R, t \geq 0,$ if the following inequality holds,

$\begin{matrix} | G_{2} (t, x_{1}) - G_{2} (t, x_{2}) | \leq L (t) | x_{1} - x_{2} |, \end{matrix}$

in which, when t ∈ [0, + ∞), L(t) is not only bounded but also integrable. Then the nonlinear uncertain differential system

$d Z_{t} = σ_{t} Z_{t} d t + G_{2} (t, Z_{t}) d C_{t},$ (5)

is attractive in Pth moment if and only if

$\int_{0}^{+ \infty} σ_{s} d s = - \infty .$

Proof. By means of the Theorem 2.3,

$\begin{matrix} Z_{1 t} = exp (\int_{0}^{t} σ_{s} d s) [Z_{10} + \int_{0}^{t} exp \\ (- \int_{0}^{s} σ_{r} d r) G_{2} (s, Z_{1 s}) d C_{s}] . \end{matrix}$

and

$\begin{matrix} Z_{2 t} = exp (\int_{0}^{t} σ_{s} d s) [Z_{20} + \int_{0}^{t} exp \\ (- \int_{0}^{s} σ_{r} d r) G_{2} (s, Z_{2 s}) d C_{s}] \end{matrix}$

Then for each γ, by using Grownwall’s inequality,

$\begin{matrix} \begin{matrix} exp (- \int_{0}^{t} σ_{s} d s) | Z_{1 t} (γ) - Z_{2 t} (γ) | \\ = | (Z_{10} - Z_{20}) + \int_{0}^{t} exp (- \int_{0}^{s} σ_{r} d r) (G_{2} (s, Z_{1 s} (γ)) - G_{2} (s, Z_{2 s} (γ))) d C_{s} | \\ \leq | Z_{10} - Z_{20} | + \int_{0}^{t} exp (- \int_{0}^{s} σ_{r} d r) | G_{2} (s, Z_{1 s} (γ)) - G_{2} (s, Z_{2 s} (γ)) | | d C_{s} (γ) | \\ \leq | Z_{10} - Z_{20} | + \int_{0}^{t} exp (- \int_{0}^{s} σ_{r} d r) L (s) K (γ) | Z_{1 s} (γ) - Z_{2 s} (γ) | d s \\ \leq | Z_{10} - Z_{20} | exp (K (γ) \int_{0}^{t} L (s) d s) . \end{matrix} \end{matrix}$

$\begin{matrix} \begin{matrix} | Z_{1 t} (γ) - Z_{2 t} (γ) | \\ \leq | Z_{10} - Z_{20} | exp (K (γ) \int_{0}^{t} L (s) d s) exp (\int_{0}^{t} σ_{s} d s) \\ \leq | Z_{10} - Z_{20} | exp (K (γ) \int_{0}^{+ \infty} L (s) d s) exp (\int_{0}^{t} σ_{s} d s) . \end{matrix} \end{matrix}$

From the Theorem 2.4, ∀ς > 0,

$M {γ \in Γ ∣ K (γ) \leq M} \geq 1 - ς .$

And based on $\int_{0}^{+ \infty} L (s) d s < + \infty$ , Then

$\begin{matrix} E [| Z_{1 t} - Z_{2 t} |^{P}] \\ \leq | Z_{10} - Z_{20} |^{P} exp (PM \int_{0}^{+ \infty} L (s) d s) exp (P \int_{0}^{t} σ_{s} d s) . \end{matrix}$

So $lim_{t \to + \infty} E [| Z_{1 t} - Z_{2 t} |^{P}] = 0$ if and only if $\int_{0}^{+ \infty} σ_{s} d s = - \infty$ .

Example 4.1. Think about the following system

$d Z_{t} = - (1 + t^{2}) Z_{t} d t + exp (- 2 t) cos Z_{t} d C_{t} .$

$\forall x_{1}, x_{2} \in R, t \geq 0,$ we can get

$\begin{matrix} | exp (- 2 t) cos x_{1} - exp (- 2 t) cos x_{2} | \\ \leq exp (- 2 t) | x_{1} - x_{2} |, \end{matrix}$

and $\int_{0}^{+ \infty} σ_{s} d s = \int_{0}^{+ \infty} - (1 + t^{2}) d s = - \infty .$ By Theorem 4.3, the system

$d Z_{t} = - (1 + t^{2}) Z_{t} d t + exp (- 2 t) cos Z_{t} d C_{t}$

is attractive in Pth moment.

4.2 Attractivity for linear u-d-s

Theorem 4.4. Assuming that A_1t, A_2t, B_1t, B_2t are all continuous functions on [0, + ∞). If $\int_{0}^{+ \infty} | B_{1 s} | d s < \frac{π}{\sqrt{3} P} .$ Then the linear u-d-s

$d Z_{t} = (A_{1 t} Z_{t} + A_{2 t}) d t + (B_{1 t} Z_{t} + B_{2 t}) d C_{t}$ (6)

is attractive in Pth moment if and only if

$\int_{0}^{+ \infty} A_{1 s} d s = - \infty .$

Proof. Suppose that Z_1t and Z_2t are any two solutions of (5) with different initial values Z₁₀ and Z₁₀, respectively. We can easily get the following equation

$Z_{1 t} - Z_{2 t} = (Z_{10} - Z_{20}) exp (\int_{0}^{t} A_{1 S} d s + \int_{0}^{t} B_{1 s} d C_{s}) .$

Then we can get

$\begin{matrix} E [| Z_{1 t} - Z_{2 t} |^{P}] = | Z_{10} - Z_{20} |^{P} exp \\ (P \int_{0}^{t} A_{1 S} d s) E [exp (P \int_{0}^{t} B_{1 s} d C_{s})] . \end{matrix}$

Since $\int_{0}^{+ \infty} | B_{1 s} | d s < \frac{π}{\sqrt{3} P}$ and $\int_{0}^{t} B_{1 s} d C_{s} \sim N (0, \int_{0}^{t} | B_{1 s} | d s)$ , we have

$\begin{matrix} E [exp (p \int_{0}^{t} B_{1 s} d C_{s})] & = \int_{0}^{1} exp (\frac{p \sqrt{3} \int_{0}^{t} | B_{1 s} | d s}{π} ln \frac{α}{1 - α}) d α \\ = \int_{0}^{1} {(\frac{α}{1 - α})}^{\frac{p \sqrt{3} \int_{0}^{t} | B_{1 s} | d s}{π}} d α \\ = \int_{0}^{1} {(\frac{1 - α}{α})}^{\frac{p \sqrt{3} \int_{0}^{t} | B_{1 s} | d s}{π}} d α \\ < \int_{0}^{1} {(\frac{1}{α})}^{\frac{p \sqrt{3} \int_{0}^{+ \infty} | B_{1 s} | d s}{π}} d α \\ < + \infty . \end{matrix}$

Obviously,

$lim_{t \to + \infty} E [| Z_{1 t} - Z_{2 t} |^{p}] = 0$

is equivalent to

$\int_{0}^{+ \infty} A_{1 s} d s = - \infty .$

Therefore the linear u-d-s (5) is attractive in Pth moment if and only if

$\int_{0}^{+ \infty} A_{1 s} d s = - \infty .$

Example 4.2. For the following u-d-s

$d Z_{t} = - exp (t) Z_{t} d t + exp (- 8 t) Z_{t} d C_{t} .$

Since A_1t = - exp(t) and B_1t = exp(-8t) , it is easy to get

$\int_{0}^{+ \infty} A_{1 s} d s = - \infty and \int_{0}^{+ \infty} | B_{1 s} | d s = \frac{1}{8} < \frac{π}{14 \sqrt{3}} .$

Thus dZ_t = - exp(t) Z_t dt + exp(-8t) Z_t dC_t is attractive in 14th moment.

5 Conclusion

In the article, we presented attractivity in Pth moment concept of uncertain differential systems. Based on Grownwall’s inequality and Mokov inequality, we considered attractivity in Pth moment for a kind of nonlinear u-d-s and presented the judgment conditions of attractivity in Pth moment for linear u-d-ss.

Footnotes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No.61273009).

Compliance with ethical standards

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