Stability and attractivity in pessimistic value for uncertain dynamical system

Abstract

The pessimistic value of uncertain variables is a critical value to deal with optimization problems in environments with uncertainty. In many uncertain decision problems, pessimistic values at a certain level of reliability sometimes get attention, such as the problem that the objective function is time or cost. This article introduces two definitions of pessimistic value stability and attractivity. And the corresponding judgment conditions of attractivity are presented for linear differential systems with uncertainty. Furthermore, pessimistic value stability is analyzed for three kinds of nonlinear uncertain differential systems. Then pessimistic value attractivity is considered for a kind of nonlinear differential system with uncertainty.

Keywords

Pessimistic value uncertainty stability attractivity differential systems

1 Introduction

Uncertain Dynamical System (U-D-S) was established in an uncertain theoretical environment [1], which is a differential equation involving the canonical Liu process [2]. In many areas, U-D-Ss have been applied. For instance, uncertain optimal control [3], and financial market with uncertainty [4, 5]. Recently, much research has been done (see [6 –9]).

For U-D-Ss, Liu [2] first presented stability. Then the literature [10] derived a few stability theorems. In 2014, almost sure stability was considered by Liu and Fei [11]. In 2015, stability in mean was studied by Yao et al. [12]. Next, Sheng and Gao [13] analyzed exponential stability in 2016. Soon, more attention has been paid to the stability. The main stability achievements are as follows: multi-dimensional U-D-Ss (see [14 –16]), U-D-Ss with jumps (see [17 –20]), uncertain differential delay system (see [21 –23]).

Liu [2] introduced the pessimistic value of an uncertain variable to handle pessimistic problems under uncertain environments in 2009. Following that, the article [24, 25] presented attractivity concepts of U-D-Ss.

In this paper, our aim is to consider pessimistic value stability and attractivity. Then we will analyze the corresponding judgment conditions of pessimistic value stability and attractivity. In Section 2, Several basic uncertainty definitions and theorems will be reviewed. Section 3 will introduce two pessimistic value stability and attractivity concepts. In Section 4, some corresponding judgment conditions are given.

2 Preliminary

In order to improve the theoretical chair for our research, some important concepts and theorems are reviewed.

Definition 2.1. ([2]) Let 0 < α ≤ 1. Then the α-pessimistic value to an uncertain variable ζ is defined by $ζ_{inf} (α) = inf {r ∣ M {ζ \leq r} \geq α} .$

Definition 2.2. ([4]) Let f₁ and f₂ be two given binary functions, and C_t be a canonical process. Then $d Z_{t} = f_{1} (t, Z_{t}) d t + f_{2} (t, Z_{t}) d C_{t}$ (1) is called a U-D-S

Definition 2.3. ([2]) Let Y_t and Z_t be any two solutions of (1), and their initial values be Y₀ and Z₀, respectively. The U-D-S (1) is called stability in measure if for any given number ɛ > 0, the following equation $lim_{| Y_{0} - Z_{0} | \to 0} M {| Y_{t} - Z_{t} | > ɛ} = 0, \forall t > 0$ holds.

Theorem 2.1. (Markov Inequality, [1]) Markov If ζ is a variable with uncertainty. ∀t > 0 and ∀k > 0, the following inequality holds.

$M {| ζ | \geq t} \leq \frac{E [| ζ |^{k}]}{t^{k}} .$

Theorem 2.2. ([6]) Chen-Liu Suppose that Z_t defined on [a₀, a₁] is an integrable uncertain process, C_t is a canonical process, and the sample path C_t (γ) has the Lipschitz constant K (γ). Then the following inequality holds

$| \int_{a_{0}}^{a_{1}} Z_{t} (γ) d C_{t} (γ) | \leq K (γ) \int_{a_{0}}^{a_{1}} | Z_{t} (γ) | d t .$

Theorem 2.3. ([10]) Yaoet al1 For a canonical process C_t,

$lim_{a \to + \infty} M {γ \in Γ ∣ K (γ) \leq a} = 1 .$ where K (γ) is a Lipschitz constant of C_t (γ) for each γ.

Theorem 2.4. ([26]) Let f₁ be a given binary function. Then

$d Z_{t} = f_{1} (t, Z_{t}) d t + σ_{t} Z_{t} d C_{t}$ (2) has a solution $Z_{t} = Y_{t}^{- 1} X_{t}$ where $Y_{t} = exp (- \int_{0}^{t} σ_{s} d s)$ and X_t satisfies the U-D-S $d X_{t} = Y_{t} f_{1} (t, Y_{t}^{- 1} X_{t}) d t$ with initial value X₀ = Z₀.

3 Stability and attractivity concepts

In this section, two pessimistic value stability and attractivity definitions are introduced. we suppose that Y_t and Z_t are any two solutions satisfying the U-D-S (1), and the initial values are Y₀ and Z₀, respectively.

3.1 Stability in pessimistic value concept

Definition 3.1. The U-D-S (1) is said to be stable in pessimistic value if for 0 < α ≤ 1, one has $\begin{matrix} lim_{| Y_{0} - Z_{0} | \to 0} | Y_{t} - Z_{t} |_{inf} (α) \\ = lim_{| Y_{0} - Z_{0} | \to 0} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ = 0 . \end{matrix}$ Example 3.1. Let a U-D-S have the below form $d Z_{t} = exp (t) d t + σ d C_{t} .$ Then $Y_{t} = Y_{0} + exp (t) + σ C_{t},$ $Z_{t} = Z_{0} + exp (t) + σ C_{t} .$

For any given ɛ > 0, taking $δ = \frac{ɛ}{3}$ , when |Y₀ - Z₀| < δ, $\begin{matrix} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ = inf {r ∣ M {| Y_{0} - Z_{0} | \leq r} \geq α} \\ < inf {r ∣ M {\frac{ɛ}{3} \leq r} \geq α} \\ = \frac{ɛ}{3} \\ < ɛ . \end{matrix}$ Thus the U-D-S dZ_t = exp(t) dt + σdC_t is stable in pessimistic value.

Example 3.2. Analyze a U-D-S $d Z_{t} = Z_{t} d t + σ d C_{t} .$ We immediately have $Y_{t} = Y_{0} exp (t) + σ exp (t) \int_{0}^{t} exp (- s) d C_{s},$ $Z_{t} = Z_{0} exp (t) + σ exp (t) \int_{0}^{t} exp (- s) d C_{s} .$ Let’s take ɛ₁ = 1. For any δ > 0 and |Y₀ - Z₀| < δ, here exists $t > ln \frac{2}{| Y_{0} - Z_{0} |}$ satisfying $\begin{matrix} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ = inf {r ∣ M {exp (t) | Y_{0} - Z_{0} | \leq r} \geq α} \\ > 1 = ɛ_{1} . \end{matrix}$ Thus dZ_t = Z_tdt + σdC_t is unstable in pessimistic value.

3.2 Attractivity in pessimistic value concept

Definition 3.2. The U-D-S (1) is said to be attractive in pessimistic value if here exists σ > 0 satisfying when |Y₀ - Z₀| < σ, $\begin{matrix} lim_{t \to + \infty} | Y_{t} - Z_{t} |_{inf} (α) \\ = lim_{t \to + \infty} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ = 0 \end{matrix}$ holds, where 0 < α ≤ 1.

Example 3.3. Now let us consider the following U-D-S $d Z_{t} = - Z_{t} d t + σ d C_{t} .$ We have $d Y_{t} = - Y_{t} d t + σ d C_{t} .$ $d Z_{t} = - Z_{t} d t + σ d C_{t},$ Then $d (Y_{t} - Z_{t}) = - (Y_{t} - Z_{t}) d t .$ Thus we can write $Y_{t} - Z_{t} = (Y_{0} - Z_{0}) exp (- t) .$ For any given ɛ > 0, we make σ = ɛ. When t > ln 3 and |Y₀ - Z₀| < σ, $\begin{matrix} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ = inf {r ∣ M {| Y_{0} - Z_{0} | exp (- t) \leq r} \geq α} \\ < inf {r ∣ M {\frac{ɛ}{3} \leq r} \geq α} \\ = \frac{ɛ}{3} \\ < ɛ . \end{matrix}$ Then dZ_t = - Z_tdt + σdC_t is attractive in pessimistic value.

Example 3.4. Analyze the following U-D-S $d Z_{t} = Z_{t} d t + σ d C_{t} .$ We can get $Y_{t} - Z_{t} = exp (t) (Y_{0} - Z_{0}) .$ Taking ɛ₀ = 1. For any σ > 0 and |Y₀ - Z₀| < σ, such that when $t > ln \frac{2}{| Y_{0} - Z_{0} |}$ , one has $\begin{matrix} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ = inf {r ∣ M {exp (t) | Y_{0} - Z_{0} | \leq r} \geq α} \\ > 1 = ɛ_{0} . \end{matrix}$ Thus dZ_t = Z_tdt + σdC_t is not attractive in pessimistic value.

4 Stability and attractivity analysis

4.1 Linear U-D-S

We consider the linear U-D-S $d Z_{t} = (H_{1} (t) Z_{t} + H_{2} (t)) d t + (H_{3} (t) Z_{t} + H_{4} (t)) d C_{t}$ (3) where H₁ (t), H₂ (t), H₃ (t), H₄ (t) are continuous functions on [0, + ∞). We suppose that Y_t and Z_t are any two solutions satisfying the U-D-S (3), and the initial values are Y₀ and Z₀, respectively.

4.1.1 Stability analysis

Theorem 4.1. stablein optimistic value The linear U-D-S (3) is stable in pessimistic value if

$sup_{t \geq 0} \int_{0}^{t} H_{1} (t) d s < + \infty,$ and $\int_{0}^{+ \infty} | H_{3} (t) | d s < + \infty .$

Proof. According to the system (3), it is obtained that $d (Y_{t} - Z_{t}) = H_{1} (t) (Y_{t} - Z_{t}) d t + H_{3} (t) (Y_{t} - Z_{t}) d C_{t} .$ Thus $\begin{matrix} Y_{t} - Z_{t} \\ = (Y_{0} - Z_{0}) exp (\int_{0}^{t} H_{1} (s) d s + \int_{0}^{t} H_{3} (s) d C_{s}) . \end{matrix}$ Since $\int_{0}^{t} H_{3} (s) d C_{s} \sim N (0, \int_{0}^{t} | H_{3} (s) | d s)$ and $\int_{0}^{+ \infty} | H_{3} (s) | d s < + \infty .$ Suppose that F_t (x) is the uncertainty distribution of $\int_{0}^{t} H_{3} (s) d C_{s}$ and 0 < α ≤ 1. We can obtain $F_{t}^{- 1} (α) = \frac{\sqrt{3} \int_{0}^{t} | H_{3} (s) | d s}{π} ln \frac{α}{1 - α} < + \infty,$ Then $\begin{matrix} \begin{matrix} M {| Y_{t} - Z_{t} | \leq r} \\ = M {| Y_{0} - Z_{0} | exp (\int_{0}^{t} H_{1} (s) d s \\ + \int_{0}^{t} H_{3} (s) d C_{s}) \leq r} \\ = M {\int_{0}^{t} H_{3} (s) d C_{s} \leq ln \frac{r}{| Y_{0} - Z_{0} |} - \int_{0}^{t} H_{1} (s) d s} \\ = F_{t} (ln \frac{r}{| Y_{0} - Z_{0} |} - \int_{0}^{t} H_{1} (s) d s) . \end{matrix} \end{matrix}$ Since $F_{t}^{- 1} (α) < + \infty$ and $sup_{t \geq 0} \int_{0}^{t} H_{1} (s) d s < + \infty$ , we can get $\begin{matrix} \begin{matrix} lim_{| Y_{0} - Z_{0} | \to 0} | Y_{t} - Z_{t} |_{inf} (α) \\ = lim_{| Y_{0} - Z_{0} | \to 0} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ = lim_{| Y_{0} - Z_{0} | \to 0} inf {r ∣ F_{t} (ln \frac{r}{| Y_{0} - Z_{0} |} \\ - \int_{0}^{t} H_{1} (s) d s) \geq α} \\ = lim_{| Y_{0} - Z_{0} | \to 0} inf {r ∣ r \geq | Y_{0} - Z_{0} | exp (F_{t}^{- 1} (α) \\ + \int_{0}^{t} H_{1} (s)) d s)} \\ = 0, \end{matrix} \end{matrix}$ Then the linear U-D-S (3) is stable in pessimistic value.

Example 4.1. Analyze the U-D-S $d Z_{t} = exp (- t) Z_{t} d t + \frac{1}{1 + t^{2}} Z_{t} d C_{t} .$ Since H₁ (t) = exp(- t) and $H_{3} (t) = \frac{1}{1 + t^{2}}$ , it is clear that $sup_{t \geq 0} \int_{0}^{t} H_{1} (s) d s = 1 < + \infty$ and $\int_{0}^{+ \infty} | H_{3} (s) | d s = \frac{π}{2} < + \infty .$ Thus the linear U-D-S $d Z_{t} = exp (- t) Z_{t} d t + \frac{1}{1 + t^{2}} Z_{t} d C_{t}$ is stable in pessimistic value.

4.1.2 Attractivity analysis

Theorem 4.2. The linear U-D-S (ref3) is attractive in pessimistic value if and only if

$\int_{0}^{+ \infty} H_{1} (s) d s = - \infty .$

Proof. From the linear U-D-S (ref3), we have $\begin{matrix} Y_{t} - Z_{t} \\ = (Y_{0} - Z_{0}) exp (\int_{0}^{t} H_{1} (s) d s + \int_{0}^{t} H_{3} (s) d C_{s}) . \end{matrix}$ Note that $\int_{0}^{t} H_{3} (s) d C_{s} \sim N (0, \int_{0}^{t} | H_{3} (s) | d s)$ and $\int_{0}^{+ \infty} | H_{3} (s) | d s < + \infty$ . Then $F_{t}^{- 1} (α) = \frac{\sqrt{3} \int_{0}^{t} | H_{3} (s) | d s}{π} ln \frac{α}{1 - α} < + \infty,$ Then $\begin{matrix} \begin{matrix} M {| Y_{t} - Z_{t} | \leq r} \\ = M {| Y_{0} - Z_{0} | exp (\int_{0}^{t} H_{1} (s) d s \\ + \int_{0}^{t} H_{3} (s) d C_{s}) \leq r} \\ = M {\int_{0}^{t} H_{3} (s) d C_{s} \leq ln \frac{r}{| Y_{0} - Z_{0} |} - \int_{0}^{t} H_{1} (s) d s} \\ = F_{t} (ln \frac{r}{| Y_{0} - Z_{0} |} - \int_{0}^{t} H_{1} (s) d s), \end{matrix} \end{matrix}$ one has

$\begin{matrix} lim_{t \to + \infty} | Y_{t} - Z_{t} |_{inf} (α) \\ = lim_{t \to + \infty} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ = lim_{t \to + \infty} inf {r ∣ \\ F_{t} (ln \frac{r}{| Y_{0} - Z_{0} |} - \int_{0}^{t} H_{1} (s) d s) \geq α} \\ = lim_{t \to + \infty} inf {r ∣ r \\ \geq | Y_{0} - Z_{0} | exp (F_{t}^{- 1} (α) + \int_{0}^{t} H_{1} (s) d s)} . \end{matrix}$ Since $F_{t}^{- 1} (α) < + \infty$ , $lim_{t \to + \infty} | Y_{t} - Z_{t} |_{inf} (α) = 0$ if and only if $\int_{0}^{+ \infty} H_{1} (s) d s = - \infty$ . Thus we complete the proof.

Example 4.2. For the linear U-D-S with following form $d Z_{t} = - \frac{1}{t + 1} Z_{t} d t + exp (- t) Z_{t} d C_{t} .$ Since $H_{1} (t) = - \frac{1}{t + 1}$ and H₃ (t) = exp(- t), we obtain $\int_{0}^{+ \infty} H_{1} (s) d s = - \infty$ and $\int_{0}^{+ \infty} | H_{3} (s) | d s = 1 < + \infty .$ Thus ${dZ}_{t} = - \frac{1}{t + 1} Z_{t} d t + exp (- t) Z_{t} d C_{t}$ is attractive in pessimistic value.

4.2 Stability and attractivity for nonlinear U-D-S

4.2.1 Stability for dX_t = f₁ (t, X_t) dt + σ_tdC_t

We study the nonlinear U-D-S $d Z_{t} = f_{1} (t, Z_{t}) d t + σ_{t} d C_{t}$ (4) Suppose that Y_t and Z_t are any two solutions of the U-D-S (4) with initial values Y₀ and Z₀, respectively.

Theorem 4.3. The nonlinear U-D-S (4) is stable in pessimistic value if $\forall x, y \in R, t \geq 0$ , the coefficient f₁ satisfies Lipschitz condition

$| f_{1} (t, x) - f_{1} (t, y) | \leq L (t) | x - y |,$ where L (t) is bounded and integrable on [0, + ∞).

Proof. For any γ, one has $\begin{matrix} d | Y_{t} (γ) - Z_{t} (γ) | \\ \leq | f_{1} (t, Y_{t} (γ)) - f_{1} (t, Z_{t} (γ)) | d t \\ \leq L (t) | Y_{t} (γ) - Z_{t} (γ) | d t . \end{matrix}$ Then $\begin{matrix} | Y_{t} (γ) - Z_{t} (γ) | \\ \leq | Y_{0} - Z_{0} | exp (\int_{0}^{+ \infty} L (s) d s) . \end{matrix}$ Since $\int_{0}^{+ \infty} L (s) d s < + \infty$ , $exp (\int_{0}^{+ \infty} L (s) d s) < + \infty$ . Note that $exp (\int_{0}^{+ \infty} L (s) d s) = M$ . For any given ɛ > 0, let $σ = \frac{ɛ}{2 M}$ . When |Z₀ - Y₀| < σ, $\begin{matrix} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ \leq inf {r ∣ M {| Y_{0} - Z_{0} | exp (\int_{0}^{+ \infty} L (s) d s) \leq r} \geq α} \\ < inf {r ∣ M {\frac{ɛ}{2 M} M \leq r} \geq α} \\ = \frac{ɛ}{2} \\ < ɛ, \end{matrix}$ where 0 < α ≤ 1. Then the U-D-S (4) is stable in pessimistic value.

Example 4.3. Let us analyze the nonlinear uncertain dynamical system $d Z_{t} = exp (- t) sin Z_{t} d t + t^{2} d C_{t} .$ Since f₁ (t, x) = exp(- t) sin x, $\forall x, y \in R, t \geq 0$ , we immediately have $| f_{1} (t, x) - f_{1} (t, y) | \leq exp (- t) | x - y | .$ By Theorem 4.2.1, the nonlinear U-D-S $d Z_{t} = exp (- t) sin Z_{t} d t + t^{2} d C_{t}$ is stable in pessimistic value.

4.2.2 Stability for dX_t = σ_tdt + f₂ (t, X_t) dC_t

We study the nonlinear U-D-S $d X_{t} = σ_{t} d t + f_{2} (t, X_{t}) d C_{t} .$ (5)

Theorem 4.4. The nonlinear U-D-S (5) is stable in pessimistic value if the coefficient f₂ satisfies Lipschitz condition

$| f_{2} (t, x) - f_{2} (t, y) | \leq L (t) | x - y |, \forall x, y \in R, t \geq 0 .$ for some bounded and integrable L (t) on [0, + ∞).

Proof. For any γ, one has $\begin{matrix} d | Y_{t} (γ) - Z_{t} (γ) | \\ \leq | f_{2} (t, Y_{t} (γ)) - f_{2} (t, Z_{t} (γ)) d C_{t} | \\ \leq L (t) K (γ) | Y_{t} (γ) - Z_{t} (γ) | d t, \end{matrix}$ Then $\begin{matrix} | Y_{t} (γ) - Z_{t} (γ) | \\ \leq | Y_{0} - Z_{0} | exp (K (γ) \int_{0}^{+ \infty} L (s) d s), \end{matrix}$ By Theorem2, $M {γ \in Γ | K (γ) < + \infty} = 1$ , and according to $\int_{0}^{+ \infty} L (s) d s < + \infty$ , we get $\begin{matrix} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ \leq inf {r ∣ \\ M {| Y_{0} - Z_{0} | exp (K (γ) \int_{0}^{+ \infty} L (s) d s) \leq r} \geq α} \\ \to 0, \end{matrix}$ as |Y₀ - Z₀|→0, where α ∈ (0, 1]. Then the U-D-S (5) is stable in pessimistic value.

Example 4.4. Now let us consider the following U-D-S $d Z_{t} = \frac{1}{1 + t^{2}} d t + exp (- t) Z_{t} d C_{t} .$ $\forall x, y \in R, t \geq 0,$ we can get easily $\begin{matrix} | f_{2} (t, x) - f_{2} (t, y) | \\ \leq exp (- t) | x - y | . \end{matrix}$ By Theorem 4.2.2, the U-D-S $d Z_{t} = \frac{1}{1 + t^{2}} d t + exp (- t) Z_{t} d C_{t}$ is stable in pessimistic value.

4.2.3 Stability for dX_t = f₁ (t, X_t) dt + σ_tX_tdC_t

We consider the nonlinear U-D-S $d X_{t} = f_{1} (t, X_{t}) d t + σ_{t} X_{t} d C_{t} .$ (6)

Theorem 4.5. The nonlinear U-D-S (6) is stable in pessimistic value if the coefficient f₁ satisfies Lipschitz condition

$\begin{matrix} | f_{1} (t, x) - f_{1} (t, y) | \\ \leq L (t) | x - y |, \forall x, y \in R, t \geq 0 \end{matrix}$ for some bounded and integrable L (t) on [0, + ∞) and $sup_{t > 0} \int_{0}^{t} | σ_{s} | d s < + \infty$ .

Proof. According to Theorem 2, one has $\begin{matrix} Y_{t} = Y_{0} exp (\int_{0}^{t} σ_{s} d C_{s}) \\ + exp (\int_{0}^{t} σ_{s} d C_{s}) \int_{0}^{t} exp (- \int_{0}^{s} σ_{s} d C_{s}) f_{1} (s, Y_{s}) d s . \end{matrix}$ and $\begin{matrix} Z_{t} = Z_{0} exp (\int_{0}^{t} σ_{s} d C_{s}) \\ + exp (\int_{0}^{t} σ_{s} d C_{s}) \int_{0}^{t} exp (- \int_{0}^{s} σ_{s} d C_{s}) f_{1} (s, Z_{s}) d s \end{matrix}$ For any γ, by Gronwall inequality $\begin{matrix} exp (- \int_{0}^{t} σ_{s} d C_{s}) | Y_{t} (γ) - Z_{t} (γ) | \\ \leq | Y_{0} - Z_{0} | \\ + \int_{0}^{t} exp (- \int_{0}^{s} σ_{s} d C_{s}) | f_{1} (s, Y_{s} (γ)) - f_{1} (s, Z_{s} (γ)) | d s \\ \leq | Y_{0} - Z_{0} | \\ + \int_{0}^{t} exp (- \int_{0}^{s} σ_{s} d C_{s}) L (s) | Y_{s} (γ) - Z_{s} (γ) | d s | \\ \leq | Y_{0} - Z_{0} | exp (\int_{0}^{t} L (s) d s) . \end{matrix}$ Then one has $\begin{matrix} | Y_{t} (γ) - Z_{t} (γ) | \\ \leq | Y_{0} - Z_{0} | exp (\int_{0}^{t} L (s) d s) exp (\int_{0}^{t} σ_{s} d C_{s}) \\ \leq | Y_{0} - Z_{0} | \\ \times exp (\int_{0}^{t} L (s) d s) exp (K (γ) sup_{s > 0} \int_{0}^{t} | σ_{s} | d s) \end{matrix}$ ∀ɛ > 0, ς > 0, $M {γ \in Γ ∣ K (γ) \leq M} \geq 1 - ς .$ Let

$\begin{matrix} δ = \frac{1}{2} exp (- \int_{0}^{+ \infty} L (s) d s) \\ \times exp (- M sup_{s > 0} \int_{0}^{t} | σ_{s} | d s) ɛ, \end{matrix}$ if |Y₀ - Z₀| < δ and K (γ) ≤ M, we can obtain $\begin{matrix} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ \leq inf {r ∣ \\ M {| Y_{0} - Z_{0} | exp (K (γ) \int_{0}^{+ \infty} L (s) d s) \leq r} \geq α} \\ \leq \frac{ɛ}{2} \\ < ɛ, \end{matrix}$ where 0 < α ≤ 1. Then the U-D-S (6) is stable in pessimistic value.

Example 4.5. Now let us study the nonlinear U-D-S $d Z_{t} = exp (- 2 t - Z_{t}^{2}) d t + exp (- t) Z_{t} d C_{t} .$ Since f₁ (t, x) = exp(-2t - x²) , $\forall x, y \in R, t \geq 0,$ we immediately have $\begin{matrix} | exp (- 2 t - x^{2}) - exp (- 2 t - y^{2}) | \\ \leq exp (- 2 t) | x - y | . \end{matrix}$ and $sup_{t > 0} \int_{0}^{t} | σ_{s} | d s = sup_{t > 0} \int_{0}^{t} | exp (- s) | d s = 1 < + \infty .$ By Theorem 4.2.3, the nonlinear U-D-S $d Z_{t} = exp (- 2 t - Z_{t}^{2}) d t + exp (- t) Z_{t} d C_{t}$ is pessimistic value stable.

4.2.4 Attractivity analysis

For nonlinear U-D-S as follows $d X_{t} = σ_{t} X_{t} d t + f_{2} (t, X_{t}) d C_{t},$ (7)

Theorem 4.6. Let f₂ satisfy Lipschitz condition

$\begin{matrix} | f_{2} (t, x) - f_{2} (t, y) | \\ \leq L (t) | x - y |, \forall x, y \in R, t \geq 0 \end{matrix}$ where L (t) is bounded and integrable on [0, + ∞). Then the nonlinear U-D-S (7) is attractive in pessimistic value if and only if $\int_{0}^{+ \infty} σ_{s} d s = - \infty .$

Proof. By Theorem 2, one can get $\begin{matrix} Y_{t} = exp (\int_{0}^{t} σ_{s} d s) [Y_{0} \\ + \int_{0}^{t} exp (- \int_{0}^{s} σ_{r} d r) f_{2} (s, Y_{s}) d C_{s}] . \end{matrix}$ and $\begin{matrix} Z_{t} = exp (\int_{0}^{t} σ_{s} d s) [Z_{0} \\ + \int_{0}^{t} exp (- \int_{0}^{s} σ_{r} d r) f_{2} (s, Z_{s}) d C_{s}] \end{matrix}$ Then for each γ, by using Grownwall’s inequality, $\begin{matrix} exp (- \int_{0}^{t} σ_{s} d s) | Y_{t} (γ) - Z_{t} (γ) | \\ = | (Y_{0} - Z_{0}) \\ + \int_{0}^{t} exp (- \int_{0}^{s} σ_{r} d r) (f_{2} (s, Y_{s} (γ)) - f_{2} (s, Z_{s} (γ))) d C_{s} | \\ \leq | Y_{0} - Z_{0} | \\ + \int_{0}^{t} exp (- \int_{0}^{s} σ_{r} d r) | f_{2} (s, Y_{s} (γ)) - f_{2} (s, Z_{s} (γ)) | | d C_{s} (γ) | \\ \leq | Y_{0} - Z_{0} | \\ + \int_{0}^{t} exp (- \int_{0}^{s} σ_{r} d r) L (s) K (γ) | Y_{s} (γ) - Z_{s} (γ) | d s \\ \leq | Y_{0} - Z_{0} | exp (K (γ) \int_{0}^{t} L (s) d s) . \end{matrix}$ So $\begin{matrix} | Y_{t} (γ) - Z_{t} (γ) | \\ \leq | Y_{0} - Z_{0} | exp (K (γ) \int_{0}^{t} L (s) d s) exp (\int_{0}^{t} σ_{s} d s) \\ \leq | Y_{0} - Z_{0} | exp (K (γ) \int_{0}^{+ \infty} L (s) d s) exp (\int_{0}^{t} σ_{s} d s) . \end{matrix}$ By Theorem2, we have $M {γ \in Γ | K (γ) < + \infty} = 1$ . And according to $\int_{0}^{+ \infty} L (s) d s < + \infty$ , we can get $\begin{matrix} inf {r ∣ M {| Y_{t} - Z_{t} | \leq r} \geq α} \\ \leq inf {r ∣ M {| Y_{0} - Z_{0} | \\ \times exp (K (γ) \int_{0}^{+ \infty} L (s) d s) exp (\int_{0}^{t} σ_{s} d s) \leq r} \geq α} . \end{matrix}$ It’s easy to know that $lim_{t \to + \infty} | Y_{t} - Z_{t} |_{inf} (α) = 0$ if and only if $\int_{0}^{+ \infty} σ_{s} d s = - \infty$ .

Example 4.6. Consider the U-D-S $d Z_{t} = - \frac{1}{1 + t} Z_{t} d t + exp (- t) cos Z_{t} d C_{t} .$ $\forall x, y \in R, t \geq 0,$ it is easy to see that $\begin{matrix} | exp (- t) cos x - exp (- t) cos y | \\ \leq exp (- t) | x - y |, \end{matrix}$ and $\int_{0}^{+ \infty} σ_{s} d s = \int_{0}^{+ \infty} - \frac{1}{1 + t} d s = - \infty .$ By Theorem 4.2.4, the U-D-S $d Z_{t} = - \frac{1}{1 + t} Z_{t} d t + exp (- t) cos Z_{t} d C_{t}$ is attractive in pessimistic value.

5 Conclusion

This article studied pessimistic value stability and attractivity for U-D-S For linear U-D-S, sufficient conditions for stability and attractivity were given, respectively. Besides, the stability was analyzed for three kinds of nonlinear U-D-Ss. And the attractivity was considered for a type of nonlinear U-D-S Future work will focus on the application of stability and attractivity

Footnotes

Acknowledgments

This work is supported by the Key Scientific Research Projects of Henan Higher Education Institutions (No.18B110012).

Compliance with ethical standards

Conflict of interest: The authors declare that they have no competing interests.

Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors.

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Stability and attractivity in pessimistic value for uncertain dynamical system

Abstract

Keywords

1 Introduction

2 Preliminary

3.1 Stability in pessimistic value concept

3.2 Attractivity in pessimistic value concept

4 Stability and attractivity analysis

4.1 Linear U-D-S

4.1.2 Attractivity analysis

4.2 Stability and attractivity for nonlinear U-D-S

4.2.1 Stability for dX t = f1 (t, X t ) dt + σ t dC t

Footnotes

Acknowledgments

Compliance with ethical standards

References

4.2.1 Stability for dX_t = f₁ (t, X_t) dt + σ_tdC_t