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On Time-Varying Delayed Stochastic Differential Systems with Non-Markovian Switching Parameters

Xinyu Wu, Zidong Wang, Wenlian Lu

Abstract

This paper focuses on time-varying delayed stochastic differential systems with stochastically switching parameters formulated by a unified switching behavior combining a discrete adapted process and a Cox process. Unlike prior studies limited to stationary and ergodic switching scenarios, our research emphasizes non-Markovian, non-stationary, and non-ergodic cases. It arrives at more general results regarding stability analysis with a more rigorous methodology. The theoretical results are validated through numerical examples.

On Time-Varying Delayed Stochastic Differential Systems with Non-Markovian Switching Parameters

Abstract

This paper focuses on time-varying delayed stochastic differential systems with stochastically switching parameters formulated by a unified switching behavior combining a discrete adapted process and a Cox process. Unlike prior studies limited to stationary and ergodic switching scenarios, our research emphasizes non-Markovian, non-stationary, and non-ergodic cases. It arrives at more general results regarding stability analysis with a more rigorous methodology. The theoretical results are validated through numerical examples.
Paper Structure (10 sections, 9 theorems, 35 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Solution of Cauchy Problem and Generalized Dynkin's formula
  4. Independent switching
  5. Homogeneous Markovian switching
  6. Hidden Markov process
  7. Stability Analysis
  8. Stability of Switching Stochastic Nonlinear Systems
  9. Illustrative Examples
  10. Conclusions

Key Result

Theorem 1

Assuming that Hypothesis $\mathbf H_{1}$ is valid and $\tau(t)$ and $r(t)$ are defined as in Section sec2, with the event rate function $\mu(\cdot)$ measurable and satisfying $\sup_{\xi\in\Omega}\mu(\xi)\le\mu_{0}$ for a positive $\mu_{0}$, equation (ds) with the initial function $\phi\in C_{\mathca

Figures (1)

  • Figure 1: Panel (a): Attractor dynamic behaviors of the subsystem of (\ref{['nn']}) with r=0; the curve is plotted by disregarding the initial time interval $[0,500]$ from the whole interval $[0,1000]$. Panel (b): Convergent dynamics behaviors of (\ref{['nn']}) with $\tau(t)\equiv 1$ and initial value $[-0.4,0.6]$; the curve is plotted with time interval $[0,100]$. Panel (c): Convergent dynamics behaviors of (\ref{['nn']}) with $\tau(t)= 0.1t+1$ and initial value $[-0.4,0.6]$; the curve is plotted by disregarding the initial time interval $[0,10]$ from the whole interval $[0,100]$.

Theorems & Definitions (27)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 2
  • Theorem 1
  • proof
  • Lemma 1
  • Lemma 2
  • ...and 17 more