Journal of East China Normal University(Natural Science) >

2023 , Vol. 2023 >Issue 4: 11 - 23

DOI: https://doi.org/10.3969/j.issn.1000-5641.2023.04.002

Mathematics

LaSalle’s invariance principle for delay differential equations driven by α-stable processes

  • Zhenzhong ZHANG ,
  • Xu CHEN ,
  • Jinying TONG
Expand
  • College of Science, Donghua University, Shanghai　201620, China

Received date: 2021-09-24

  Online published: 2023-07-25

Fold

Abstract

LaSalle’s invariance principle is an important tool for studying the stability of stochastic systems. Considering the influence of time delay and pure-jump path on the stability of the system and using the convergence theorem for special semi-martingale, the LaSalle’s invariance principle for a class of stochastic delay differential equations driven by $\alpha$ -stable processes is established in this study. The sufficient conditions for the asymptotic stability of a class of delay equations are given by LaSalle’s invariance principle.

Key words： LaSalle’s invariance principle; special semi-martingale; asymptotic stability in probability; stability in probability

Cite this article

Zhenzhong ZHANG , Xu CHEN , Jinying TONG . LaSalle’s invariance principle for delay differential equations driven by α-stable processes[J]. Journal of East China Normal University(Natural Science), 2023 , 2023(4) : 11 -23 . DOI: 10.3969/j.issn.1000-5641.2023.04.002

References

1 MAO X R. Stochastic versions of the LaSalle theorem. Journal of Differential Equations, 1999, 153 (1): 175- 195.
2 YIN G G, ZHU C. Hybrid Switching Diffusions: Properties and Applications [M]. New York: Springer, 2010.
3 LI X Y, MAO X R. The improved LaSalle-type theorems for stochastic differential delay equations. Stochastic Analysis and Applications, 2012, 30 (4): 568- 589.
4 SHEN Y, LUO Q, MAO X R. The improved LaSalle-type theorems for stochastic functional differential equations. Journal of Mathematical Analysis and Applications, 2006, 318 (1): 134- 154.
5 ZHUANG G M, XIA J W, ZHANG W H, et al. State feedback control for stochastic Markovian jump delay systems based on LaSalle-type theorem. Journal of the Franklin Institute, 2018, 355 (5): 2179- 2196.
6 MAO X R. A note on the LaSalle-type theorems for stochastic differential delay equations. Journal of Mathematical Analysis and Applications, 2002, 268 (1): 125- 142.
7 ZHANG X C. Derivative formulas and gradient estimates for SDEs driven by α-stable processes . Stochastic Processes and Their Applications, 2013, 123, 1213- 1228.
8 LIPTSER R S, SHIRYAEV A N. Theory of Martingales [M]. Boston: Kluwer Academic Publishers, 1989.
9 ZHANG Z Z, TONG J Y, MENG Q T, et al. Population dynamics driven by truncated stable processes with Markovian switching. Journal of Applied Probability, 2021, 58 (2): 505- 532.
10 KHASMINSKⅡ R. Stochastic Stability of Differential Equations [M]. Berlin: Springer, 2011.
Options
Outlines

/

〈 〉