Stabilization in distribution of hybrid stochastic differential delay equations with Lévy noise by discrete-time state feedback controls

Jingjing Yang; Jianqiu Lu; Jingjing Yang; Jianqiu Lu

doi:10.3934/math.2025160

AIMS Mathematics

2025, Volume 10, Issue 2: 3457-3483. doi: 10.3934/math.2025160

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Research article

Stabilization in distribution of hybrid stochastic differential delay equations with Lévy noise by discrete-time state feedback controls

Jingjing Yang ,
Jianqiu Lu ^,

Department of Mathematics and Statistics, Donghua University, Shanghai, China

Received: 08 December 2024 Revised: 28 January 2025 Accepted: 14 February 2025 Published: 24 February 2025
MSC : 93C55, 93D15, 93E03

This paper was concerned with stabilization in distribution by feedback controls based on discrete-time state observations for a class of nonlinear stochastic differential delay equations with Markovian switching and Lévy noise (SDDEs-MS-LN). Compared with previous literature, we employed Lévy noise in the discussion about stabilization in distribution for hybrid stochastic delay systems and we considered using a discrete-time linear feedback control which is more realistic and costs less. In addition, by constructing a new Lyapunov functional, stabilization in distribution of controlled systems can be achieved with the coefficients satisfying globally Lipschitz conditions. In particular, we discussed the design of feedback controls in two structure cases: state feedback and output injection. At the same time, the lower bound for the duration between two consecutive observations $ \tau $ ($ \tau^* $) was obtained as well. Finally, a numerical experiment with some computer simulations was given to illustrate the new results.
- stochastic differential equations,
- Markov chain,
- Lévy noise,
- stability in distribution,
- discrete-time feedback controls
Citation: Jingjing Yang, Jianqiu Lu. Stabilization in distribution of hybrid stochastic differential delay equations with Lévy noise by discrete-time state feedback controls[J]. AIMS Mathematics, 2025, 10(2): 3457-3483. doi: 10.3934/math.2025160

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Abstract

This paper was concerned with stabilization in distribution by feedback controls based on discrete-time state observations for a class of nonlinear stochastic differential delay equations with Markovian switching and Lévy noise (SDDEs-MS-LN). Compared with previous literature, we employed Lévy noise in the discussion about stabilization in distribution for hybrid stochastic delay systems and we considered using a discrete-time linear feedback control which is more realistic and costs less. In addition, by constructing a new Lyapunov functional, stabilization in distribution of controlled systems can be achieved with the coefficients satisfying globally Lipschitz conditions. In particular, we discussed the design of feedback controls in two structure cases: state feedback and output injection. At the same time, the lower bound for the duration between two consecutive observations $ \tau $ ($ \tau^* $) was obtained as well. Finally, a numerical experiment with some computer simulations was given to illustrate the new results.

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