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The almost sure stability for uncertain delay differential equations based on normal lipschitz conditions

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  • Gao, Yin
  • Gao, Jinwu
  • Yang, Xiangfeng

Abstract

Uncertain delay differential equations (UDDEs) can be applied to model the feedback control systems concerning both the present and the past states, such as the engineer system, the pharmacokinetic system, and the microbial batch fermentation system. The almost sure stability of its solution plays a significant role in the applications. By means of the strong Lipschitz condition, which relates to the present state, the almost sure stability of UDDEs is successfully explored. In this paper, the normal Lipschitz conditions concerning both the present and the past states are proposed. In other words, if the UDDEs satisfy the strong Lipschitz condition, it must satisfy the normal Lipschitz conditions. Conversely, it may not be established. Based on the normal Lipschitz conditions, two theorems for the almost sure stability of UDDEs and their equivalence relations are proved. Finally, a class of UDDEs being the almost sure stability is verified without any limited condition.

Suggested Citation

  • Gao, Yin & Gao, Jinwu & Yang, Xiangfeng, 2022. "The almost sure stability for uncertain delay differential equations based on normal lipschitz conditions," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    • Handle: RePEc:eee:apmaco:v:420:y:2022:i:c:s0096300321009863
    DOI: 10.1016/j.amc.2021.126903
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