Fixed-time stability theorem of stochastic nonlinear systems

ABSTRACT

To solve the fixed-time stability problem for stochastic systems, which has not been investigated due to the difficulty caused by the existence of stochastic, the fixed-time stability is investigated for stochastic nonlinear systems described by the It$\widehat{o}$ differential equation in this note. The main contributions exist in: (1) A new concept, fixed-time stable in probability, is proposed, and the detailed definition is given. (2) A criterion theorem is proposed to judge the fixed-time stable in probability, in addition, several corollaries are given for proving the fixed-time stability in probability for stochastic nonlinear systems. Two simulation examples are given to verify the proposed theoretical results in this paper.

KEYWORDS:

• fixed-time stability
• stochastic nonlinear systems
• infinitesimal generator
• stochastic settling time function
• Lyapunov function

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China [grant number 41674037], [grant number 61503080], [grant number 61374126] and Fundamental Research Funds for the Central Universities of China [grant number 3132017131].