Fixed-time stability and predefined-time stability for stochastic differential equations with Lévy noise

In this paper, the fixed-time stability and predefined-time stability of stochastic differential equations with Lévy noise are discussed. Firstly, the fixed-time stability criterion of stochastic differential equations with Lévy noise is given based on Itô formula. Secondly, a special form predefined-time stability criterion and its settling-time function for stochastic differential equations with Lévy noise are established, and a scheme for the optimal-time stability is given. This stability criterion provides a theoretical foundation for predefined-time and optimal-time control for stochastic systems with Lévy noise. Finally, an example to verify the validity of the theoretical analysis.