Stabilisation for discrete‐time mean‐field stochastic Markov jump systems with multiple delays

Abstract

Abstract In this paper, the operator spectrum theory is applied to study the general stabilisation issues for mean‐field stochastic Markov jump systems (MF‐SMJSs), where multiple delays, multiplicative noises and homogeneous Markov chain exist simultaneously. The innovative contributions are described as follows. On the one hand, a feasible model augmented strategy is adopted to transform the dynamics into an auxiliary delay‐free form. By introducing a delay‐dependent linear Lyapunov operator (DDLLO), the Lyapunov/spectrum stabilising conditions are constructed, which are both necessary and sufficient. On the other hand, in terms of spectral analysis technique, the notions of interval stabilisation and essential destabilisation are generalised to MF‐SMJSs for the first time. The necessary and sufficient stabilisation conditions are derived, respectively, which can be verified availably by LMI feasibility tests. To confirm the effectiveness of the theoretic results, two illustrative examples are included.

Keywords

• asymptotic stability
• delay systems
• Markov processes
• mathematical operators
• stochastic systems