Lie algebra approach in the study of the stability of stochastic linear hybrid systems

The problem of the stability of a class of stochastic linear hybrid systems with a special structure of matrices and a multiplicative excitation is considered. Sufficient conditions of the exponential $ p$-th mean stability and the almost sure stability for a class of stochastic linear hybrid systems with the Markovian switching are derived. Also, sufficient conditions of the exponential mean-square stability for a class of stochastic linear hybrid systems satisfying the Lie algebra conditions with any switching are found. The obtained results are illustrated by examples and simulations.