Necessary and Sufficient Conditions for $2p$th Moment Stability of Several Classes of Linear Stochastic Systems

This technical note presents necessary and sufficient conditions for $2p$ th moment stability of several widely used linear stochastic systems. By the matrix derivative operator and vectorial Ito's formula, we first derive two extended systems: one is described by stochastic differential equation (SDE) and the other by ordinary differential equation (ODE). It is shown that the stochastic system is $2p$ th-moment exponentially stable if and only if the SDE is mean-square exponentially stable or the ODE is exponentially stable. Subsequently, combining the quasi-periodic homogeneous polynomial Lyapunov function methods with the proposed techniques, a series of nonconservative stability criteria are established for linear impulsive, sampled-data, and switched stochastic systems under dwell-time constraints. A numerical example illustrates the proposed theoretical results.