MeansquareStability
MeansquareStability is a concept in the study of stochastic dynamical systems, referring to the behavior of the second moment of the state. In broad terms, a system is mean-square stable if the expected value of the squared state remains bounded for all time, and it is often called asymptotically mean-square stable when this second moment tends to zero under appropriate conditions. The notion is widely used in control theory, signal processing, and estimation when random disturbances affect system evolution.
In a discrete-time setting, consider x_{k+1} = A x_k + w_k, where w_k is zero-mean noise with covariance
In continuous time, for xdot = A x + w(t) with w(t) white noise and covariance rate Q,
Lyapunov methods provide practical tools for proving MSS: finding a positive definite P that satisfies a suitable