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 W. The system is mean-square stable if sup_k E[||x_k||^2] < ∞ for all initial states x_0 with finite second moment. A common sufficient condition is that the deterministic part A is Schur stable (all eigenvalues inside the unit circle). When this holds, E[||x_k||^2] converges to a finite limit that solves the discrete-time Lyapunov equation P = A^T P A + W for some P > 0.

In continuous time, for xdot = A x + w(t) with w(t) white noise and covariance rate Q, mean-square stability typically requires A to be Hurwitz (all eigenvalues with negative real parts). Then a stationary second moment exists and is obtained from the continuous-time Lyapunov equation A^T P + P A + Q = 0.

Lyapunov methods provide practical tools for proving MSS: finding a positive definite P that satisfies a suitable inequality ensures bounded second moments and guides stability analysis. Applications include stochastic control design, filtering, networked systems, and robust estimation.