Stability, stochastic stationarity, and generalized Lyapunov equations for two-point boundary-value descriptor systems

The authors introduce the concept of internal stability for two-point boundary-value descriptor systems (TPBVDSs). Since TPBVDSs are defined only over a finite interval, the concept of stability is not easy to formulate. The definition which is used consists of requiring that as the length of the interval of definition increases, the effect of boundary conditions on states located close to the center of the interval should go to zero. Stochastic TPBVDSs are studied, and the property of stochastic stationarity is characterized in terms of a generalized Lyapunov equation satisfied by the variance of the boundary vector. A second generalized Lyapunov equation satisfied by the state variance of a stochastically stationary TPBVDS is also introduced, and the existence and uniqueness of positive definite solutions to this equation are then used to characterize the property of internal stability