Stability, Stochastic Stationarity and Generalized Lyapunov Equations for Two-Point Boundary-Value Descriptor Systems.

摘要

This reprint introduces the concept of internal stability for two-point boundary-value descriptor systems (TPBVD's). Since TPBVD's are defined only over a finite interval, the concept of stability is not easy to formulate for these systems. The definition which is used here 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 TPBVD's 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 is then used to characterize the property of internal stability. (jhd)