\cal H-Representation and Applications to Generalized Lyapunov Equations and Linear Stochastic Systems.

This paper introduces an \cal H-representation method to express an n^2\times \,1 vector \overrightarrow X as \overrightarrow X=H\widetilde X. Based on the introduced \cal H-representation approach, several topics are extensively discussed, including the generalized Lyapunov equations (GLEs) arising from stochastic control, stochastic observability, generalized \cal D-stability and \cal D-stabilization, weak stability, and stabilization. A necessary and sufficient condition for the existence and uniqueness of the symmetric and skew-symmetric solutions of GLEs is presented, respectively. Moreover, the solution structure of GLEs is also clarified. Through the \cal H-representation method, several necessary and sufficient conditions are also obtained for stochastic observability, generalized \cal D-stability and \cal D-stabilization, weak stability, and stabilization. [ABSTRACT FROM AUTHOR]