${cal H}$-Representation and Applications to Generalized Lyapunov Equations and Linear Stochastic Systems

摘要

This paper introduces an ${cal H}$-representation method to express an $n^{2}times ,$1 vector $overrightarrow {X}$ as $overrightarrow {X}=Hwidetilde {X}$. 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.