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

This paper introduces an H-representation method to express an n² × 1 vector X^→ as X^→=H[(X)tilde]. Based on the introduced H-representation approach, several topics are extensively discussed, including the generalized Lyapunov equations (GLEs) arising from stochastic control, stochastic observability, generalized D-stability and 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 H-representation method, several necessary and sufficient conditions are also obtained for stochastic observability, generalized D-stability and D-stabilization, weak stability, and stabilization.