The lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space

This paper is devoted to a study of the properties of the equationA ^*FA–F=–G, where FL() is unknown, AL(), GL() is positive and is a Hilbert space. It is shown that necessary and sufficient (in some sense) conditions for the existence of positive definite solutions of this equation are directly connected with the stability of infinite dimensional linear systemx _k+1=Ax _k . The relationships between stability of such a system and stability of a continuous-time system generated by a strongly continuous semigroup are given also. As an example the case of the delayed system in Rⁿ is considered.This work was supported in part by the Polish Academy of Sciences under the contract Problem Miedzyresortowy I.1, Grupa Tematyczna 3 This paper was written while the author was with the Instytut Automatyki, the same university.