The infinite-time admissibility of observation operators and operator Lyapunov equations

Abstract

Let Σ(A, −, C) be an abstract dynamical system withA being the generator of aC 0-semigroup on a Hilbert spaceH, C:D(A)→Y a linear operator,Y another Hilbert space. In this paper, some sufficient and necessary conditions are obtained for the observation operatorC to be infinite-time admissible. For a control system Σ(A, B, −), due to duality argument, some sufficient and necessary conditions are also given for the control operatorB to be extended admissible. It is wellknown that observation operatorC is admissible if and only if the operator Lyapunov equation associated with the system has a nonnegative solution. In this paper, all nonnegative solutions to this equation are represented parametrically.