Connections between the algebraic Riccati equation and the Hamiltonian for Riesz-spectral systems

The algebraic Riccati equation (ARE) has been studied in great detail for finite-dimensional systems. For infinite-dimensional systems almost all results concentrate on the relation with the linear quadratic optimal control problem. The object of this paper is to consider solutions of the ARE for infinite-dimensional systems from a more general point of view. The relation between linear, bounded solutions of the ARE and the eigenvectors of the Hamiltonian will be studied, in the case when the Hamiltonian is a Riesz-spectral operator. We present a general form of all possible linear, bounded solutions of the ARE in terms of the eigenvectors of the Hamiltonian. Characterizations for self-adjoint, nonnegative and stabilizing solutions are given as well. The derived results shall be applied to the heat equation.