Hamiltonians with Riesz Bases of Generalised Eigenvectors and Riccati Equations

TL;DR

This paper investigates algebraic Riccati equations for unbounded operators, demonstrating the existence of infinitely many solutions using Riesz bases of generalized eigenvectors, with applications to differential operator Riccati equations.

Contribution

It introduces a novel approach using Riesz bases to analyze Hamiltonians with unbounded operators and characterizes solutions to Riccati equations in this context.

Findings

Existence of infinitely many selfadjoint solutions for unbounded operator Riccati equations.

Construction of invariant graph subspaces via Riesz bases of generalized eigenvectors.

Representation of all bounded solutions under additional assumptions.

Abstract

An algebraic Riccati equation for linear operators is studied, which arises in systems theory. For the case that all involved operators are unbounded, the existence of infinitely many selfadjoint solutions is shown. To this end, invariant graph subspaces of the associated Hamiltonian operator matrix are constructed by means of a Riesz basis with parentheses of generalised eigenvectors and two indefinite inner products. Under additional assumptions, the existence and a representation of all bounded solutions is obtained. The theory is applied to Riccati equations of differential operators.