# Dichotomous Hamiltonians and Riccati equations for systems with unbounded control and observation operators

## TL;DR

This paper investigates the control algebraic Riccati equation for systems with unbounded control and observation operators, utilizing a dichotomy property of Hamiltonian operators to construct solutions and analyze stability.

## Contribution

It introduces a novel approach using Hamiltonian operator dichotomy to find solutions to Riccati equations in systems with unbounded operators.

## Key findings

- Constructed invariant graph subspaces for Hamiltonian operators.
- Established boundedness of the nonnegative Riccati solution.
- Proved exponential stability of the feedback system with compact resolvent.

## Abstract

The control algebraic Riccati equation is studied for a class of systems with unbounded control and observation operators. Using a dichotomy property of the associated Hamiltonian operator matrix, two invariant graph subspaces are constructed which yield a nonnegative and a nonpositive solution of the Riccati equation. The boundedness of the nonnegative solution and the exponential stability of the associated feedback system is proved for the case that the generator of the system has a compact resolvent.