# The geometry of the generalized algebraic Riccati equation and of the singular Hamiltonian system

## TL;DR

This paper explores the geometric properties of the generalized algebraic Riccati equation and singular Hamiltonian systems, revealing subspace structures that can aid in stabilizing control systems even without traditional solutions.

## Contribution

It introduces a geometric perspective on the generalized Riccati equation, identifying subspaces that enable stabilization beyond standard solutions.

## Key findings

- Identification of subspaces facilitating stabilization
- Insights into the structure of singular Hamiltonian systems
- Potential methods for control stabilization without Riccati solutions

## Abstract

This paper analyzes the properties of the solutions of the generalized continuous algebraic Riccati equation from a geometric perspective. This analysis reveals the presence of a subspace that may provide an appropriate degree of freedom to stabilize the system in the related optimal control problem even in cases where the Riccati equation does not admit a stabilizing solution.