# Minimal-norm static feedbacks using dissipative Hamiltonian matrices

## TL;DR

This paper introduces a new characterization of static feedbacks for stabilizing linear systems using dissipative Hamiltonian matrices, leading to algorithms for minimal-norm feedbacks and static-output feedback, with demonstrated effectiveness.

## Contribution

It provides a novel parametrization of stabilizing feedbacks via dissipative Hamiltonian matrices and develops algorithms for minimal-norm static and output feedbacks.

## Key findings

- Algorithm computes minimal-norm static feedbacks efficiently.
- Method extends to static-output feedback problem.
- Numerical examples show improved performance over existing methods.

## Abstract

In this paper, we characterize the set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair using dissipative Hamiltonian matrices. This characterization results in a parametrization of feedbacks in terms of skew-symmetric and symmetric positive semidefinite matrices, and leads to a semidefinite program that computes a static-state stabilizing feedback. This characterization also allows us to propose an algorithm that computes minimal-norm static feedbacks. The theoretical results extend to the static-output feedback (SOF) problem, and we also propose an algorithm to tackle this problem. We illustrate the effectiveness of our algorithm compared to state-of-the-art methods for the SOF problem on numerous numerical examples from the COMPLeIB library.