# On approximating the nearest \Omega-stable matrix

## TL;DR

This paper introduces a convex optimization approach to approximate matrices with eigenvalues in specific regions of the complex plane, relevant for system stability, using dissipative Hamiltonian matrices and linear matrix inequalities.

## Contribution

It formulates the nearest -stable matrix problem within a convex framework using LMI and dissipative Hamiltonian parametrization, enabling efficient solutions.

## Key findings

- Convex reformulation of the approximation problem.
- Application of block coordinate descent for solutions.
- Illustrative examples demonstrating effectiveness.

## Abstract

In this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region \Omega, within the complex plane. More precisely, we consider three types of regions and their intersections: conic sectors, vertical strips and disks. We refer to this problem as the nearest \Omega-stable matrix problem. This includes as special cases the stable matrices for continuous and discrete time linear time-invariant systems. In order to achieve this goal, we parametrize this problem using dissipative Hamiltonian matrices and linear matrix inequalities. This leads to a reformulation of the problem with a convex feasible set. By applying a block coordinate descent method on this reformulation, we are able to compute solutions to the approximation problem, which is illustrated on some examples.