Characterizing matrices with eigenvalues in an LMI region: A dissipative-Hamiltonian approach

In this paper, we provide a dissipative Hamiltonian (DH) characterization for the set of matrices whose eigenvalues belong to a given LMI region. This characterization is a generalization of that of Choudhary et al. (Numer. Linear Algebra Appl., 2020) to any LMI region. It can be used in various contexts, which we illustrate on the nearest $\mathrm{\Omega }$-stable matrix problem: given an LMI region $\mathrm{\Omega }\subseteq \mathbb{C}$ and a matrix $A\in {\mathbb{C}}^{n,n}$, find the nearest matrix to $A$ whose eigenvalues belong to $\mathrm{\Omega }$. Finally, we generalize our characterization to more general regions that can be expressed using LMIs involving complex matrices.