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 omega-stable matrix problem: given an LMI region Omega subset of C and a matrix A is an element of R-nxn, find the nearest matrix to A whose eigenvalues belong to Omega. Finally, we generalize our characterization to more general regions that can be expressed using LMIs involving complex matrices.