Nearest structured polynomial matrix having an eigenvalue with prescribed geometric multiplicity

This paper addresses the problem of determining the closest structured regular polynomial matrix to a given structured polynomial matrix having an eigenvalue that satisfies a constraint on its geometric multiplicity. Initially, we approach this problem for the linearly structured unimodular polynomial matrices without imposing any specific constraints on the multiplicity of any eigenvalues of the perturbed non-unimodular polynomial matrix. Subsequently, we extend our investigation for any linearly structured polynomial matrices by introducing constraints related to the eigen-structure of the perturbed polynomial matrix. Our primary focus is to compute the nearest structured regular polynomial matrix that has an eigenvalue with a specific lower bound of the geometric multiplicity. We begin by considering the eigenvalue as any complex number, and then restrict our analysis to the case where it is real. In addressing these challenges, we use an optimization-based approach, reshaping the problem into a constrained optimization scenario. We conclude the paper by providing numerous numerical case studies, demonstrating the efficacy and applicability of our proposed approach.