Theory and numerics of some matrix nearness problems with applications in systems and control

Structured matrices naturally arise in different fields of sciences since they are the keys of smart and efficient solutions. Several properties of physical or engineering models can be usually characterized by constraints on structured matrices (e.g. rank constraints). There- fore it is usually interesting to study how far a given model is from another one satisfying a given property. The thesis focuses on some problems which can be restated as (structured) matrix near- ness problems, and in particular we are interested in computing which is the closest matrix to a given one which satisfy a certain rank constraint and preserves the structure at the same time. Such problems, also known in the literature as Structured Low-Rank Approxi- mation problems, are typically nonconvex optimization problems which do not allow analytic solution. We consider three main problems: the computation of approximate Greatest Common Divisors for scalar polynomials and matrix polynomials (first part) and Hankel (and mosaic Hankel) low-rank approximation (second part). The basic idea of the numerical approach for solving the problems is similar, however every problem presents its numerical issues and the computational strategies are slightly different. The theory and the numerical algorithm for the solution of each problem is followed by some applications in system theory, control theory and signal processing in order to motivate the usefulness of the proposed problems.