Solving Matrix Nearness Problems via Hamiltonian Systems, Matrix Factorization, and Optimization

The main goal of these lecture notes is to survey a series of recent works (Gillis and Sharma, Automatica 85:113–121, 2017; SIAM J. Numer. Anal. 56(2):1022–1047, 2018; Linear Algebra Appl. 623:258–281, 2021; Gillis et al., Numerical Linear Algebra Appl. 25(5):e2153, 2018; Linear Algebra Appl. 573:37–53, 2019; Appl. Numer. Math. 148:131–139, 2020; Choudhary et al., Numerical Linear Algebra Appl. 27(3):e2282, 2020) that aim at solving several nearness problems for a given system. As we will see, these problems can be written as distance problems of matrices or matrix pencils. To solve them, this series of recent works rely on a two-step approach. The first step parametrizes the system using a Port-Hamiltonian representation where stability is guaranteed via convex constraints on the parameters. The second step uses standard non-linear optimization algorithms to optimize these parameters, minimizing the distance between the given system and the sought parametrized stable system. In these lecture notes, we will illustrate this strategy in order to find the nearest stable continuous-time and discrete-time systems, the nearest stable matrix pair, and the nearest positive-real system, as well as generalizations when the eigenvalues need to belong to some set Omega (which is referred to as Omega stability).