Structured Linear Stability Problems

We study problems of robustness of linear stability under structured matrix perturbations. Perturbations are restricted to lie in a prescribed structure space, which can be an arbitrary complex- or real-linear subspace of complex \(n\times n\) matrices. Cases of interest include real matrices or matrices with a given sparsity pattern or matrices with given range and co-range. We derive algorithms for computing the structured stability radius, which measures the robustness of asymptotic stability under structured perturbations, and a related algorithm for computing the structured \(\varepsilon \) -stability radius for a given \(\varepsilon >0\) , which yields robust transient bounds proportional to \(1/\varepsilon \) under structured matrix perturbations for solutions of homogeneous and inhomogeneous linear differential equations. These problems belong to a large class of matrix nearness problems in matrix analysis and control theory and graph theory that are closely related to eigenvalue optimization problems. Their extremizers are found to be orthogonal projections of rank-1 matrices onto the structure. The proposed algorithms make use of this fact via discretized rank-1 matrix differential equations that derive from a norm- and structure-constrained gradient flow.