Extended matrix cube theorems with applications to μ-theory in control

In this paper, we study semi-infinite systems of Linear Matrix Inequalities which are generically NP-hard. For these systems, we introduce computationally tractable approximations and derive quantitative guarantees of their quality. As applications, we discuss the problem of maximizing a Hermitian quadratic form over the complex unit cube and the problem of bounding the complex structured singular value. With the help of our complex Matrix Cube Theorem we demonstrate that the standard scaling upper bound on μ(M) is a tight upper bound on the largest level of structured perturbations of the matrix M for which all perturbed matrices share a common Lyapunov certificate for the (discrete time) stability.