Dynamical System Stability Criteria Based on the Frobenius Norm

It is well known that the position of the Jacobian matrix spectrum in the left-half complex plane provides the local asymptotic stability of a nonlinear dynamical system, but it is also well known that for large matrices, computing its eigenvalues just to see their position is computationally prohibitive. Instead, it is recommended to check if a given matrix belongs to the H-matrix class and has negative diagonal entries. Since confirming the H-matrix property is computationally costly, the preference is to work with its subclasses, which are defined by simpler conditions. In this paper, we develop and investigate a new subclass of H-matrices via the Frobenius matrix norm, which generalizes the recently introduced classes. We support its significance with real-life examples and clarify its relationship to some well-known block H-matrices based on the Euclidean matrix norm. The main novelty in this paper is that when a fast and inexpensive answer about the stability of a dynamical system is required, and the system matrix has a natural block structure, we develop a simple tool to check whether this structure, along with the additional condition of negative diagonal elements, ensures stability. This is especially important when the matrix does not belong to any previously known H-matrix subclasses.