Positive Definiteness and Stability of Interval Tensors

Abstract

In this paper, we focus on the positive definiteness and Hurwitz stability of interval tensors. First, we introduce auxiliary tensors $\mathcal{A}^z$ and establish equivalent conditions for the positive (semi-)definiteness of interval tensors. That is, an interval tensor is positive definite if and only if all $\mathcal{A}^z$ are positive (semi-)definite. For Hurwitz stability, it is revealed that the stability of the symmetric interval tensor $\mathcal{A}_s^I$ can deduce the stability of the interval tensor $\mathcal{A}^I$, and the stability of symmetric interval tensors is equivalent to that of auxiliary tensors $\tilde{\mathcal{A}}^z$. Finally, taking $4$th order $3$-dimensional interval tensors as examples, the specific sufficient conditions are built for their positive (semi-)definiteness.