Positive Definiteness and Stability of Interval Tensors

In this paper, we focus on the positive definiteness and Hurwitz stability of interval tensors. First, we introduce auxiliary tensors Aᶻ 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 Aᶻ are positive (semi-)definite. For Hurwitz stability, it is revealed that the stability of the symmetric interval tensor AₛI can deduce the stability of the interval tensor AI, and the stability of symmetric interval tensors is equivalent to that of auxiliary tensors Ãᶻ. Finally, taking $4$th order $3$-dimensional interval tensors as examples, the specific sufficient conditions are built for their positive (semi-)definiteness.