Stability and Stabilization of Kronecker-Sum-Coupled Multilinear Systems

Overview & Synopsis

Coupled dynamical systems with multi-dimensional state structures arise in many applications such as spatiotemporal processes and large-scale interconnected systems. However, existing studies primarily focus on matrix-based coupled systems or tensor-based dynamical systems without couplings. The properties of coupled tensor-based systems remain largely unexplored. In this article, we study Kronecker-sum-coupled multilinear systems under the transform-based tensor product framework. We establish fundamental tensor spectral properties showing that stability can be characterized through lower-dimensional subsystem dynamics together with slice-wise matrix stability in the transform domain. We further develop optimization conditions for state-feedback stabilization and show that the stabilization problem can be decomposed into independent subsystem problems. Finally, we analyze the computational complexity of the proposed framework and demonstrate its substantial computational advantages over unfolding-based approaches using both synthetic systems and multi-inertia motor systems.