On Controllability of Multilinear Dynamical Systems

Multilinear dynamical systems (MLDSs) are crucial for modeling multidimensional data across various fields, including neuroscience, social sciences, and signal processing, as they effectively capture higher-order interactions and complex tensorial structures beyond the scope of linear systems. However, analyzing and controlling MLDSs poses significant challenges due to their inherently high-dimensional, multilinear characteristics. In this article, we examine the controllability of MLDSs where the systems’ evolution is governed by a multilinear operator based on Tucker decomposition. Notably, we extend the classical Kalman’s rank condition to MLDSs by leveraging the factor matrices in the Tucker decomposition. Additionally, we explore the weak and strong controllability of MLDSs in the presence of stochastic noise. Numerical examples are provided to demonstrate the effectiveness and efficiency of our proposed framework.