From t-Product to t-STP of Cubic Matrices With Application to Hyper-Networked Systems

Abstract: From the perspective of dynamic (control) systems over cubic matrices, this paper proposes some new products on cubic matrices and uses them to construct linear and nonlinear control systems over cubic matrices. First, the t-product of cubic matrices is surveyed, and some algebraic properties are investigated. When t-product based dynamic (control) systems are considered, its two weaknesses are found as follows: (i) it has rigorous dimension restriction; (ii) it is unable to formulate nonlinear (control) systems. Then the dimension keeping semi-tensor product (DK-STP) of cubic matrices is proposed, which overcomes the aforementioned weaknesses. Unfortunately, all subsystems, corresponding to every frontal slides, of a DK-STP based dynamic (control) system are decoupled, which makes it unsuitable for formulating systems with coupled subsystems. Finally, the t-STP, a combination of t-product with DK-STP, is presented. The t-STP related algebraic structures over cubic matrices, such as group, ring, module, and Lie group/Lie algebra, etc. are investigated. Finally, t-STP based dynamic (control) systems over cubic matrices are obtained, and its application to hyper-networked evolutionary games is investigated.