Solving Tensor Equation Systems under the Semi-Tensor Product: A t-Product Approach

This paper addresses the solvability of the tensor system A⋉X = B, X ⋉C = D using thesemi-tensor product framework. Initially, we examine the associated tensor-vector systemunder the semi-tensor product operation, deriving necessary and sufficient conditions forthe existence of solutions and propose explicit solution methods. Furthermore, we introduce suitable tensor partitioning techniques that enable the reduction of the tensor-vectorsystem to a linear system formulated via the t-product. This approach facilitates a comprehensive study of the solvability of the tensor system when the unknown tensor X is avector, matrix or a higher-order tensor under the semi-tensor product. We apply the STPframework to model and control Boolean gene regulatory networks with pinning controlinputs. Several illustrative examples are provided to demonstrate the applicability and effectiveness of the proposed approaches. Our study includes the tensor equation A⋉X = Band the matrix system A ⋉ X = B, X ⋉ C = D as special cases.

کلیدواژه ها:

Tensor equation ، Semi-tensor product ، Toeplitz tensor ، t-product