Decoupling Control Of Boolean Networks And Solutions Of Matrix Equation With Semi-tensor Product

Boolean network is a powerful tool for describing gene regulatory net-work. With the rapid development of systems biology, Boolean network becomes a hot research topic. Since coupling and disturbance generally exist in practice, they also appear in gene regulatory networks, cell differentiation, and other biological dynamics. Decoupling control problems of Boolean net-works should be considered. In the research of Boolean networks, game, and other issues, some control problems can be converted into solving matrix e-quation with respect to semi-tensor product. This dissertation investigates decoupling control of Boolean control networks, and studies matrix equation with respect to semi-tensor product. The main results and contributions are as follows:1. Input-output decomposition problem of Boolean control networks is studied. The concept of comparable output-friendly subspaces is proposed, and the problem is converted into two issues:finding comparable output-friendly subspaces; designing a control such that the system with designed control can be expressed as p parallel subsystems. Firstly, a necessary and sufficient condition for the solvability of the first issue is obtained, and spe-cific method of calculating comparable output-friendly subspaces is supplied. Then, the open-loop control and the state feedback control are both consid-ered. It is shown how to find proper controllers to solve the second issue. Based on this, necessary and sufficient conditions are derived for the solv-ability of the problem with open-loop control and state feedback control, respectively.2. Some further research on disturbance decoupling of Boolean control networks is carried out. A necessary and sufficient condition is developed for the solvability of the problem. Furthermore, an algorithm is proposed to find all disturbance decoupling controllers.3. The matrix equation A × X = B with respect to semi-tensor product is investigated. Firstly, the matrix-vector equation A × X= B with semi-tensor product is discussed. Compatible conditions are established for the matrices, and a necessary and sufficient condition for the solvability of the matrix-vector equation is proposed. In addition, concrete solving methods are provided. Based on this, the solvability of the matrix equation A ×X= B with semi-tensor product is studied. Compatible conditions, solvability con-ditions, and concrete solving methods of the matrix equation are developed as well.4. Least squares solution of the matrix equation A × X= B with respect to semi-tensor product is considered. Firstly, the matrix-vector equation A × X= B is studied. Then the matrix equation A × X= B is discussed. Least squares solution with specific form is obtained.