Containment problem of finite-field networks with fixed and switching topology

•The containment problem is generalized to FFNs, which enriches the theory of FFNs and extends the application of STP.•Using STP, we propose some new necessary and sufficient conditions to achieve containment by analyzing the property of structural matrix. The conditions listed in this paper are easily to check via the MATLAB toolbox.•Comparison with the existing works: From the point of systems, the state variables in [14,16,17,20] is 1-dimensional, while the state variables in this paper is k-dimensional; From the point of research methods, the conventional matrix theory and graph theory are the main mathematical tool in [14–17,20], including characteristic polynomial, eigenvalue and connectivity, while the STP method is more convenient to check via MATLAB toolbox. The containment problem of finite-field networks (FFNs) with fixed topology (FT-FFNs) and switching topology (ST-FFNs) is studied. Firstly, the concept of containment for FFNs is proposed. Secondly, by semi-tensor product (STP) of matrices, the logical updating rule of FFNs with multiple leaders can be equivalently converted into an algebraic form. Thirdly, the containment problem is converted into the problem of set stability, and an easily verifiable necessary and sufficient condition is presented to achieve containment. In addition, the obtained results are applied to the containment problem of ST-FFNs. Finally, two examples support the effectiveness of the main results.