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In this paper, comparison of topological structures between Boolean control networks and nominal Boolean networks is investigated. First, transient periods and attractor basins of Boolean control networks are introduced, and three necessary and sufficient conditions are derived in order to determine transient periods and attractor basins. From two aspects of attractors and transient periods, topological structures of Boolean control networks and nominal Boolean networks are compared, and the existences of different relations are illustrated via some examples. Finally, four necessary and sufficient conditions are proposed for judging relations of topological structures between Boolean control networks and nominal Boolean networks.

Two kinds of optimal control problems of Boolean control networks are investigated in this paper. For these two optimal control problems, existences of optimal control sequences are proved firstly. To find the optimal control sequences, weighted directed graphs are built. Then graph-theoretical approaches are proposed based on the Dijkstra's algorithm for the shortest path problem. Finally, an example is given to show the feasibility of the proposed approaches.

Nominal Boolean network is a kind of Boolean network, which is obtained from a Boolean control network with all controls disconnected. In this paper, the relationship between the transition matrix of the original Boolean control network and the transition matrix of the corresponding nominal Boolean control network is investigated. Due to the fact that transition matrices can be derived from structure matrices, the relationship between structure matrices of the original Boolean control network and structure matrices of the nominal Boolean network is studied. Via constant values corresponding to disconnected controls, structure matrices of the nominal Boolean network can be derived from structure matrices of the original Boolean control network. Finally, a concrete algorithm is presented to derive transition matrices of nominal Boolean networks.