Optimal Asynchronous Stabilization for Boolean Control Networks With Lebesgue Sampling

Using semitensor products (STPs) of matrices, sampled-data state-feedback control (SDSFC) with the Lebesgue sampling region $\mathcal {S}_{\tau }$ is first considered to stabilize a Boolean control network (BCN) to a fixed point, under which a necessary and sufficient condition for stabilization is obtained when the considered BCN with the Lebesgue sampling region is converted to a switching system. Meanwhile, the corresponding asynchronous SDSFC gains are designed from a sequence of reachable sets. Then, an algorithm is shown to obtain the minimal number of controlling times and all states globally stabilize to the desired state with the fastest convergence rate under the minimal number of controlling times. Besides, the results have been extended to $p$ Lebesgue sampling regions $\mathcal {S}_{\tau _{i}}, i=1,2,\ldots, p$ . And some results are presented for this situation, including the necessary and sufficient conditions stabilization under the $p$ Lebesgue sampling regions, the asynchronous SDSFC gains, and the algorithm to obtain the optimal states sampling regions. Examples are listed to show the effectiveness of our results, and the biological example indicates that the SDSFC with Lebesgue sampling is also suitable for stochastic BCNs.