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In this paper, we address the optimal control problem for Boolean control networks (BCNs). We first consider the problem of finding the input sequences that minimize a given cost function over a finite time horizon. The problem solution is obtained by means of a recursive algorithm that represents the analogue for BCNs of the difference Riccati equation for linear systems. We prove that a significant number of optimal control problems for BCNs can be easily reframed into the present setup. In particular, the cost function can be adjusted so as to include penalties on the switchings, provided that we augment the size of the BCN state variable. In the second part of the paper, we address the infinite horizon optimal control problem and we provide necessary and sufficient conditions for the problem solvability. The solution is obtained as the limit of the solution over the finite horizon 0,T, and it is always achieved in a finite number of steps. Finally, the average cost problem over the infinite horizon, investigated in “Optimal control of logical control networks” (Y. Zhao , IEEE Trans. Autom. Control, vol 56, no. 8, pp. 1766-1776, Aug. 2011), is addressed by making use of the results obtained in the previous sections.
Fornasini et al. (2014) studied this question.