Optimal control of logical control network with noisy inputs

This paper considers the optimal control of the logical control network with noisy inputs. The optimal control problem concerning the minority game (MG) with mixed population is formulated. In the game the producers are represented by the states of nodes in the Boolean control network, the speculators and the noise traders are identified as the controllable inputs and the noisy inputs of the nodes. The speculators aim at achieving the maximum profits but the noise traders make the issue complicated. The Boolean logical variables and functions are transformed into the algebraic form by using the recently developed Semi-tensor Product of matrices technique. In this manner, we calculate the optimal cycles that lead to the optimal control and analyze the effect of noisy input based on the definition of score. Although we develop our method within the context of MG, it actually can be applied to other general cases of logical networks with noisy inputs. To avoid the computational expensive calculation of the optimal cycles for all states in the input-state network, we further propose a simplified algorithm with formulas for large scale MG case. An illustrative example is included to show the validity and efficiency of the simplified algorithm.