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A Boolean network is widely used as a model of gene regulatory networks. In control of gene regulatory networks using Boolean networks, we assume that the concentration level of a part of genes is arbitrarily determined as the control input. However, there are cases that no gene satisfying this assumption exists, and it is important to consider weak control. In this paper, a Boolean network with two types of the control inputs is proposed as a model of gene regulatory networks. The first control input is the control input satisfying the above assumption. The second control input is called here a weak control input, and dynamics i.e. Boolean functions are selected among the candidates of dynamics. For example, activation/inactivation of the whole network is controlled by the weak control input. In order to solve the optimal control problem, two approaches, i.e., an integer programming approach and a polynomial optimization approach are proposed.