Optimal Target Control of Complex Networks With Selectable Inputs

Leitao Gao, Guangshe Zhao, Guoqi Li, Fanghong Guo, Fei Zeng

Abstract: Controlling a preselected subset of nodes (named “target control”) of complex networks with minimal control energy is a critically important physical issue. To address this issue, first, an energy cost function is established by designing an optimal controller, in which an input matrix B is involved as a matrix variable to be determined so as to minimize the control cost. In particular, we integrate the design equations to obtain an equivalent expression of the cost function without solving the Riccati differential equation directly. Second, based on this expression, we derive its gradient with respect to B by introducing some methods for matrix differentiation. Two different constraints, i.e., trace and positive element constraints, which are imposed on the matrix variable B of the cost function, are considered. Last but not least, we propose two corresponding algorithms to solve these two different constraint optimization problems. Numerical examples in directed networks are provided to show the effectiveness of the proposed methods. This work suggests that the target control of complex networks can reduce both the minimum number of external control sources and the control cost.