Linear Quadratic Optimal Control of Time-Invariant Linear Networks With Selectable Input Matrix

Optimal control of networks is to minimize the cost function of a network in a dynamical process with an optimal control strategy. For the time-invariant linear systems, ẋ(t)=Ax(t)+Bu(t), and the traditional linear quadratic regulator (LQR), which minimizes a quadratic cost function, has been well established given both the adjacency matrix A and the control input matrix B. However, this conventional approach is not applicable when we have the freedom to design B. In this article, we investigate the situation when the input matrix B is a variable to be designed to reduce the control cost. First, the problem is formulated and we establish an equivalent expression of the quadratic cost function with respect to B, which is difficult to obtain within the traditional theoretical framework as it requires obtaining an explicit solution of a Riccati differential equation (RDE). Next, we derive the gradient of the quadratic cost function with respect to the matrix variable B analytically. Further, we obtain three inequalities of the cost functions, after which several possible design (optimization) problems are discussed, and algorithms based on gradient information are proposed. It is shown that the cost of controlling the LTI systems can be significantly reduced when the input matrix becomes ``designable.'' We find that the nodes connected to input sources can be sparsely identified and they are distributed as evenly as possible in the LTI networks if one wants to control the networks with the lowest cost. Our findings help us better understand how the LTI systems should be controlled through designing the input matrix.