Figure 1: An example network and control energy reduction with fewer targets.
From: Energy scaling of targeted optimal control of complex networks
(a) A sample network with seven nodes and colour-coded input signals (blue) and output sensors (pink). Note that each control input is directly connected to a single node, and each output sensor receives the state of a single node. Nodes directly connected to the pink outputs are target nodes, that is, they have a prescribed final state that we wish to achieve in finite time, tf. The corresponding vector y(tf) is defined in terms of the states as well. Nodes directly receiving a signal from a blue node are called input nodes and the remaining nodes are neither input nodes nor target nodes. (b) We examine a three-node network where every node is a target node (pink nodes) and one node receives a control input (blue). The edge weights are shown and the self-loop magnitude k=1. (c) The state evolution is shown where the initial condition is the origin and the final state for each target node is yi(tf)=1, i=1,2,3. (d) The square of the magnitude of the control input is also shown. The energy, or the control effort, is found by integrating the square of the magnitude of the control input. For this case, E=|u(t)|2≈382 (a.u.). (e) The same network as in b but now only one node is declared a target node. (f) The state evolution is shown where the initial condition remains the origin but the final condition is only defined for y3(tf)=1. (g) The square of the magnitude of the control input is also shown. Note the different vertical axis scale as compared to d. For the second case, E=|u(t)|2≈66.3 (a.u.).
