Pinning Control of Dynamical Networks with Optimal Cost

Complex networks of coupled dynamical systems, are effective models of numerous distributed systems. The desire to control such systems has led to the pinning control techniques where a subset of nodes in the network is controlled. A problem of interest is to optimize the pinning node selection and control gain design, while minimizing the associated total control cost. Here this optimal control optimization problem has been reformulated as a standard semidefinite programming problem. Solving the resultant problem, the analytical solution for the optimal feedback gain and pinning nodes are derived. An algorithm for determining the optimal feedback gain for a set of pinned nodes is developed. For a number of topologies, closed-forms of the optimal results are provided. Interestingly, increasing the number of pinned nodes in the network reduces the total pinning cost, i.e., the performance index. For a network of Lorenz systems, optimal results for all possible pinning node selections are provided.