Optimal adaptive pinning control of network systems onto an unstable equilibrium

When controlling network systems through pinning, distributed adaptive strategies have been devised to increase the coupling and control gains until convergence onto the desired equilibrium configuration is attained. However, in the presence of persistent perturbations, the gains would indefinitely increase, making control energetically inefficient. Here, we propose a novel adaptation law that overcomes this limitation by optimally selecting the setpoint for the coupling and control gains, so that the network rejects local perturbations with the minimum control energy. Extensive numerical simulations on paradigmatic networks of nonlinear systems demonstrate the robustness of the proposed approach to large perturbations from the desired equilibrium.