Foresight and relaxation enable efficient control of nonlinear complex systems

There exist numerous effective techniques for influencing linear systems to desired outcomes, but control of general nonlinear complex systems remains an open problem. One promising technique, which exploits our deep understanding of linear systems, involves linearizing at the current position and then applying linear optimal control to move the system to a local target from which it should be easier to reach the desired final state. However, nonlocal trajectories are often required to influence linear systems even to local targets, meaning that local linearization-based strategies can lead to inefficient jagged trajectories with high path length and large control energy cost. To address these limitations, here we propose an alternative control strategy with two innovations. The first is avoiding fixation on varying local targets. Instead, we exercise foresight by consistently planning a complete route to the final target, moving a short distance along the planned route, and then updating the plan according to the new local conditions. The second refinement, which we term relaxation, discourages overinvestment in control strategies which are optimal according to the linearization at the current point but could be inefficient according to dynamical conditions encountered at later times. We evaluate our strategy on complex systems from neuroscience and statistical mechanics, showing that our innovations substantially increase the success rate of control for a given path length or energy expenditure, and that these advantages persist as system size increases.