  We develop a principled mathematical framework for controlling nonlinear,
networked dynamical systems. Our method integrates dimensionality reduction,
bifurcation theory and emerging model discovery tools to find low-dimensional
subspaces where feed-forward control can be used to manipulate a system to a
desired outcome. The method leverages the fact that many high-dimensional
networked systems have many fixed points, allowing for the computation of
control signals that will move the system between any pair of fixed points. The
sparse identification of nonlinear dynamics (SINDy) algorithm is used to fit a
nonlinear dynamical system to the evolution on the dominant, low-rank subspace.
This then allows us to use bifurcation theory to find collections of constant
control signals that will produce the desired objective path for a prescribed
outcome. Specifically, we can destabilize a given fixed point while making the
target fixed point an attractor. The discovered control signals can be easily
projected back to the original high-dimensional state and control space. We
illustrate our nonlinear control procedure on established bistable,
low-dimensional biological systems, showing how control signals are found that
generate switches between the fixed points. We then demonstrate our control
procedure for high-dimensional systems on random high-dimensional networks and
Hopfield memory networks.
