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.

中文翻译：

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网络动力系统的非线性控制


我们开发了一个用于控制非线性、网络动力系统的原则性数学框架。我们的方法集成了降维、分岔理论和新兴模型发现工具，以找到低维子空间，在其中可以使用前馈控制来操纵系统达到期望的结果。该方法利用了许多高维网络系统具有许多固定点的事实，允许计算控制信号，从而使系统在任何一对固定点之间移动。非线性动力学稀疏辨识 (SINDy) 算法用于使非线性动力学系统适应主导低秩子空间上的演化。然后，这使我们能够使用分叉理论来找到恒定控制信号的集合，这些信号将为规定的结果产生所需的客观路径。具体来说，我们可以使给定固定点不稳定，同时使目标固定点成为吸引子。发现的控制信号可以很容易地投影回原始的高维状态和控制空间。我们在已建立的双稳态、低维生物系统上说明了我们的非线性控制程序，展示了如何找到控制信号来生成固定点之间的切换。然后，我们在随机高维网络和 Hopfield 记忆网络上演示了高维系统的控制程序。