Optimal control of interacting active particles on complex landscapes

Active many-body systems composed of many interacting degrees of freedom often operate out of equilibrium, giving rise to non-trivial emergent behaviors which can be functional in both evolved and engineered contexts. This naturally suggests the question of control to optimize function. Using navigation as a paradigm for function, we deploy the language of stochastic optimal control theory to formulate the inverse problem of shepherding a system of interacting active particles across a complex landscape. We implement a solution to this high-dimensional problem using an Adjoint-based Path Integral Control (APIC) algorithm that combines the power of recently introduced continuous-time back-propagation methods and automatic differentiation with the classical Feynman-Kac path integral formulation in statistical mechanics. Numerical experiments for controlling individual and interacting particles in complex landscapes show different classes of successful navigation strategies as a function of landscape complexity, as well as the intrinsic noise and drive of the active particles. However, in all cases, we see the emergence of paths that correspond to traversal along the edges of ridges and ravines, which we can understand using a variational analysis. We also show that the work associated with optimal strategies is inversely proportional to the length of the time horizon of optimal control, a result that follows from scaling considerations. All together, our approach serves as a foundational framework to control active non-equilibrium systems optimally to achieve functionality, embodied as a path on a high-dimensional manifold.