Optimal Control of Branched Flow

Branched flow is a striking phenomenon in which waves traveling through a complex or disordered medium spontaneously form intricate, tree-like branching patterns. This behavior appears across a wide range of systems, from electrons moving in quantum materials to ocean waves refracting over sandbars. In this talk, I will present a model, based on the nonlinear Schrödinger equation, that captures the emergence of branched flow and discuss how it can be used to understand and predict these patterns.

Building on the scientific model, I will present a mathematical framework for controlling branched flow in disordered media. We formulate an illumination problem where the goal is to shape initial conditions so that the wave pattern matches a desired target. We prove the problem admits a maximizer using calculus of variations, and we derive a projected gradient flow and a Newton method on the L2-sphere from first- and second-order optimality conditions. We establish sharp upper and lower bounds on achievable wave concentration, showing how fundamental energy, geometric, and bandwidth constraints set intrinsic limits that distinguish the true capabilities of the dynamics from algorithmic performance. Then, to address optimization robustness, we introduce a minimax formulation based on randomized media and density matrices, providing guarantees on worst-case performance. Finally, we use the Wigner transform to develop phase-space fingerprints of branching that reveal statistical features of the medium and guide the design of effective controls.

Throughout the talk, I will highlight both the mathematical ideas (well-posedness, compactness, controllability) and their physical implications, connecting theory, computation, and visualization of this fascinating wave phenomenon.