# Control of complex systems with applications in collective behavior and fluid dynamics

# Control of complex systems with applications in collective behavior and fluid dynamics

Collective behavior and self-organization emerge in many phenomena, from natural swarms of birds and bees to the more surprising, economics of wealth distribution, and social phenomena of formation of opinions. In the first part of this talk, I will discuss methods for controlling collective behavior of large groups of interacting particles. In general, I am interested in the question, given particle-level dynamics, how must we design inter-particle interactions so that we are able to steer the particles to a desired distribution. Specifically, I will focus on two applications: swarm robotics and biological networks of cell species. In both models, the particles evolve stochastically, thus the macroscopic model is governed by a (deterministic) Markov process. I will demonstrate how to design control strategies to render arbitrary probability distributions globally asymptotically stable, with minimal assumptions on the regularity on the distributions. I will show how despite the infinite dimensional nature of the problem and failure of classical dynamical systems tools such as Lyapunov functionals, one can use a monotonicity argument to establish global stability. In the second part of the talk, I will focus on optimal control of partial differential equations (PDE), particularly the thin-film equation which arises in fluid applications. I will consider a situation in which uncontrolled thin film dynamics is inherently unstable, but it is required to strategically vary the flux at the boundary of the PDE to shape the downstream behavior of the flow. I pose this as an optimization problem with PDE constraints to identify such a strategy. I will present numerical simulations that support this theory.