Necessary Conditions for Optimality for a Distributed Optimal Control Problem

Abstract—This paper presents a novel optimal control prob-lem formulation and new optimality conditions, referred to as distributed optimal control, for systems comprised of many dynamic agents that can each be described by the same ordi-nary differential equations (ODEs). The macroscopic system performance is represented by an integral cost function of a restriction operator comprised of the probability density function of the individual agents ’ state variables, and of their control laws. It is shown that, under proper assumptions, the macroscopic cost can be optimized subject to a hyperbolic partial differential equation (PDE) that describes the evolution of the macroscopic state over larger spatial and temporal scales. This methodology extends the capabilities of optimal control to complex systems described by numerous interacting dynamical systems. The approach is demonstrated on a simulated network of distributed sensors installed on autonomous underwater vehicles, and deployed to provide track coverage over a region of interest. I