Averaging of ordinary differential equations and averaged optimization

Averaging ordinary differential equations provides a method of approximating the solutions of highly oscillatory equations, or equations with coupled fast and slow variables. Modern methods of averaging fast-slow control systems, introduced in works of Artstein, Gaitsgory and Leizarowitz, employ occupational/empirical measures to provide a variational limit, whose solution approximates the slow variable. Occupational measures are probability measures that provide statistical information of a curve in space, and we study the structure of the set of occupational measures through infinite-horizon optimal control, with an averaged cost.

Both averaging and infinite horizon optimization have many applications in physics, engineering and operations research, we present new results in both fields, as well as in averaged shape optimization.