Peak Time-Windowed Mean Estimation using Convex Optimization

This paper presents an algorithmic approach towards bounding the peak time-windowed average value attained by a state function along trajectories of a dynamical system. An example includes the maximum average current flowing across a power line in any 5 -minute window. The peak time-windowed mean estimation task may be posed as a finite-dimensional but nonconvex optimization problem in terms of an initial condition and stopping time. This problem can be lifted into an infinite-dimensional linear program in occupation measures, where no conservatism is introduced under compactness and dynamical regularity assumptions. The peak time-windowed mean estimation linear program is in turn truncated into a convergent sequence of semidefinite programs using the moment-Sum-of-Squares hierarchy. Bounds of the time-windowed mean are computed for example systems.