Optimal regulation in a periodic environment: insights from a simple model

TL;DR

This paper analyzes a simple mathematical model for optimal periodic regulation of a process, proving existence, uniqueness, and characterizing solutions, with numerical insights into how solutions vary with data.

Contribution

It provides a rigorous analysis of the measure-based optimal regulation problem, including existence, uniqueness, explicit solution formulas, and numerical exploration.

Findings

Existence and uniqueness of the optimal solution in the measure space

Explicit formula for the solution's density under regularity conditions

Solution measures can include atomic components when data is discontinuous

Abstract

We perform a detailed study of a simple mathematical model addressing the problem of optimally regulating a process subject to periodic external forcing, which is interesting both in view of its direct applications and as a prototype for more general problems. In this model one must determine an optimal time-periodic `effort' profile, and the natural setting for the problem is in a space of periodic non-negative measures. We prove that there exists a unique solution for the problem in the space of measures, and then turn to characterizing this solution. Under some regularity conditions on the problem's data, we prove that its solution is an absolutely continuous measure, and provide an explicit formula for the measure's density. On the other hand, when the problem's data is discontinuous, the solution measure can also include atomic components. Complementing our analytical results, we carry out numerical computations to obtain solutions of the problem in various instances, which enable us to examine the interesting ways in which the solution's structure varies as the problem's data is varied.