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

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.

Figures (6)

• Figure 1: Optimal effort profile ($\eta_{opt}(t)$), cumulative effort profile ($\alpha_{opt}(t)$) and the resulting function $S_{opt}(t)$ for inflow rate $c(t)=1-0.9\cos(2\pi t)$, weight $w(t)\equiv 1$, $\delta=1$, and three values of $\bar{\eta}$. For comparison, dashed black lines show corresponding results for a uniform effort profile $\eta(t)=\bar{\eta}$. The dashed red line in the second row display the analytical expression \ref{['eq:etaopt1']}.
• Figure 2: Optimal effort profile ($\eta_{opt}(t)$), cumulative effort profile ($\alpha_{opt}(t)$) and the resulting function $S_{opt}(t)$ for inflow rate $c(t)$ given by \ref{['eq:ceeo']}, weight $w(t)\equiv 1$, $\delta=1$, and three values of $\bar{\eta}$.
• Figure 3: Optimal cumulative effort profile ($\alpha_{opt}(t)$) for inflow rate given by \ref{['eq:ceeo']}, weight $w(t)\equiv 1$, $\delta=1$, and four values of $\bar{\eta}$. The bottom panel shows the size of the jump in $\alpha_{opt}$ occurring at $t=0.5$.
• Figure 4: Optimal effort profile ($\eta_{opt}(t)$), cumulative effort profile ($\alpha_{opt}(t)$) and the resulting function $S_{opt}(t)$ for inflow rate $c(t)$ given by \ref{['eq:ceeo1']}, weight $w(t)\equiv 1$, $\delta=1$, and three values of $\bar{\eta}$.
• Figure 5: Optimal effort profile ($\eta_{opt}(t)$), cumulative effort profile ($\alpha_{opt}(t)$) and the resulting function $S_{opt}(t)$ for inflow rate $c(t)$ given by \ref{['eq:cee']}, weight $w(t)\equiv 1$, $\delta=1$, and three values of $\bar{\eta}$.

Theorems & Definitions (52)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['th:existence']}
• Theorem 2
• Example 1
• Theorem 3
• Example 2