Analytical characterization of the battery-threshold structure

Characterize analytically the threshold-based structure of the AoA-optimal transmission policy with respect to the receiver battery level in the energy-harvesting actuation system.

Background

The paper proves that the AoA-optimal policy has a threshold structure with respect to the age of actuation for each fixed battery level. The authors also investigate whether an analogous threshold structure holds as the battery level varies while the age of actuation is fixed.

The available monotonicity results do not establish the battery-level threshold property. Nevertheless, numerical results consistently exhibit such a structure, leaving its analytical proof or characterization unresolved for the finite-battery energy-harvesting actuation model. The conclusion reiterates this as an open question.

References

Similarly, we investigate a threshold-based structure of the optimal policy w.r.t. the battery level ($B$), in Appendix \ref{App:threshold_B}. But the monotonicity results do not establish a threshold-based structure along the battery level dimension. That is, given two states $s_{1}\equiv s_{(A,B_{1})}$ and $s_{2}\equiv s_{(A,B_{2})}$, the threshold structure \begin{equation} \label{structure_B} \forall A: \ d_{2}{*}=\begin{cases} 1 & (d_{1}{*}=1) \land (B_{1} \leq B_{2}),\ 0 & (d_{1}{*}=0) \land (B_{1} \geq B_{2}), \end{cases} \end{equation} cannot be established. However, such a structure is consistently observed in numerical results. Thus, it remains an analytical question for the present model.

— When Fresh Information Is Not Enough: Age-of-Actuation-Optimal Scheduling for Energy-Harvesting Actuators  (2610.22304 - Nikkhah et al., 15 Oct 2026) in Remark 2 (labeled \ref{theo:threshold_B}), Section 3, “Structural Property of the Results”