Bilinear control of age--space structured populations
Abstract
We study constrained bilinear optimal control for nonlocal age--space structured population equations with renewal boundary conditions and endogenous surveillance feedback. The control acts as a coefficient in a mixed transport--diffusion equation, while a scalar observable generated by the state enters both the interior dynamics and the renewal law. This produces a nonlinear closed-loop control-to-state map and a feedback-dependent adjoint system. Using a characteristic mild formulation rather than a standard Lions--Magenes argument, we establish closed-loop well-posedness and Fréchet differentiability. We then derive the reduced and feedback-corrected adjoint equations. The feedback derivative is identified as a low-rank perturbation ℓy̅,u̅(p)(t)χ(a,x); in the Volterra-kernel regime, the associated transfer operator is quasinilpotent, yielding an explicit resolvent representation of the adjoint. Finally, we prove first-order optimality conditions and decompose the switching function into reduced and feedback-induced components.
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BibTeX
@misc{yu2026bilinear,
title = {{Bilinear control of age--space structured populations}},
author = {Jiguang Yu and Louis Shuo Wang and Ye Liang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.01347},
eprint = {2607.01347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.01347},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bilinear control of age--space structured populations
AU - Jiguang Yu
AU - Louis Shuo Wang
AU - Ye Liang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2607.01347
UR - https://arxiv.org/abs/2607.01347
AB - We study constrained bilinear optimal control for nonlocal age--space structured population equations with renewal boundary conditions and endogenous surveillance feedback. The control acts as a coefficient in a mixed transport--diffusion equation, while a scalar observable generated by the state enters both the interior dynamics and the renewal law. This produces a nonlinear closed-loop control-to-state map and a feedback-dependent adjoint system. Using a characteristic mild formulation rather than a standard Lions--Magenes argument, we establish closed-loop well-posedness and Fréchet differentiability. We then derive the reduced and feedback-corrected adjoint equations. The feedback derivative is identified as a low-rank perturbation ℓ_y̅,u̅(p)(t)χ(a,x); in the Volterra-kernel regime, the associated transfer operator is quasinilpotent, yielding an explicit resolvent representation of the adjoint. Finally, we prove first-order optimality conditions and decompose the switching function into reduced and feedback-induced components.
ER -