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arXiv · 2607.01347

Bilinear control of age--space structured populations

Jiguang Yu·Louis Shuo Wang·Ye Liang
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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 Fr\'echet differentiability. We then derive the reduced and feedback-corrected adjoint equations. The feedback derivative is identified as a low-rank perturbation \(\ell_{\bar y,\bar u}(p)(t)\chi(a,x)\); 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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Jiguang Yu, Louis Shuo Wang, Ye Liang. 2026-07-01. Bilinear control of age--space structured populations. https://arxiv.org/abs/2607.01347

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