{"ID":5675934,"CreatedAt":"2026-07-03T01:40:09.565152011Z","UpdatedAt":"2026-07-04T17:54:59.62573241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.01347","arxiv_id":"2607.01347","title":"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 Frechet differentiability. We then derive the reduced and feedback-corrected adjoint equations. The feedback derivative is identified as a low-rank perturbation $\\ell_{\\bar y,\\bar u}(p)(t)χ(a,x)$; 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.","short_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 interi...","url_abs":"https://arxiv.org/abs/2607.01347","url_pdf":"https://arxiv.org/pdf/2607.01347v1","authors":"[\"Jiguang Yu\",\"Louis Shuo Wang\"]","published":"2026-07-01T18:05:22Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[\"Diffusion Model\"]","has_code":false}