Bilinear control of age--space structured populations

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 $\ell_{\bar y,\bar u}(p)(t)\chi(a,x)$; 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.