年龄-空间结构化群体的双线性控制
Bilinear control of age--space structured populations
- College of Engineering, Boston University（波士顿大学工程学院）
- Department of Mathematics, Northeastern University（东北大学数学系）
- College of Engineering, The University of Iowa（爱荷华大学工程学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究具有更新边界条件和内源性监测反馈的非局部年龄-空间结构化种群方程的双线性最优控制，通过特征温和公式建立闭环适定性和弗雷歇可微性，推导降阶和反馈校正伴随方程，并证明一阶最优性条件。
AI中文摘要：
我们研究了具有更新边界条件和内源性监测反馈的非局部年龄-空间结构化种群方程的约束双线性最优控制。控制作为混合输运-扩散方程中的系数，而由状态生成的标量可观测量同时进入内部动力学和更新律。这产生了一个非线性闭环控制到状态映射和一个依赖于反馈的伴随系统。使用特征温和公式而非标准的Lions-Magenes论证，我们建立了闭环适定性和弗雷歇可微性。然后推导了降阶和反馈校正的伴随方程。反馈导数被识别为低秩扰动$\ell_{\bar y,\bar u}(p)(t)\chi(a,x)$；在Volterra核机制下，关联的传递算子是拟幂零的，从而得到伴随的显式预解表示。最后，我们证明了一阶最优性条件，并将切换函数分解为降阶和反馈诱导分量。
英文摘要：
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 \(\ell_{\bar y,\bar u}(p)(t)χ(a,x)\); 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.
