发表机构
College of Engineering, Boston University; Northeastern University（波士顿大学工程学院; 东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究非线性尺寸结构输运方程，内生标量输出反馈到速度和死亡率，冻结反馈路径转化为非自治线性演化，通过质量平衡和范数收缩证明非负弱解唯一性等。
AI 中文摘要
我们研究一个非线性尺寸结构输运方程，其中内生标量输出\(E(t)=\int_{l_0}^{l_m}\chi(l)x(t,l)\,dl\)反馈到速度和死亡率中。这种主系数反馈排除了半线性扰动框架。冻结反馈路径产生一个非自治线性演化，将闭环问题简化为一个标量Volterra不动点\(E = \mathcal K(E)\)。质量平衡提供了一个内在反馈区间，而Bielecki范数收缩确保了唯一的非负弱解。平稳平衡点满足一个标量封闭方程\(E = \Phi(E)\)。我们证明在尖锐边界\(1 - \Phi'(E)>0\)以下的唯一性，并将\(\Phi'(E)=1\)识别为一个非退化折叠阈值。线性化产生一个具有特征方程\(\mathcal E(\lambda)=1\)的有限记忆更新方程，其根集决定了反馈谱和稳定性。最后，平稳收获伴随简化为一个秩一扰动公式。在零折扣时，我们建立了等式\(\mathcal E(0)=\Phi'(E^*)=B(0)\)，将封闭共振、谱交叉和伴随回路增益联系起来。
英文摘要
We study a nonlinear size-structured transport equation where the endogenous scalar output $E(t)=\int_{l_0}^{l_m}χ(l)x(t,l)\,dl$ feeds back into velocity and mortality. This principal-coefficient feedback precludes a semilinear perturbation framework. Freezing the feedback path yields a non-autonomous linear evolution, reducing the closed-loop problem to a scalar Volterra fixed point $E=\mathcal K(E)$. Mass balance provides an intrinsic feedback interval, while a Bielecki-norm contraction ensures unique nonnegative weak solutions. Stationary equilibria satisfy a scalar closure equation $E=Φ(E)$. We prove uniqueness below the sharp margin $1-Φ'(E)>0$ and identify $Φ'(E)=1$ as a nondegenerate fold threshold. Linearization yields a finite-memory renewal equation with characteristic equation $\mathcal E(λ)=1$, whose root set determines the feedback spectrum and stability. Finally, the stationary harvesting adjoint reduces to a rank-one perturbation formula. At zero discount, we establish the identity $\mathcal E(0)=Φ'(E^*)=B(0)$, linking closure resonance, spectral crossing, and adjoint loop gain.
