AI 中文总结
该研究针对线性块演化系统提出算子理论框架，引入有限时间轨迹稳定约化，推导扰动估计，通过示例区分正型与强制性，将未解决影响的受控近似作为核心约化问题。
AI 中文摘要
降维会移除变量，但可靠的降维不应抹去这些变量的动力学影响。我们针对已解决和未解决希尔伯特空间上的线性块演化系统研究这一原理。精确消除未解决分量会得到一个带有记忆核的沃尔泰拉方程，以及携带隐藏初始数据和力项的有效强迫项；在拉普拉斯域中，同一操作是动态舒尔补。我们引入有限时间轨迹稳定约化，并推导算子值核的扰动估计，再将其推广到隐藏传播子、耦合项、隐藏初始状态和隐藏强迫的扰动。对于被动斜自伴耦合，约化记忆算子具有正型和精确存储恒等式，且范数收敛的被动实现同时保留轨迹和存储-耗散泛函。兼容的有限谱截断提供了保结构内部变量约化。两个基础示例将正型与瞬时L²-强制性区分开，表明小的隐藏振幅不一定意味着小的长期影响。该框架将未解决影响的受控近似（而非单纯消除）确定为核心约化问题。
英文摘要
Dimension reduction removes variables, but a reliable reduction should not erase their dynamical influence. We study this principle for linear block evolution systems on resolved and unresolved Hilbert spaces. Exact elimination of the unresolved component yields a Volterra equation with a memory kernel and an effective forcing term carrying hidden initial data and forcing; in the Laplace domain the same operation is a dynamic Schur complement. We introduce finite-horizon trajectory-stable reduction and derive perturbation estimates for operator-valued kernels, then lift them to perturbations of the hidden propagator, couplings, hidden initial state, and hidden forcing. For passive skew-adjoint couplings, the reduced memory operator has positive type and an exact storage identity, and norm-convergent passive realisations preserve both trajectories and the storage--dissipation functional. Compatible finite spectral truncations provide structure-preserving internal-variable reductions. Two elementary examples separate positive type from instantaneous $L^2$-coercivity and show that small hidden amplitude need not imply small long-time influence. The framework isolates the controlled approximation of unresolved influence, rather than elimination alone, as the central reduction problem.
