发表机构
University of California, San Diego（加州大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对一维标量双曲PDE与二维快速输运-扩散PDE的耦合对，提出基于时间尺度分离的反步镇定方法，证明低于显式阈值时指数稳定。
AI 中文摘要
一个标量双曲型偏微分方程（PDE），在一个边界上受驱动，与一个二维空间中的快速 PDE 耦合：在轴向坐标和内部坐标中具有输运，并且可能在内部坐标中具有扩散。对于耦合双曲系统、其系综及其连续体，反步（backstepping）设计已经存在，但在所有这些设计中，快速子系统的第二个变量不携带输运或扩散；当它携带时，没有可用的设计，并且从标量输入无法期望对二维状态的精确控制。这里通过时间尺度分离来镇定该 PDE 对：在准稳态极限下，二维子系统坍缩为一维简化被控对象内的空间 Volterra 算子，反步法适用于该对象，而二维子系统则留作稳定的边界层。一个定理确立了：对于低于显式阈值的每个时间尺度比率，该 PDE 对都是指数稳定的，并且一致地适用于内部扩散系数直至纯输运情形。
英文摘要
A scalar hyperbolic PDE, actuated at one boundary, is coupled with a fast PDE in two spatial dimensions: transport in the axial coordinate and in an internal coordinate, and possibly diffusion in the internal coordinate. Backstepping designs exist for coupled hyperbolic systems, for their ensembles, and for their continua, but in all of these the second variable of the fast subsystem carries no transport or diffusion; when it does, no design is available, and from a scalar input no exact control of the two-dimensional state is to be expected. The pair is stabilized here by time-scale separation: in the quasi-steady limit the two-dimensional subsystem collapses into a spatial Volterra operator inside a one-dimensional reduced plant, backstepping applies there, and the two-dimensional subsystem is left to be a stable boundary layer. One theorem establishes exponential stability of the pair for every time-scale ratio below an explicit threshold, uniformly in the internal diffusion coefficient down to the pure-transport case.
