AI 中文总结
针对带局域内部控制的一维粘性伯格斯方程二次最优跟踪问题，本文建立了其全局指数 turnpike 性质，这是该方程的首批相关结果，证明结合了局部指数 turnpike 与抛物耗散论证。
AI 中文摘要
我们针对具有局域内部控制的一维粘性伯格斯方程所描述的二次最优跟踪问题，建立了全局指数 turnpike 性质。对于每个初始数据，当周期跟踪目标足够小时，有限时间域最优解会趋近于唯一的最优周期状态；零目标情形在原点处产生全局稳态 turnpike，且对初始数据无小性假设。据我们所知，这是粘性伯格斯方程的首批全局指数 turnpike 结果。证明结合了通过严格凸性和周期 Riccati 理论得到的局部指数 turnpike，以及提供与时间域无关的吸收时间的抛物耗散论证。
英文摘要
We establish global exponential turnpike properties for quadratic optimal tracking problems governed by the one-dimensional viscous Burgers equation with localized internal control. For every initial datum, finite-horizon optimal solutions approach the unique optimal periodic regime when the periodic tracking target is sufficiently small; the zero-target case yields a global steady turnpike at the origin, with no smallness assumption on the initial datum. To our knowledge, these are the first global exponential turnpike results for the viscous Burgers equation. The proof combines a local exponential turnpike, obtained through strict convexity and periodic Riccati theory, with a parabolic dissipation argument that provides an absorbing time independent of the horizon.
