一般多维区域上一类不稳定热方程的边界稳定性与可观测性
Boundary Stabilization and Observation of an Unstable Heat Equation in a General Multi-dimensional Domain
摘要：
本文结合有限维谱截断技术和最近发展起来的动力学补偿方法，研究了一般多维区域上一类不稳定热方程的指数镇定和可观测性.相对于不稳定的一维偏微分方程（PDE），如输运方程、波动方程和热方程，可以用著名的PDE反推方法来处理，一般多维域上不稳定偏微分方程的镇定仍然是一个具有挑战性的问题。我们分别处理稳定化和观测问题。首先提出了一种动态状态反馈控制律，使不稳定的热传导方程指数镇定，然后通过边界测量设计了一种状态观测器。证明了闭环系统的稳定性和观测器的适定性。部分理论结果得到了数值模拟的验证。
英文摘要：
In this paper, we consider the exponential stabilization and observation of an unstable heat equation in a general multi-dimensional domain by combining the finite-dimensional spectral truncation technique and the recently developed dynamics compensation approach. In contrast to the unstable one-dimensional partial differential equation (PDE), such as the transport equation, wave equation and the heat equation, that can be treated by the well-known PDE backstepping method, stabilization of unstable PDE in a general multi-dimensional domain is still a challenging problem. We treat the stabilization and observation problems separately. A dynamical state feedback law is proposed firstly to stabilize the unstable heat equation exponentially and then a state observer is designed via a boundary measurement. Both the stability of the closed-loop system and the well-posedness of the observer are proved. Some of the theoretical results are validated by the numerical simulations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
