Topological obstructions in the control of partial differential equations

由偏微分方程控制的控制系统具有重要的理论意义和重要的工业应用价值。这样的系统在实际应用中比比皆是，从船闸和航道中航行的水流控制到工业过程中的热流控制。由偏微分方程控制的系统的控制性质与控制偏微分方程本身的性质密切相关，这与有限维的情况形成了鲜明的对比，在有限维的情况下，唯一相关的分界线是(时不变的)线性系统和非线性系统。与系统控制有关的两个主要问题是可控性和稳定性；而对于有限维系统，这些问题已经得到了很好的解决和理解，而对于由非线性偏微分方程控制的系统，情况远非如此。特别地，虽然已知通过正则反馈律使非线性有限维系统的局部渐近镇定存在拓扑障碍，但对于非线性偏微分方程的反馈镇定尚无这种障碍。类似地，虽然有限维系统的能控性已经被很好地理解，但到目前为止，对非线性偏微分方程能控性的拓扑障碍只针对非常有限的非线性偏微分方程类。目前的提议旨在弥合有限维控制系统和无限维控制系统之间的差距。具体地说，它有两个目标：主要目标是利用适当规则的静态反馈律来发现由非线性偏微分方程控制的系统镇定的障碍。即使在这种情况下，由于问题的各向同性群，预计障碍也是拓扑性质的。这项建议的次要目标是发现阻碍非线性偏微分方程可控性的因素，而不是文献中所考虑的那类。这一建议中考虑的问题对于加深我们对偏微分方程组的理解具有基本的理论意义；由于它们在现实世界中的许多应用，它们也具有基本的实际意义。

Control systems governed by partial differential equations are of key theoretical interest as well as of key importance for industrial applications . Such systems abound in real-world applications, from the control of water flow for navigation in locks and channels to the control of heat flow in industrial processes. The control properties of systems governed by partial differential equations is intimately tied to the nature of the governing partial differential equations themselves, and this stands in stark contrast to the finite-dimensional case, where the only relevant line of divide is between (time-invariant) linear and nonlinear systems. Two major problems related to the control of systems are controllability and stabilization; Whereas these are quite well settled and understood for finite-dimensional systems, this is far from being the case for systems governed by nonlinear partial differential equations. In particular, whereas it is now known that there are topological obstructions to local asymptotic stabilization of nonlinear finite dimensional systems via regular feedback laws, no such obstruction is yet known for feedback stabilization of nonlinear partial differential equations. Similarly, whereas the controllability properties of finite-dimensional systems are very well understood, topological obstructions to controllability of nonlinear partial differential equations have been uncovered so far only for very restricted classes of nonlinear partial differential equations. The current proposal is aiming at bridging this gap between finite-dimensional and infinite-dimensional control systems. In particular, it has two goals: The primary goal is to uncover obstructions to stabilization of a system governed by a nonlinear partial differential equation using a suitably regular static state feedback law. Even in this setting, the obstructions are expected to be of a topological nature owing to the isotropy group of the problem. The secondary goal of this proposal is to uncover obstructions to the controllability of nonlinear partial differential equations beyond the classes that have been considered so far in the literature. The problems considered in this proposal are of fundamental theoretical importance in furthering our understanding of systems governed by partial differential equations; they are of fundamental practical significance as well, owing to their numerous real-world applications.