Topological obstructions in the control of partial differential equations

偏微分方程控制中的拓扑障碍

• 批准号: RGPIN-2022-03832
• 负责人: Mansouri, AbdolReza
• 金额: $ 1.75万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759055
• 关键词: Topological; obstructions; control; partial; differential

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.

由偏微分方程控制的系统是一个重要的理论问题，在工业应用中也具有重要意义。这种系统在现实世界的应用中比比皆是，从船闸和渠道中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特性与控制偏微分方程本身的性质密切相关，这与有限维情况形成鲜明对比，其中唯一相关的分界线是（时不变）线性和非线性系统之间。与系统控制相关的两个主要问题是可控性和稳定性;虽然这些问题对于有限维系统已经很好地解决和理解，但对于由非线性偏微分方程控制的系统，情况远非如此。特别是，而现在已知的是，有拓扑障碍的局部渐近稳定的非线性有限维系统通过定期反馈法律，没有这样的障碍是已知的反馈稳定的非线性偏微分方程。类似地，虽然有限维系统的可控性已经被很好地理解，但非线性偏微分方程可控性的拓扑障碍迄今为止只在非常有限的非线性偏微分方程类中被发现。目前的建议是为了弥合有限维和无限维控制系统之间的差距。特别是，它有两个目标：的主要目标是发现障碍的非线性偏微分方程使用适当的定期静态状态反馈法的系统稳定。即使在这种情况下，由于问题的各向同性群，障碍物也被认为是拓扑性质的。这个建议的第二个目标是发现障碍物的可控性的非线性偏微分方程超出了类，迄今已被认为是在文献中。 在这个建议中考虑的问题是基本的理论上的重要性，在进一步了解系统的偏微分方程，他们是基本的实际意义，以及，由于其众多的现实世界的应用。