Geometric and topological approaches in nonlinear control theory

非线性控制理论是控制理论的一个重要分支，其应用范围从核磁共振医学成像系统到飞机制导和导航系统。非线性控制系统的复杂性和多样性给控制理论家带来了巨大的挑战，甚至围绕一般非线性控制系统的可控性和稳定性的基本问题仍然没有完全理解。我们提出的研究方向是为了进一步了解非线性控制系统，揭示：拓扑学在镇定中的作用，局部几何不变量与小时间局部能控性之间的可能关系，亚黎曼几何的热核及其渐近性，以及在通过能量成形的机械系统的稳定中发生的偏微分方程的性质。 我们提出的研究的重要性源于这样一个事实，即它涉及非线性控制理论的基本问题。我们相信，我们希望通过我们的研究实现的对非线性控制系统的更好理解，也将通过指导改进控制器和稳定控制器的设计，在所有使用非线性控制系统的众多行业中产生实际利益。 我们预计许多HQP将受益于我们拟议的研究计划提供的培训;特别是，我们预计至少有4个硕士，2个博士和2个PDF HQP的直接参与，以及每年至少有2个夏季研究本科HQP的参与。这些HQP将在他们的纯数学和应用数学领域获得丰富的经验，并将能够随后为该领域做出贡献。控制系统是许多高技术产业的重要组成部分，像加拿大这样的工业化国家只能从这种培训中受益。

Nonlinear control theory is an important branch of control theory, with numerous applications ranging from Nuclear Magnetic Resonance medical imaging systems to airplane guidance and navigation systems. The complexity and the diversity of nonlinear control systems yield substantial challenges for the control theorist, and even basic issues surrounding controllability and stabilizability of general nonlinear control systems are still not completely understood. The research directions we are proposing to investigate are meant to further our understanding of nonlinear control systems by shedding light on: The role of topology in stabilization, the possible relations between local geometric invariants and Small-Time Local Controllability, heat kernels of sub-Riemannian geometries and their asymptotics, as well as properties of the partial differential equations that occur in the stabilization of mechanical systems via energy shaping. The importance of our proposed research stems from the fact that it deals with fundamental issues in nonlinear control theory. We believe that the better understanding of nonlinear control systems which we hope to achieve through our research will also yield practical benefits in all the numerous industries that make use of nonlinear control systems, by guiding the design of improved controllers and stabilizing controllers. We anticipate numerous HQPs to benefit from the training afforded by our proposed research program; in particular, we anticipate the direct involvement of at least 4 MSc, 2 PhD, and 2 PDF HQPs, as well as the involvement of at least 2 summer research undergraduate HQPs every year. These HQPs will gain deep experience in their area of pure and applied mathematics, and will be able to subsequently contribute to the field. Control systems form an essential part of numerous high-technology industries, and an industrialized country such as Canada can only benefit from such training.