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，从核磁共振医学成像系统到飞机制导和导航系统都有广泛的应用。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通过揭示拓扑在稳定中的作用，局部几何不变量与小时局部可控性之间的可能关系，亚黎曼几何的热核及其渐近性，以及通过能量整形在机械系统稳定中出现的偏微分方程的性质来进一步了解非线性控制系统。

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.