Geometric, topological, and stochastic approaches in nonlinear control theory

非线性控制理论是控制理论的一个重要分支，广泛应用于电网、航空航天系统、生物医学系统、纳米工程、过程控制、化学工程等领域和行业。它以其问题的丰富性和它所连接的数学领域的多样性而脱颖而出。后者中值得注意的是亚黎曼几何，它在我们对非线性控制系统的理解中继续发挥着关键作用。线性控制理论将自己限制在一个狭窄的、表现良好的控制系统类别中，因此它拥有丰富而完善的理论工具和技术体系，主要来自线性泛函分析。相比之下，非线性控制理论远未达到这种成熟程度，并且已经在设计新的工具和方法方面投入了大量的努力。

Nonlinear control theory is an important branch of control theory, with applications in areas and industries as diverse as power grids, aerospace systems, biomedical systems, nano-engineering, process control, chemical engineering, and many others. It stands out by the richness of its problems and the diversity of mathematical areas that it connects to. Worthy of note among the latter is sub-Riemannian geometry, which continues to play a key role in our understanding of nonlinear control systems. By virtue of restricting itself to a narrow and well-behaved class of control systems, linear control theory enjoys a rich and well-established body of theoretical tools and techniques, drawn principally from linear functional analysis. By contrast, nonlinear control theory is far from enjoying this level of maturity, and intensive efforts have been deployed in devising new tools and approaches.