可控性的二次障碍：从常微分方程到偏微分方程
Quadratic obstructions to controllability: from ODEs to PDEs
摘要：
研究了系统在平衡点附近的小时间局部能控性。给定一个小的时间，一个初始数据和一个接近平衡的最终数据，是否有可能找到一个控制（源项），在给定的时间将解从初态引导到所希望的最终状态？自然的方法是从研究线性化系统在平衡点附近的可控性开始。当这个系统是不可控的，它是必要的，继续扩展与二次order.In本说明中，我们突出了不同的最近的结果之间的联系围绕这个主题，在特定的情况下，控制是一个单一的标量输入。这些结果倾向于证明，在这种特殊情况下，二次阶只能对可控性产生障碍。特别评述了与Karine Beauchard合作在有限维中得到的一个详尽结果和作者论文中的一个结果，其中涉及到无限维中的一种新现象.
英文摘要：
We investigate the small-time local controllability of systems in the vicinity of an equilibrium. Given a small time, an initial data and a final data close from the equilibrium, is it possible to find a control (a source term) that guides the solution from the initial state to the wished final state at the given time? The natural method is to start by studying the controllability of the linearized system near the equilibrium. When this system is not controllable, it is necessary to continue the expansion with the quadratic order.In this note, we highlight the links between different recent results around this topic, in the particular case where the control is a single scalar input. These results tend to prove that, in this particular case, the quadratic order can only yield obstructions to controllability. We especially comment an exhaustive result obtained in collaboration with Karine Beauchard in finite dimension and a result contained in the thesis of the author, which involves a new kind of phenomenon in infinite dimension.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
