AI 中文总结
研究伯格斯方程在特定情况下的局部可控性，通过确定控制分布条件及与迭代李括号的关系，利用分部积分等方法证明，构造出在各负整数阶产生障碍的光滑空间分布。
AI 中文摘要
我们考虑在一种线性化系统不可控的情况下，通过固定空间分布作用的标量控制的伯格斯方程。我们确定了控制分布上导致小时间局部零可控性的二次障碍的自然条件，由任意整数阶控制的负索伯列夫范数量化。我们表明这些条件与迭代李括号有关。证明依赖于二次展开中的分部积分和非线性余项估计。最后，我们构造了在每个负整数阶都产生障碍的光滑空间分布。
英文摘要
We consider the Burgers equation with a scalar control acting through a fixed spatial profile, in a regime where the linearized system is not controllable. We identify natural conditions on the control profile leading to quadratic obstructions to small-time local null-controllability, quantified by negative Sobolev norms of the control of arbitrary integer order. We show that these conditions are related to iterated Lie brackets. The proof relies on repeated integrations by parts in the quadratic expansion combined with nonlinear remainder estimates. Finally, we construct smooth spatial profiles that produce obstructions at every negative integer order.
CommentsCompanion paper to arXiv:2606.28106
