标量输入系统的一些四次控制结果
Some quartic control results for scalar-input systems
AI总结：
针对标量输入系统的小时间局部可控性，研究四次项的作用，提出新可控性充分条件、确定四次阻碍类，解答Kawski1987年关于构造区分四次括号的Hall基的开放问题。
AI中文摘要：
我们研究四次项在标量输入系统的小时间局部可控性中的作用。首先，我们证明了一个新的可控性充分条件，该条件比先前结果同时利用了更多有益的四次李括号。其次，我们确定了一类对可控性的四次阻碍，这类阻碍依赖于第二类坐标为控制的正定泛函的李括号。第三，我们表明这些结果的互补性可视为对Kawski在1987年提出的关于构造能以某种方式区分有益与有害四次括号的Hall基的开放问题的解答。我们给出了例子和注记，阐明这些概念的一些复杂性。
英文摘要：
We investigate the role of quartic terms in the small-time local controllability of scalar-input systems. First, we prove a new sufficient condition for controllability, which exploits simultaneously more good quartic Lie brackets than previous results. Second, we identify a family of quartic obstructions to controllability, relying on Lie brackets whose coordinates of the second kind are positive-definite functionals of the control. Third, we show that the complementarity of these results can be seen as an answer to Kawski's 1987 open problem concerning the construction of a Hall basis which somehow separates good and bad quartic brackets. We give examples and remarks illustrating some of the intricacies of these notions.
