论文标题
标量输入系统的小型局部可控性障碍的统一方法
A unified approach of obstructions to small-time local controllability for scalar-input systems
论文作者
论文摘要
我们提出了一种统一的方法，用于确定和证明标量输入控制系统的小型局部可控性的障碍。我们的方法认为障碍物是由于与系统的正式谎言括号相关的功能之间的插值不平等所致。 使用这种方法，我们给出了所有已知必要条件的紧凑型统一证据，我们证明了1986年由于kawski的猜想，我们得出了全新的障碍。我们的工作使以前所知的必要条件数量增加了一倍，这些条件均在1980年代建立。特别是，对于第三个二次支架，我们得出了一种新的必要条件，该条件与Agrachev-Gamkrelidze足够的互补。 我们依靠最近的Magnus型代表公式，这是两个发电机上的自由谎言代数的新大厅，适当使用Sussmann的无限产品来计算Magnus的扩张以及Gagliardo-Nirenberg插值不平等。
We present a unified approach for determining and proving obstructions to small-time local controllability of scalar-input control systems. Our approach views obstructions to controllability as resulting from interpolation inequalities between the functionals associated with the formal Lie brackets of the system. Using this approach, we give compact unified proofs of all known necessary conditions, we prove a conjecture of 1986 due to Kawski, and we derive entirely new obstructions. Our work doubles the number of previously-known necessary conditions, all established in the 1980s. In particular, for the third quadratic bracket, we derive a new necessary condition which is complementary to the Agrachev-Gamkrelidze sufficient ones. We rely on a recent Magnus-type representation formula for the state, a new Hall basis of the free Lie algebra over two generators, an appropriate use of Sussmann's infinite product to compute the Magnus expansion, and Gagliardo-Nirenberg interpolation inequalities.
