科研之友
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摘要
If the vector fields f(1), f(2) are locally Lipschitz, the classical Lie bracket [f(1), f(2)] is defined only almost everywhere. However, it has been shown that, by means of a set- valued Lie bracket [f(1), f(2)] set (which is defined everywhere), one can generalize classical results like the Commutativity theorem and Frobenius' theorem, as well as a Chow-Rashevski's theorem involving Lie brackets of degree 2 (we call 'degree' the number of vector fields contained in a formal bracket). As it might be expected, these results are consequences of the validity of an asymptotic formula similar to the one holding true in the regular case. Aiming to more advanced applications-say, a general Chow-Rashevski's theorem or higher order conditions for optimal controls-we address here the problem of defining, for any m > 2 and any formal bracket B of degree m, a Lie bracket B(f(1),..., f(m)) corresponding to vector fields (f(1),..., f(m)) lacking classical regularity requirements. A major complication consists in finding the right extension of the degree 2 bracket, namely a notion of bracket which admits an asymptotic formula. In fact, it is known that a mere iteration of the construction performed for the case m = 2 is not compatible with the validity of an asymptotic formula. We overcome this difficulty by introducing a set-valued bracket x -> B-set(f(1),..., f(m))(x), defined at each point x as the convex hull of the set of limits along suitable d-tuples of sequences of points converging to x. The number d depends only on the formal bracket B and is here called the diff-degree of B. It counts the maximal order of differentiations involved in B(g(1),..., g(m)) (for any smooth m-tuple of vector fields (g(1),..., g(m))). The main result of the paper is an asymptotic formula valid for the bracket Bset(f(1),..., f(m))(x).
- 出版日期2017-12
