Integral representation for bracket-generating multi-flows

Abstract

If $f_1,f_2$ are smooth vector fields on an open subset of an Euclidean space and $[f_1,f_2]$ is their Lie bracket, the asymptotic formula $${\Psi }_{[f_1,f_2]}(t_1,t_2)(x)-x=t_1{t}_2[f_1,f_2](x)+o(t_1{t}_2),$$ where we have set ${\Psi }_{[f_1,f_2]}(t_1,t_2)(x):=\exp (-t_2{f}_2)\circ \exp (-t_1{f}_1)\circ \exp (t_2{f}_2)\circ \exp (t_1{f}_1)(x)$, is valid for all $t_1,t_2$ small enough. In fact, the integral, exact formula \begin{equation}\label{abstractform} Ψ_{[f_1,f_2]}(t_1,t_2)(x) - x = \int_0^{t_1}\int_0^{t_2}[f_1,f_2]^{(s_2,s_1)} (Ψ(t_1,s_2)(x))ds_1\,ds_2 , \end{equation} where $[f_1,f_2{]}^{(s_2,s_1)}(y):=D(\exp (s_1{f}_1)\circ \exp (s_2{f}_2){)}^{-1}\cdot [f_1,f_2](\exp (s_1{f}_1)\circ \exp (s_2{f}_2)(y)),$ with $y=\Psi (t_1,s_2)(x)$ has also been proven. Of course the integral formula can be regarded as an improvement of the asymptotic formula. In this paper we show that an integral representation holds true for any iterated bracket made from elements of a family of vector fields $f_1,\dots ,f_k$. In perspective, these integral representations might lie at the basis for extensions of asymptotic formulas involving nonsmooth vector fields.

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• Nonsmooth Hormander vector fields and their control balls
• Almost exponential maps and integrability results for a class of horizontally regular vector fields
• A Frobenius-type theorem for singular Lipschitz distributions
• Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields
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Cited by in corpus (2)

• Lyapunov-like functions involving Lie brackets
• HJ inequalities involving Lie brackets and feedback stabilizability with cost regulation