September  2015, 35(9): 4345-4366. doi: 10.3934/dcds.2015.35.4345

Integral representations for bracket-generating multi-flows

1. 

Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63 - 35121 - Padova (PD), Italy, Italy

Received  May 2014 Revised  September 2014 Published  April 2015

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 \begin{equation}\label{abstract:EQ} \Psi_{[f_1,f_2]}(t_1,t_2)(x) - x =t_1t_2 [f_1,f_2](x) +o(t_1t_2), \,                                         (1) \end{equation} where we have set $\Psi_{[f_1,f_2]}(t_1,t_2)(x) \overset{\underset{\mathrm{def}}{}}{=} \exp(-t_2 f_2)\circ \exp(-t_1f_1) \circ \exp(t_2f_2) \circ \exp(t_1f_1)(x)$, is valid for all $t_1,t_2$ small enough. In fact, the integral, exact formula \begin{equation}\label{abstract:EQ} \Psi_{[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)} (\Psi(t_1,s_2)(x))ds_1\,ds_2 ,                                  (2) \end{equation} where $[f_1,f_2]^{(s_2,s_1)}(y) \overset{\underset{\mathrm{def}}{}}{=} D (\exp(s_1f_1) \circ \exp(s_2f_2)))^{-1}(y) \cdot [f_1,f_2](\exp (s_1f_1) \circ \exp(s_2f_2)(y) ), $ has also been proven. Of course (2) can be regarded as an improvement of (1). In this paper we show that an integral representation like (2) holds true for any iterated Lie bracket made of elements of a family ${f_1,\dots,f_m}$ of vector fields. In perspective, these integral representations might lie at the basis for extensions of asymptotic formulas involving non-smooth vector fields.
Citation: Ermal Feleqi, Franco Rampazzo. Integral representations for bracket-generating multi-flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4345-4366. doi: 10.3934/dcds.2015.35.4345
References:
[1]

A. A. Agračev and R. V. Gamkrelidze, Exponential representation of flows and a chronological enumeration,, Mat. Sb. (N.S.), 107 (1978), 467.   Google Scholar

[2]

A. A. Agračev and R. V. Gamkrelidze, Chronological algebras and nonstationary vector fields,, in Problems in geometry, 11 (1980), 135.   Google Scholar

[3]

M. Bramanti, L. Brandolini and M. Pedroni, Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality,, Forum Math., 25 (2013), 703.  doi: 10.1515/form.2011.133.  Google Scholar

[4]

A. Montanari and D. Morbidelli, Nonsmooth Hörmander vector fields and their control balls,, Trans. Amer. Math. Soc., 364 (2012), 2339.  doi: 10.1090/S0002-9947-2011-05395-X.  Google Scholar

[5]

A. Montanari and D. Morbidelli, Almost exponential maps and integrability results for a class of horizontally regular vector fields,, Potential Anal., 38 (2013), 611.  doi: 10.1007/s11118-012-9289-6.  Google Scholar

[6]

A. Montanari and D. Morbidelli, Step-$s$ involutive families of vector fields, their orbits and the Poincaré inequality,, J. Math. Pures Appl. (9), 99 (2013), 375.  doi: 10.1016/j.matpur.2012.09.005.  Google Scholar

[7]

A. Montanari and D. Morbidelli, Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields,, J. Geom. Anal., 24 (2014), 687.  doi: 10.1007/s12220-012-9351-z.  Google Scholar

[8]

F. Rampazzo and H. J. Sussmann, Set-valued differentials and a nonsmooth version of Chow-Rashevski's theorem,, in Proceedings of the 40th IEEE Conference on Decision and Control, (2001), 2613.   Google Scholar

[9]

F. Rampazzo and H. J. Sussmann, Commutators of flow maps of nonsmooth vector fields,, J. Differential Equations, 232 (2007), 134.  doi: 10.1016/j.jde.2006.04.016.  Google Scholar

[10]

E. T. Sawyer and R. L. Wheeden, Hölder continuity of weak solutions to subelliptic equations with rough coefficients,, Mem. Amer. Math. Soc., 180 (2006).  doi: 10.1090/memo/0847.  Google Scholar

show all references

References:
[1]

A. A. Agračev and R. V. Gamkrelidze, Exponential representation of flows and a chronological enumeration,, Mat. Sb. (N.S.), 107 (1978), 467.   Google Scholar

[2]

A. A. Agračev and R. V. Gamkrelidze, Chronological algebras and nonstationary vector fields,, in Problems in geometry, 11 (1980), 135.   Google Scholar

[3]

M. Bramanti, L. Brandolini and M. Pedroni, Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality,, Forum Math., 25 (2013), 703.  doi: 10.1515/form.2011.133.  Google Scholar

[4]

A. Montanari and D. Morbidelli, Nonsmooth Hörmander vector fields and their control balls,, Trans. Amer. Math. Soc., 364 (2012), 2339.  doi: 10.1090/S0002-9947-2011-05395-X.  Google Scholar

[5]

A. Montanari and D. Morbidelli, Almost exponential maps and integrability results for a class of horizontally regular vector fields,, Potential Anal., 38 (2013), 611.  doi: 10.1007/s11118-012-9289-6.  Google Scholar

[6]

A. Montanari and D. Morbidelli, Step-$s$ involutive families of vector fields, their orbits and the Poincaré inequality,, J. Math. Pures Appl. (9), 99 (2013), 375.  doi: 10.1016/j.matpur.2012.09.005.  Google Scholar

[7]

A. Montanari and D. Morbidelli, Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields,, J. Geom. Anal., 24 (2014), 687.  doi: 10.1007/s12220-012-9351-z.  Google Scholar

[8]

F. Rampazzo and H. J. Sussmann, Set-valued differentials and a nonsmooth version of Chow-Rashevski's theorem,, in Proceedings of the 40th IEEE Conference on Decision and Control, (2001), 2613.   Google Scholar

[9]

F. Rampazzo and H. J. Sussmann, Commutators of flow maps of nonsmooth vector fields,, J. Differential Equations, 232 (2007), 134.  doi: 10.1016/j.jde.2006.04.016.  Google Scholar

[10]

E. T. Sawyer and R. L. Wheeden, Hölder continuity of weak solutions to subelliptic equations with rough coefficients,, Mem. Amer. Math. Soc., 180 (2006).  doi: 10.1090/memo/0847.  Google Scholar

[1]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[2]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307

[3]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[4]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

[5]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[6]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

[7]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[8]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[9]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[10]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328

[11]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

[12]

Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109

[13]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117

[14]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[15]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[16]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[17]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[18]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[19]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[20]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]