Calculus of Variations and Geometric Measure Theory

S. Kryzhevich - E. Stepanov

Constructive controllability for incompressible vector fields

created by stepanov on 01 Dec 2021
modified on 03 Dec 2023


Published Paper

Inserted: 1 dec 2021
Last Updated: 3 dec 2023

Journal: Tchemisova, T.V., Torres, D.F.M., Plakhov, A.Y. (eds) Dynamic Control and Optimization. DCO 2021. Springer Proceedings in Mathematics & Statistics
Volume: 407
Pages: 3-18
Year: 2021

ArXiv: 2203.14120 PDF


We give a constructive proof of a global controllability result for an autonomous system of ODEs guided by bounded locally Lipschitz and divergence free (i.e.\ incompressible) vector field, when the phase space is the whole Euclidean space and the vector field satisfies so-called vanishing mean drift condition. For the case when the ODE is defined over some smooth compact connected Riemannian manifold, we significantly strengthen the assertion of the known controllability theorem in absence of nonholonomic constraints by proving that one can find a control steering the state vector from one given point to another by using the observations of only the state vector, i.e., in other words, by changing slightly the vector field, and such a change can be made small not only in uniform, but also in Lipschitz (i.e. $C^1$) topology.

Keywords: global controllability, Poisson stable points, Poincar\'{e} recurrentce theorem