Parsimonious feedback controls for implementing diffeomorphisms

Develop time-varying feedback control laws for driftless control-affine systems on a smooth compact manifold M with bracket-generating vector fields that implement a given diffeomorphism ψ ∈ Diff₀(M) while substantially reducing the number of switchings between time-invariant flows compared to constructions based on the Agrachev–Caponigro representation and smooth fragmentation, thereby lowering implementation complexity.

Background

The paper studies controllability of the Liouville equation via the implementation of diffeomorphisms using controlled dynamical systems. For nonlinear systems on smooth manifolds, the authors review and quantify a structural result by Agrachev and Caponigro showing that diffeomorphisms near the identity can be realized by switching among finitely many time-invariant flows, yielding time-varying feedback controls that are piecewise constant in time.

They prove a quantitative lower bound on the number of switchings required by this construction, noting that the bound is not optimal and that actual implementation complexity may be much higher due to reliance on Sussmann’s orbit theorem, smooth fragmentation, and covering arguments. In the conclusion, they explicitly identify the need for more efficient control designs—controls that implement diffeomorphisms with fewer switchings—as an open problem.