The asymptotic stability of a control system

Let a control system be described by the following nonlinear ordinary differential equations:

$$(1)\quad \frac{dx(t)}{dt}=f_0(x(t))+\sum_{j=1}^m f_j(x(t))u_j,\;\;x(t)\in D\subset\mathbb R^n,\;\;u\in\mathbb R^m,$$

where $$x$$ is the stae and $$u=(u_1,u_2,\dots,u_m)$$ is the control, $$f_0(0)=0$$, $$0\in int D$$, and the vector fields $$f_j:D\to\mathbb R^n$$ are smooth.

Assume that the system (1) is small-time locally controllable at $$x=0$$ (so that the Lie algebra rank condition holds at $$x=0$$).

The problem is to find an $$N\ge 0$$ and functions $$a_{jk}(x)$$ such that $$a_{jk}(0)=0$$ and the solution

$$(2)\quad u_j=\sum_{k=-N}^Na_{jk}(x)\exp\big(\tfrac{2\pi ki}\varepsilon t\big)$$

is asymptotically stable in the sense of Lyapunov, provided that $$\varepsilon>0$$ is small enough.

(The problem was asked on 13.08.2017 by Prof. Alexander Zuyev, see pages 57 and 58 in Lviv Scottish Book).

• Prof. Alexander Zuyev sent a paper to arXiv in July 2019 which may close this question (or at least, it seems to be related to): arxiv.org/abs/1907.05694 Jul 16, 2019 at 15:43
• @Shamisen: you can post this (a bit expanded...) as an answer. Aug 23, 2019 at 16:32

• Alexander Zuyev, the author of the problem, wrote me that he knows this paper (and also the authors of this paper), but the cited work neither deals with the stabilization problem by a feedback control, nor with the class of control-affine systems with drift (for which $f_0(x)$ is not identically zero). Dec 4, 2018 at 14:03
It can be seen in Theorem 1 of this paper that if (1) satisfy the hypothesis of bounded time-varying drift and that the control vector fields of (1) together with their iterated Lie brackets satisfy Hormander's condition in a neighborhood of the origin, then there is a control of the form (2) ensuring the practical exponential stability of the closed-loop system, provided that the period $$\varepsilon$$ is small enough.