The asymptotic stability of a control system

Question

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).

Answer 1

Here's a relevant reference: JP Gauthier, B. Jakubczyk, V. Zakalyukin, Motion planning and fastly oscillating controls, SIAM Journ. On Control and Opt, Vol. 48 (5), pp. 3433-3448, 2010.

Answer 2

Victoria Grushkovskaya and Alexander Zuyev sent a paper to arXiv in July 2019 solving a related problem. This paper has been accepted for publication in the Proceedings of the Joint 8th IFAC Symposium on Mechatronic Systems and 11th IFAC Symposium on Nonlinear Control Systems (MECHATRONICS & NOLCOS 2019): https://arxiv.org/abs/1907.05694

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.