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Stabilization of Nonholonomic Systems 195

n
3
m
where s d evolves in the finite set {1, 2}, k : R ×{1, 2}→ R is
n
continuous in x for each fixed s d ; k d : R ×{1, 2}→{1, 2}, and ¯s d
s
is defined as ¯ d (t) = lim s<t s d (s);
• Time varying, state feedback, sampled-data, control laws described
by equations of the form
u = u T (x(kT), kT) (5.4)
n
m
where T > 0 is the sampling time, and u T : R × R → R is a
continuous function of its arguments.
Remark 5.1 Whenever we deal with discontinuous control laws, functions
which are not defined at some points, for example, are unbounded at x = 0,
are allowed. In particular the term discontinuous will be used throughout this
chapter to denote functions which are unbounded, hence undefined, in a certain
set; for example, the function 1 is discontinuous at x = 0.
x
The purpose of the control law is to guarantee that each initial state in a given
set converges asymptotically to the origin. However, as we use different control
laws, we will need different definitions of stability.

Definition 5.1 [20] A control law described by equations of the form (5.2)
4
5
almost stabilizes the system (5.1) in the region 0 if the following holds:
(i) For all initial states x 0 ∈ 0 the closed loop system admits a unique
(forward) solution
(ii) For all initial states x 0 ∈ 0 one has, along the trajectories of the
closed loop system, lim t→∞ x(t) = 0
Moreover, the control law almost exponentially stabilizes the system (5.1)
in the region 0 if in addition
(iii) There exist positive constant c 0 and λ 0 such that for all initial states
x 0 ∈ 0 and for all t ≥ 0 one has, along the trajectories of the
closed loop system, x(t) ≤ c 0 exp −λ 0 t

Hybrid and sampled-data control laws are discussed in relation with robust
stabilization problems. To discuss the properties of hybrid control laws we need
to introduce a notion of robustness with respect to small noise. To this end,

3 For this controller to make sense we equip {1, 2} with the discrete topology, that is, every set is
an open set.
4 This terminology differs from that introduced in Reference 37. Note also that stability has to be
understood as Lagrange stability.
5 The set 0 does not need to be a neighborhood of the origin, but may be an open and dense set
with the origin at its boundary.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 195 — #9