![](bg4.png)
Theorem 3. Let k ∈ N if d is odd or k ∈ N \{1} if
d is even and fix ` := b
d
2
c+ k + 1; we use the Radial
Basis Function ψ(r) = φ
`,k
(cr) with c > 0, where φ
`,k
denotes the Wendland function. Set τ = k + (d + 1)/2
and σ = dτe.
Consider the dynamical system defined by (1),
where f ∈ C
σ
(R
d
,R
d
). Let x
0
be an exponentially
stable equilibrium of (1). Let f be bounded in A(x
0
)
and denote by V ∈W
τ
2
(A(x
0
),R) the Lyapunov func-
tion satisfying V
0
(x) = −kx −x
0
k
2
2
. Let Ω ⊂ A(x
0
) be
a bounded domain with Lipschitz continuous bound-
ary.
The reconstruction v of the Lyapunov function V
with respect to the operator (6) and a set X
N
⊂ Ω \
{x
0
} satisfies
kv
0
−V
0
k
L
∞
(Ω)
≤Ch
k−
1
2
kV k
W
k+(d+1)/2
2
(Ω)
(13)
where h := sup
x∈Ω
min
x
j
∈X
N
kx −x
j
k
2
denotes the fill
distance.
Remark 3. If the collocation points are sufficiently
dense, then the right-hand of the error estimate (13)
is smaller than a given ε > 0. This implies that
v
0
(x) ≤V
0
(x) + ε ≤ −kx −x
0
k
2
+ ε < 0
for all x ∈ Ω \B
√
ε
(x
0
). Hence, v has negative or-
bital derivative in Ω apart from a small neighborhood
of x
0
. One can use the Lyapunov function of the lin-
earized system, the so-called local Lyapunov function,
to deal with this small neighborhood of x
0
, for details
see (Giesl, 2007), or a modified method, see (Giesl,
2008). In this paper, we will not deal with this local
problem in more detail, but we exclude a small neigh-
borhood E ⊃ B
√
ε
(x
0
) of x
0
in our consideration.
The error estimate in Theorem 3 establishes that
by choosing the collocation points sufficiently dense,
we will construct a Lyapunov function. However,
there remain two questions: firstly, how do we choose
the collocation points to achieve a negative orbital
derivative with as few collocation points as possible?
The error estimate only gives information about the
error to V
0
based on the fill distance. We, however,
only require v
0
to be negative, so the error could be
larger further away from the equilibrium. The advan-
tage of meshfree collocation is to be able to use scat-
tered points, so it is natural to start with a coarse grid
and introduce a refinement algorithm, see Section 4.
Secondly, the quantities on the right-hand side of
(13) are not explicitly computable, so it remains to
show, after having obtained an approximation v, that
v
0
(x) is negative for all x. This will be done in Section
5.
4 THE REFINEMENT
ALGORITHM
In this section, we will combine the construction
method with a grid refinement algorithm, aiming
for a successful construction of Lyapunov functions
with fewer collocation points and less computation
time than the original method. For more details of
the refinement algorithm see (Mohammed and Giesl,
2015).
Our proposed algorithm is iterative and uses
Voronoi diagrams. In each step, given a set of colloca-
tion points, we generate a Voronoi diagram for these
points. Then, we run a test on each Voronoi vertex
and decide whether we add it to the set of collocation
points or not. A point is added if the orbital derivative
is non-negative.
We have used Voronoi vertices as potential new
collocation points since they are equidistant to three
or more previous grid points, and thus lie “in be-
tween” the grid points. Hence, we avoid collocation
points too close to each other, which would result in a
(nearly) singular interpolation matrix A of the linear
system (8).
4.1 Voronoi Diagrams
A Voronoi diagram is a geometric structure that di-
vides a d-dimensional space into cells based on the
distance between sets of points in the space (Preparata
and Shamos, 1985). Many algorithms for computing
Voronoi diagrams have been proposed, however, we
are going to explain the structure of Voronoi diagrams
via a very simple but less efficient algorithm using
perpendicular hyperplanes.
Let S = {s
1
,s
2
,.. .,s
n
} ⊂ R
d
be a set of n arbi-
trarily distributed and distinct sites (points) in R
d
.
The perpendicular bisector algorithm works as fol-
lows: for each pair of sites in S we construct a hy-
perplane perpendicular to the line segment joining
these sites, which intersects the line segment in the
middle. At the end of this process, we will have in-
tersections of finitely many hyperplanes which build
up cells, with a convex polygon structure, known as
Voronoi regions. The boundaries of each region are
called Voronoi edges and the intersections of Voronoi
edges are called Voronoi vertices. For more details
see (Berg et al., 2008; Klein, 1989; Iyengar et al.,
2014).
Mathematically, the Voronoi region of a point s
i
in
S is defined by
V
i
=
n
\
j=1, j6=i
n
x ∈R
d
|kx −s
i
k
2
< kx −s
j
k
2
o
,
CTDE 2018 - Special Session on Control Theory and Differential Equations
572