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2 COMPLETE LYAPUNOV
FUNCTIONS
A complete Lyapunov function for the general ODE
(1) is a function V : R
n
→ R which is not increasing
along solutions of (1). If V is sufficiently smooth, e.g.
C
1
, then this can be expressed by V
0
(x) ≤ 0, where
V
0
(x) = ∇V (x)·f(x) denotes the orbital derivative, i.e.
the derivative along solutions of (1).
A constant function would trivially satisfy this as-
sumption, and thus a complete Lyapunov function is
required to only be constant on each connected com-
ponent of the chain-recurrent set, including local at-
tractors and repellers, and be strictly decreasing else-
where. A point is in the chain-recurrent set, if an ε-
trajectory through it comes back to it after any given
time. An ε-trajectory is arbitrarily close to a true so-
lution of the system. This indicates recurrent motion;
for a precise definition see, e.g. (Conley, 1978). The
dynamics outside the chain-recurrent set are similar
to a gradient system, i.e. a system (1) where the right-
hand side f(x) = ∇W (x) is given by the gradient of a
function W : R
n
→ R.
The values and level sets of the complete Lya-
punov function provide additional information about
the dynamics and the long-term behaviour of the sys-
tem, e.g. an asymptotically stable equilibrium is a
local minimum of a complete Lyapunov function.
Moreover, the (constant) values of a complete Lya-
punov function on each connected component of the
chain-recurrent set describe the dynamics between
them.
Summarizing, a smooth complete Lyapunov func-
tion satisfies
V
0
(x) < 0 for x ∈ G, (3)
V
0
(x) = 0 for x ∈ C, (4)
where G denotes the gradient-flow like set and C
the chain recurrent set.
The first proof of the existence of a complete Lya-
punov function for dynamical systems was given by
Conley (Conley, 1978). This proof holds for a com-
pact metric space. It considers each corresponding
attractor-repeller pair and constructs a function which
is 1 on the repeller, 0 on the attractor and decreas-
ing in between. Then these functions are summed up
over all attractor-repeller pairs. Later, Hurley gen-
eralized these ideas to more general spaces (Hurley,
1992; Hurley, 1998). These functions, however, are
just continuous functions.
3 PREVIOUS CONSTRUCTION
METHODS
Computational approaches to construct complete Lya-
punov functions were proposed in (Kalies et al., 2005;
Ban and Kalies, 2006; Goullet et al., 2015). The state
space was subdivided into cells, defining a discrete-
time system given by the multivalued time-T map
between them, which was computed using the com-
puter package GAIO (Dellnitz et al., 2001). Hence, an
approximate complete Lyapunov function was con-
structed using graph algorithms (Ban and Kalies,
2006). This approach requires a high number of cells
even for low dimensions.
In (Bj
¨
ornsson et al., 2015), a complete Lyapunov
function was constructed as a continuous piecewise
affine (CPA) function, affine on a fixed simplicial
complex. However, it is assumed that information
about local attractors is available, while we do not re-
quire any information.
In (Arg
´
aez et al., 2017; Arg
´
aez et al., 2018a;
Arg
´
aez et al., 2018b) a complete Lyapunov function
was computed by approximately solving the PDE
V
0
(x) = −1 (5)
using meshfree collocation, in particular using Ra-
dial Basis Functions; for a detailed description of the
method see Section 4. This is inspired by construct-
ing classical Lyapunov functions for an equilibrium
(Giesl, 2007; Giesl and Wendland, 2007). However,
(5) cannot be fulfilled at all x ∈ C. Meshfree col-
location still constructs an approximation, but error
estimates are not available, as they compare the ap-
proximation to the solution of the problem, which
does not exist. In practice, the method works well on
the gradient-like part and is able to detect the chain-
recurrent set as the area of the state space where the
approximation fails.
Collocation points where the approximation fails
are detected by comparing the orbital derivative of the
approximation v to the prescribed value −1 in test
points y near the collocation point x
j
. A collocation
point x
j
is classified as failing if v
0
(y) ≥ −γ holds
for at least one of the test points y near x
j
, where
0 > −γ > −1 is a threshold parameter. Thus, the set
of collocation points X is separated into X = X
−
∪X
0
,
where X
0
denotes the failing points and X
−
the re-
maining ones. X
0
is an approximation of the chain-
recurrent set, while X
−
approximates the gradient-
like set. In a subsequent step, the problem
V
0
(x) =
−1 for x ∈ X
−
0 for x ∈ X
0
(6)
is solved using meshfree collocation. This step is iter-
ated until no points are moved from X
−
to X
0
. Further
Construction of a Complete Lyapunov Function using Quadratic Programming
561