Proceedings of the
46th IEEE Conference on Decision and Control
New Orleans, LA, USA, Dec. 12-14, 2007
FrC03.1
Lyapunov and Converse Lyapunov Theorems for Semistability
Qing Hui, Wassim M. Haddad, and Sanjay P. Bhat
Abstract— This paper develops Lyapunov and converse Lyapunov theorems for semistable nonlinear dynamical systems.
Semistability is the property whereby the solutions of a dynamical system converge to (not necessarily isolated) Lyapunov
stable equilibrium points determined by the system initial
conditions. Specifically, we provide necessary and sufficient
conditions for semistability and show that semistability implies
the existence of a smooth Lyapunov function that decreases
along the dynamical system trajectories such that the Lyapunov
function satisfies inequalities involving the distance to the set
of equilibria.
I. I NTRODUCTION
The aim of this paper is to develop Lyapunov and converse Lyapunov results for semistability. Semistability is the
property of a dynamical system whereby its trajectories converge to (not necessarily isolated) Lyapunov stable equilibria.
Semistability, rather than asymptotic stability, is the appropriate notion of stability for systems having a continuum
of equilibria. Examples of such systems arise in chemical
kinetics [1], adaptive control [2], compartmental modeling
[3], thermodynamics [4] and, more recently, collaborative
control of a network of autonomous agents [5], [6]. In all
these examples, trajectories converge to limit points that
depend continuously on the system initial conditions.
It is important to note that semistability is not merely
equivalent to asymptotic stability of the set of equilibria.
Indeed, it is possible for a trajectory to converge to the set
of equilibria without converging to any one equilibrium point
as examples in [2] show. Conversely, semistability does not
imply that the equilibrium set is asymptotically stable in any
accepted sense. This is because stability of sets is defined
in terms of distance (especially in case of noncompact sets),
and it is possible to construct examples in which the system
is semistable, but the domain of semistability contains no
ε-neighborhood (defined in terms of the distance) of the
(noncompact) equilibrium set, thus ruling out asymptotic
stability of the equilibrium set. Hence, semistability and set
stability of the equilibrium set are independent notions.
For linear systems, semistability was originally defined
in [7] and applied to matrix second-order systems in [8].
References [2] and [9] extended the notion of semistability
to nonlinear systems and gave Lyapunov results for semistability. However, converse Lyapunov results for semistability
have not been considered in the literature.
As a property, semistability lies in between Lyapunov stability and asymptotic stability in the sense that an asymptotically stable equilibrium is also semistable, while a semistable
This research was supported in part by the Air Force Office of Scientific
Research under Grant FA9550-06-1-0240.
Q. Hui and W. M. Haddad are with the School of
Aerospace Engineering, Georgia Institute of Technology, Atlanta,
GA
30332-0150,
USA
(qing [email protected]),
([email protected]).
S. P. Bhat is with the Department of Aerospace Engineering, Indian Institute of Technology, Powai, Mumbai 400076, India
([email protected]).
1-4244-1498-9/07/$25.00 ©2007 IEEE.
equilibrium is also Lyapunov stable. While a converse Lyapunov theorem for Lyapunov stability is possible only with
a lower semicontinuous Lyapunov function [10], converse
Lyapunov theory for asymptotic stability is well developed.
In particular, Massera [11] proved a converse Lyapunov
theorem under the assumption that the vector field f of the
nonlinear dynamical system is locally Lipschitz continuous.
For locally Lipschitz continuous vector fields, it has been
shown that asymptotic stability implies the existence of a
smooth (i.e., infinitely differentiable) Lyapunov function.
Kurzweil [12] proved the existence of smooth Lyapunov
functions for asymptotic stability under the assumption of
f only being continuous. Wilson [13] simplified Kurzweil’s
technique to give a shorter proof of a converse Lyapunov
theorem for asymptotic stability of sets under the assumption
that the flow of the dynamical system is unique in both
backward and forward time. The authors in [14] filled some
gaps in Wilson’s proof and extended the results to the case
where the existence of a smooth Lyapunov function is global.
Recently, the authors in [15] proved a converse Lyapunov
theorem for asymptotically stable differential inclusions by
assuming that the differential inclusions are bounded.
Unlike converse theorems for Lyapunov stable and asymptotically stable systems where the existence of a lower
semicontinuous and a continuous Lyapunov function, respectively, is ensured, converse Lyapunov theorems for semistability have only recently been addressed in the literature.
Specifically, the authors in [5] conjectured the existence of
a smooth Lyapunov function for a semistable system governed by dynamics that are locally Lipschitz away from the
equilibrium set. In this paper, we prove that this conjecture
is correct. In Section III, we provide converse results which
show that semistability implies the existence of a continuous Lyapunov function that decreases along the dynamical
system trajectories such that the Lyapunov function satisfies
inequalities involving the distance to the set of equilibria. We
begin by reviewing the necessary mathematical preliminaries
in the next section.
II. M ATHEMATICAL P RELIMINARIES
The notation used in this paper is fairly standard. Specifically, R denotes the set of real numbers, R+ denotes the set
of nonnegative real numbers, R+ denotes the set of positive
real numbers, Rn denotes the set of n×1 real column vectors,
and “◦” denotes the composition operator. Furthermore, S
denotes the closure of the subset S ⊂ Rn . We write · for the Euclidean vector norm, Bε (α), α ∈ Rn , ε > 0, for
the open ball centered at α with radius ε, dist(p, M) for
the smallest distance from a point p to the set M, that is,
dist(p, M) inf x∈M p − x, and V (x) for the Fréchet
derivative of V at x.
In this paper, we consider nonlinear dynamical systems of
the form
ẋ(t) = f (x(t)),
x(0) = x0 ,
n
t ∈ Ix0 ,
(1)
where x(t) ∈ D ⊆ R , t ∈ Ix0 , is the system state vector,
D is an open set, f : D → Rn is Lipschitz continuous
on D\f −1 (0) and is continuous on f −1 (0), f −1 (0) 5870
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
{x ∈ D : f (x) = 0} is nonempty, and Ix0 = [0, τx0 ),
0 ≤ τx0 ≤ ∞, is the maximal interval of existence for the
solution x(·) of (1). A continuously differentiable function
x : Ix0 → D is said to be a solution of (1) on the
interval Ix0 ⊂ R if x satisfies (1) for all t ∈ Ix0 . The
continuity of f implies that, for every x0 ∈ D, there exist
τ0 < 0 < τ1 and a solution x(·) of (1) defined on (τ0 , τ1 )
such that x(0) = x0 . A solution x is said to be right
maximally defined if x cannot be extended on the right
(either uniquely or nonuniquely) to a solution of (1). Here,
we assume that for every initial condition x0 ∈ D, (1) has
a unique right maximally defined solution, and this unique
solution is defined on [0, ∞). Under these assumptions on
f , the solutions of (1) define a continuous global semiflow
on D, that is, s : [0, ∞) × D → D is a jointly continuous
function satisfying the consistency property s(0, x) = x and
the semi-group property s(t, s(τ, x)) = s(t + τ, x) for every
x ∈ D and t, τ ∈ [0, ∞). Given t ∈ [0, ∞) we denote
the flow s(t, ·) : D → D of (1) by st (x0 ) or st . Likewise,
given x ∈ D we denote the solution curve or trajectory
s(·, x) : R+ → D of (1) by sx (t) or sx .
A set M ⊂ Rn is positively invariant if st (M) ⊆ M for
all t ≥ 0. The set M is negatively invariant if, for every
z ∈ M and every t ≥ 0, there exists x ∈ M such that
s(t, x) = z and s(τ, x) ∈ M for all τ ∈ [0, t]. Finally, the
set M is invariant if st (M) = M for all t ≥ 0. Note that a
set is invariant if and only if it is positively and negatively
invariant.
III. LYAPUNOV AND C ONVERSE LYAPUNOV T HEOREMS
FOR S EMISTABILITY
In this section, we develop necessary and sufficient conditions for semistability. These results extend some of the
results in [10–14], [16–19]. Converse Lyapunov theorems for
asymptotic stability were extensively studied in [11], [12]. In
particular, Massera [11] proved a converse Lyapunov theorem under the assumption that the vector field f is locally
Lipschitz continuous. For locally Lipschitz continuous vector
fields, it has been proved that asymptotic stability implies
the existence of a smooth (i.e., infinitely differentiable)
Lyapunov function. Kurzweil [12] proved the existence of
smooth Lyapunov functions for asymptotic stability under the
assumption of f only being continuous. Unlike asymptotic
stability, Lyapunov stability for autonomous dynamical systems does not imply the existence of a continuous Lyapunov
function. However, semistability does imply the existence of
a smooth Lyapunov function.
The following definitions and key propositions are necessary for the main results of this section.
Definition 3.1: An equilibrium point x ∈ D of (1) is
Lyapunov stable if for every open subset Nε of D containing
x, there exists an open subset Nδ of D containing x such
that st (Nδ ) ⊂ Nε for all t ≥ 0. An equilibrium point x ∈ D
of (1) is semistable if it is Lyapunov stable and there exists
an open subset Q of D containing x such that for all initial
conditions in Q, the trajectory of (1) converges to a Lyapunov
stable equilibrium point, that is, limt→∞ s(t, x) = y, where
y ∈ D is a Lyapunov stable equilibrium point of (1) and
x ∈ Q. If, in addition, Q = D = Rn , then an equilibrium
point x ∈ D of (1) is a globally semistable equilibrium.
The system (1) is said to be Lyapunov stable if every
equilibrium point of (1) is Lyapunov stable. The system (1)
is said to be semistable if every equilibrium point of (1) is
semistable. Finally, (1) is said to be globally semistable if
(1) is semistable and Q = D = Rn .
Definition 3.2: The domain of semistability is the set of
points x0 ∈ D such that if x(t) is a solution to (1) with
FrC03.1
x(0) = x0 , t ≥ 0, then x(t) converges to a Lyapunov stable
equilibrium point in D.
Note that if (1) is semistable, then its domain of semistability contains the set of equilibria in its interior. The following
proposition gives a sufficient condition for a trajectory of
(1) to converge to a limit. For this result, Dc ⊆ D denotes
a positively invariant set with respect to (1) so that the orbit
Ox of (1) is contained in Dc for all x ∈ Dc .
Proposition 3.1: Consider the nonlinear dynamical system (1) and let x ∈ Dc . If the positive orbit Ox+ of (1)
contains a Lyapunov stable equilibrium point y, then y =
limt→∞ s(t, x), that is, Ox+ = {y}.
Proof. The proof of the result appears in [2]. For completeness of exposition, we provide a proof here. Suppose
y ∈ Ox+ is Lyapunov stable and let Nε ⊆ Dc be an
open neighborhood of y. Since y is Lyapunov stable, there
exists an open neighborhood Nδ ⊂ Dc of y such that
st (Nδ ) ⊆ Nε for every t ≥ 0. Now, since y ∈ Ox+ , it
follows that there exists τ ≥ 0 such that s(τ, x) ∈ Nδ .
Hence, s(t + τ, x) = st (s(τ, x)) ∈ st (Nδ ) ⊆ Nε for
every t > 0. Since Nε ⊆ Dc is arbitrary, it follows that
y = limt→∞ s(t, x). Thus, limn→∞ s(tn , x) = y for every
+
sequence {tn }∞
n=1 , and hence, Ox = {y}.
Next, we present alternative equivalent characterizations
of semistability of (1).
Proposition 3.2: Consider the nonlinear dynamical system G given by (1). Then following statements are equivalent:
i) G is semistable.
ii) For each xe ∈ f −1 (0), there exist class K and L
functions α(·) and β(·), respectively, and δ = δ(xe ) >
0, such that if x0 − xe < δ, then x(t) − xe ≤
α(x0 − xe ), t ≥ 0, and dist(x(t), f −1 (0)) ≤ β(t),
t ≥ 0.
iii) For each xe ∈ f −1 (0), there exist class K functions α1 (·) and α2 (·), a class L function β(·), and
δ = δ(xe ) > 0, such that if x0 − xe < δ, then
dist(x(t), f −1 (0)) ≤ α1 (x(t) − xe )β(t) ≤ α2 (x0 −
xe )β(t), t ≥ 0.
Proof. To show that i) implies ii), suppose (1) is
semistable and let xe ∈ f −1 (0). It follows from Lemma
4.5 of [20] that there exists δ = δ(xe ) > 0 and a
class K function α(·) such that if x0 − xe ≤ δ, then
x(t) − xe ≤ α(x0 − xe ), t ≥ 0. Without loss of
generality, we may assume that δ is such that Bδ (xe ) is
contained in the domain of semistability of (1). Hence, for
every x0 ∈ Bδ (xe ), limt→∞ x(t) = x∗ ∈ f −1 (0) and,
consequently, limt→∞ dist(x(t), f −1 (0)) = 0.
For each ε > 0 and x0 ∈ Bδ (xe ), define Tx0 (ε) to be the
infimum of T with the property that dist(x(t), f −1 (0)) < ε
for all t ≥ T , that is, Tx0 (ε) inf{T : dist(x(t), f −1 (0)) <
ε, t ≥ T }. For each x0 ∈ Bδ (xe ), the function Tx0 (ε) is
nonnegative and nonincreasing in ε, and Tx0 (ε) = 0 for
sufficiently large ε.
Next, let T (ε) sup{Tx0 (ε) : x0 ∈ Bδ (xe )}. We claim
that T is well defined. To show this, consider ε > 0 and
x0 ∈ Bδ (xe ). Since dist(s(t, x0 ), f −1 (0)) < ε for every
t > Tx0 (ε), it follows from the continuity of s that, for
every η > 0, there exists an open neighborhood U of x0
such that dist(s(t, z), f −1 (0)) < ε for every z ∈ U. Hence,
lim supz→x0 Tz (ε) ≤ Tx0 (ε) implying that the function
x0 → Tx0 (ε) is upper semicontinuous at the arbitrarily
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
chosen point x0 , and hence on Bδ (xe ). Since an upper
semicontinuous function defined on a compact set achieves
its supremum, it follows that T (ε) is well defined. The
function T (·) is the pointwise supremum of a collection
of nonegative and nonincreasing functions, and is hence
nonegative and nonincreasing. Moreover, T (ε) = 0 for every
ε > max{α(x0 − xe ) : x0 ∈ Bδ (xe )}.
ε
Let ψ(ε) 2ε ε/2 T (σ)dσ + 1ε ≥ T (ε) + 1ε . The
function ψ(ε) is positive, continuous, strictly decreasing, and
ψ(ε) → 0 as ε → ∞. Choose β(·) = ψ −1 (·). Then β(·) is
positive, continuous, strictly decreasing, and β(σ) → 0 as
σ → ∞. Furthermore, T (β(σ)) < ψ(β(σ)) = σ. Hence,
dist(x(t), f −1 (0)) ≤ β(t), t ≥ 0.
Next, to show that ii) implies iii), suppose ii) holds and let
xe ∈ f −1 (0). Then it follows from Lemma 4.5 of [20] that xe
is Lyapunov stable. Choosing x0 sufficiently close to xe , it
follows from the inequality x(t) − xe ≤ α(x0 − xe ),
t ≥ 0, that trajectories of (1) starting sufficiently close
to xe are bounded, and hence, the positive limit set of
(1) is nonempty. Since limt→∞ dist(x(t), f −1 (0)) = 0, it
follows that the positive limit set is contained in f −1 (0).
Now, since every point in f −1 (0) is Lyapunov stable, it
follows from Proposition 3.1 that limt→∞ x(t) = x∗ , where
x∗ ∈ f −1 (0) is Lyapunov stable. If x∗ = xe , then it follows
using similar arguments as above that there exists a class
L function β̂(·) such that dist(x(t), f −1 (0)) ≤ x(t) −
xe ≤ β̂(t) for every x0 satisfying x0 − xe <δ and
t ≥ 0. Hence, dist(x(t), f −1 (0)) ≤ x(t) − xe β̂(t),
t ≥ 0. Next, consider the case where x∗ = xe and
let α1 (·) be a class K function. In this case, note that
limt→∞ dist(x(t), f −1 (0))/α1 (x(t)−xe ) = 0, and hence,
it follows using similar arguments as above that there exists
a class L function β(·) such that dist(x(t), f −1 (0)) ≤
α1 (x(t) − xe )β(t), t ≥ 0. Finally, note that α1 ◦ α is
of class K (by Lemma 4.2 of [20]), and hence, iii) follows
immediately.
Finally, to show that iii) implies i), suppose iii) holds
and let xe ∈ f −1 (0). Then it follows that α1 (x(t) −
xe ) ≤ α2 (x(0) − xe ), t ≥ 0, that is, x(t) − xe ≤
◦ α2 is
α(x(0) − xe ), where t ≥ 0 and α = α−1
1
of class K (by Lemma 4.2 of [20]). It now follows from
Lemma 4.5 of [20] that xe is Lyapunov stable. Since xe was
chosen arbitrarily, it follows that every equilibrium point is
Lyapunov stable. Furthermore, limt→∞ dist(x(t), f −1 (0)) =
0. Choosing x0 sufficiently close to xe , it follows from the
inequality x(t)−xe ≤ α(x0 −xe ), t ≥ 0, that trajectories
of (1) starting sufficiently close to xe are bounded, and hence,
the positive limit set of (1) is nonempty. Since every point in
f −1 (0) is Lyapunov stable, it follows from Proposition 3.1
that limt→∞ x(t) = x∗ , where x∗ ∈ f −1 (0) is Lyapunov
stable. Hence, by definition, (1) is semistable.
FrC03.1
f −1 (0) contained in Q. Consider x ∈ V so that there exists
z ∈ f −1 (0) such that x ∈ Vz and s(t, x) ∈ Vz , t ≥ 0.
Since Vz is bounded it follows that the positive limit set of
x is nonempty and invariant. Furthermore, it follows from (2)
that V̇ (s(t, x)) ≤ 0, t ≥ 0, and hence, it follows from the
Krasovskii-LaSalle invariant set theorem that s(t, x) → M
as t → ∞, where M is the largest invariant set contained
in the set R = {y ∈ Vz : V (y)f (y) = 0}. Note that R =
f −1 (0) is invariant, and hence, M = R, which implies that
limt→∞ dist(s(t, x), f −1 (0)) = 0. Finally, since every point
in f −1 (0) is Lyapunov stable, it follows from Proposition 3.1
that limt→∞ s(t, x) = x∗ , where x∗ ∈ f −1 (0) is Lyapunov
stable. Hence, by definition, (1) is semistable.
Next, we present a slightly more general theorem for
semistability wherein we do not assume that all points in
V̇ −1 (0) are Lyapunov stable but rather we assume that all
points in the largest invariant subset of V̇ −1 (0) are Lyapunov
stable.
Theorem 3.2: Consider the nonlinear dynamical system
(1) and let Q be an open neighborhood of f −1 (0). Suppose
the orbit Ox of (1) is bounded for all x ∈ Q and assume that
there exists a continuously differentiable function V : Q →
R such that
V (x)f (x) ≤ 0,
x ∈ Q\f −1 (0).
Example 3.1: Consider the nonlinear dynamical system
given by
ẋ1 (t) =
ẋ2 (t) =
σ12 (x2 (t)) − σ21 (x1 (t)), x1 (0) = x10 , t ≥ 0,
(4)
σ21 (x1 (t)) − σ12 (x2 (t)), x2 (0) = x20 ,
(5)
where x1 , x2 ∈ R, σij (·), i, j = 1, 2, i = j, are Lipschitz
continuous, σ12 (x2 ) − σ21 (x1 ) = 0 if and only if x1 = x2 ,
and (x1 − x2 )(σ12 (x2 ) − σ21 (x1 )) ≤ 0, x1 , x2 ∈ R. Note
that f −1 (0) = {(x1 , x2 ) ∈ R2 : x1 = x2 = α, α ∈ R}. To
show that (4) and (5) is semistable, consider the Lyapunov
function candidate V (x1 , x2 ) = 12 (x1 − α)2 + 12 (x2 − α)2 ,
where α ∈ R. Now, it follows that
V̇ (x1 , x2 ) =
=
=
≤
(2)
If (1) is Lyapunov stable, then (1) is semistable.
Proof. Since (1) is Lyapunov stable by assumption, for
every z ∈ f −1 (0), there exists an open neighborhood Vz
of z such that s([0,
∞) × Vz ) is bounded and contained in
Q. The set V z∈f −1 (0) Vz is an open neighborhood of
(3)
If every point in the largest invariant subset M of {x ∈ Q :
V (x)f (x) = 0} is Lyapunov stable, then (1) is semistable.
Proof. Since every solution of (1) is bounded, it follows
from the hypotheses on V (·) that, for every x ∈ Q, the
positive limit set ω(x) of (1) is nonempty and contained in
the largest invariant subset M of {x ∈ Q : V (x)f (x) = 0}.
Since every point in M is a Lyapunov stable equilibrium,
it follows from Proposition 3.1 that ω(x) contains a single
point for every x ∈ Q and limt→∞ s(t, x) exists for every
x ∈ Q. Now, since limt→∞ s(t, x) ∈ M is Lyapunov stable
for every x ∈ Q, semistability is immediate.
Next, we present sufficient conditions for semistability.
Theorem 3.1: Consider the nonlinear dynamical system
(1). Let Q be an open neighborhood of f −1 (0) and assume
that there exists a continuously differentiable function V :
Q → R such that
V (x)f (x) < 0,
x ∈ Q.
(x1 − α)[σ12 (x2 ) − σ21 (x1 )]
+(x2 − α)[σ21 (x1 ) − σ12 (x2 )]
x1 [σ12 (x2 ) − σ21 (x1 )]
+x2 [σ21 (x1 ) − σ12 (x2 )]
(x1 − x2 )[σ12 (x2 ) − σ21 (x1 )]
0, (x1 , x2 ) ∈ R × R,
(6)
which implies that x1 = x2 = α is Lyapunov stable.
Next, let R {(x1 , x2 ) ∈ R2 : V̇ (x1 , x2 ) = 0} =
{(x1 , x2 ) ∈ R2 : x1 = x2 = α, α ∈ R}. Since R consists of
equilibrium points, it follows that M = R. Hence, for any
x1 (0), x2 (0) ∈ R, (x1 (t), x2 (t)) → M as t → ∞. Hence, it
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follows from Theorem 3.2 that x1 = x2 = α is semistable
for all α ∈ R.
Finally, we provide a converse Lyapunov theorem for
semistability. For this result, recall that for a given continuous
function V : D → R, the upper right Dini derivative of V
along the solution of (1) is defined by
1
V̇ (s(t, x)) lim sup [V (s(t + h, x)) − V (s(t, x))]. (7)
h→0+ h
It is easy to see that V̇ (xe ) = 0 for every xe ∈ f −1 (0). Also
note that it follows from (7) that V̇ (x) = V̇ (s(0, x)).
Theorem 3.3: Consider the nonlinear dynamical system
(1). Suppose (1) is semistable with the domain of semistability D0 . Then there exist a smooth nonnegative function
V : D0 → R+ and a class K function α(·) such that
−1
FrC03.1
Therefore, for each y ∈ U,
U (z) − U (y)
1 + 2t
−1
dist(s̄(t, z), f (0))
= sup
1+t
t≥0
1 + 2t
−1
− sup
dist(s̄(t, y), f (0))
1+t
t≥0
1 + 2t
−1
= sup
dist(s̄(t, z), f (0))
1+t
0≤t≤T
1 + 2t
−1
− sup
dist(s̄(t, y), f (0)) .
1+t
0≤t≤T
Hence,
V (x) = 0, x ∈ f (0),
(8)
(9)
V (x) ≥ α(dist(x, f −1 (0))), x ∈ D0 ,
V̇ (x) < 0, x ∈ D0 \f −1 (0).
(10)
Proof. For any given solution
x(t)
of
(1),
the
change
of
t
time variable from t to τ = 0 (1 + f (x(s)))ds results in
the dynamical system
f (x̄(τ ))
dx̄
=
,
dτ
1 + f (x̄(τ ))
x̄(0) = x0 ,
τ ≥ 0,
|U (z) − U (y)|
1 + 2t dist(s̄(t, z), f −1 (0))
≤ sup 0≤t≤T 1 + t
−dist(s̄(t, y), f −1 (0)) ≤ 2 sup dist(s̄(t, z), f −1 (0))
0≤t≤T
−dist(s̄(t, y), f −1 (0))
(11)
≤ 2 sup dist(s̄(t, z), s̄(t, y)),
where x̄(τ ) = x(t). With a slight abuse of notation, let
s̄(t, x), t ≥ 0, denote the solution of (11) starting from
x ∈ D0 . Note that (11) implies that
s̄(t, x) − s̄(τ, x) ≤ |t − τ |,
x ∈ D0 ,
t, τ ≥ 0.
(12)
Next, define the function U : D0 → R+ by
1 + 2t
dist(s̄(t, x), f −1 (0)) , x ∈ D0 . (13)
U (x) sup
1+t
t≥0
Note that U (·) is well defined since (11) is semistable.
Clearly, (8) holds with V (·) replaced by U (·). Furthermore,
since U (x) ≥ dist(x, f −1 (0)), x ∈ D0 , it follows that (9)
holds with V (·) replaced by U (·).
To show that U (·) is continuous on D0 \f −1 (0), define T : D0 \f −1 (0) → [0, ∞) by T (z) inf{h :
dist(s̄(t, z), f −1 (0)) < dist(z, f −1 (0))/2 for all t ≥ h >
0}, and denote
Wε {x ∈ D0 : dist(s̄(t, x), f −1 (0)) < ε, t ≥ 0}.
−1
(14)
−1
Note that Wε ⊃ f (0) is open. Consider z ∈ D0 \f (0)
and define λ dist(z, f −1 (0)) > 0 and let xe limt→∞ s̄(t, z). Since xe is Lyapunov stable, it follows
that there exists an open neighborhood V of xe such that
all solutions of (11) in V remain in Wλ/2 . Since xe is
semistable, it follows that there exists h > 0 such that
s̄(h, z) ∈ V. Consequently, s̄(h + t, z) ∈ Wλ/2 for all t ≥ 0,
and hence, it follows that T (z) is well defined.
Next, by continuity of solutions of (11) on compact time
intervals, it follows that there exists a neighborhood U of z
such that U ∩ f −1 (0) = Ø and s̄(T (z), y) ∈ V for all y ∈ U.
Now, it follows from the choice of V that s̄(T (z) + t, y) ∈
Wλ/2 for all t ≥ 0 and y ∈ U. Then, for every t > T (z)
and y ∈ U,
1 + 2t
dist(s̄(t, y), f −1 (0)) ≤ 2dist(s̄(t, y), f −1 (0))
1+t
≤ λ.
(15)
0≤t≤T
z ∈ D0 \f −1 (0),
y ∈ U.
(16)
Now, it follows from continuous dependence of solutions
s̄(·, ·) on system initial conditions (Theorem 3.4 of Chapter
I of [21]) and (16) that U (·) is continuous at z. Furthermore,
it follows from (12) and (16) that, for every sufficiently small
h > 0,
|U (s̄(h, z)) − U (z)| ≤ 2 sup s̄(t, s̄(h, z)) − s̄(t, z)
0≤t≤T
= 2 sup s̄(t + h, z) − s̄(t, z)
0≤t≤T
≤ 2h,
which implies that |U̇ (z)| ≤ 2. Since z ∈ D0 \f −1 (0) was
chosen arbitrarily, it follows that U (·) is continuous and
|U̇ (z)| ≤ 2 on D0 \f −1 (0).
To show that U (·) is continuous on f −1 (0), consider
−1
(0)
xe ∈ f −1 (0). Let {xn }∞
n=1 be a sequence in D0 \f
that converges to xe . Since xe is Lyapunov stable, it follows
from Lemma 4.5 of [20] that x(t) ≡ xe is the unique solution
to (11) with x0 = xe . By continuous dependence of solutions
s̄(·, ·) on system initial conditions (Theorem 3.4 of Chapter
I of [21]), s̄(t, xn ) → s̄(t, xe ) = xe as n → ∞, t ≥ 0.
Let ε > 0 and note that it follows from ii) of Proposition
3.1 that there exists δ = δ(xe ) > 0 such that, for every
solution of (11) in Bδ (xe ), there exists T̂ = T̂ (xe , ε) > 0
such that s̄t (Bδ (xe )) ⊂ Wε for all t ≥ T̂ . Next, note that
there exists a positive integer N1 such that xn ∈ Bδ (xe ) for
all n ≥ N1 . Now, it follows from (13) that
U (xn ) ≤ 2 sup dist(s̄(t, xn ), f −1 (0)) + 2ε, n ≥ N1 . (17)
0≤t≤T̂
Next, it follows from Lemma 3.1 of Chapter I of [21] that
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007
FrC03.1
s̄(·, xn ) converges to s̄(·, xe ) uniformly on [0, T̂ ]. Hence,
lim
n→∞
sup dist(s̄(t, xn ), f
−1
IV. C ONCLUSION
In this paper, we developed Lyapunov and converse Lyapunov theorems for semistable nonlinear dynamical systems. Namely, converse theorems for semistability are developed using smooth (i.e., infinitely differentiable) Lyapunov
functions. Extensions of converse Lyapunov theorems for
semistability of nonlinear dynamical systems with continuous vector fields are under investigation.
(0))
0≤t≤T̂
= sup dist( lim s̄(t, xn ), f −1 (0))
0≤t≤T̂
n→∞
= sup dist(xe , f −1 (0))
0≤t≤T̂
(18)
= 0,
which implies that there exists a positive integer N2 =
N2 (xe , ε) ≥ N1 such that sup0≤t≤T̂ dist (s̄(t, xn ), f −1 (0))
< ε for all n ≥ N2 . Combining (17) with the above result
yields U (xn ) < 4ε for all n ≥ N2 , which implies that
limn→∞ U (xn ) = 0 = U (xe ).
Next, we show that U (x̄(τ )) is strictly decreasing along
the solution of (11) on D\f −1 (0). Note that for every
x ∈ D0 \f −1 (0) and 0 < h ≤ 1/2 such that s̄(h, x) ∈
D0 \f −1 (0), it follows from the above argument that the
supremum defining U (s̄(h, x)) is reached at some time t̂
such that 0 ≤ t̂ ≤ T (x). Hence,
U (s̄(h, x))
=
=
≤
1 + 2t̂
1 + t̂
1
+ 2t̂ + 2h
dist(s̄(t̂ + h, x), f −1 (0))
1 + t̂ + h
h
· 1−
(1 + 2t̂ + 2h)(1 + t̂)
h
,
(19)
U (x) 1 −
2(1 + T (x))2
dist(s̄(t̂ + h, x), f −1 (0))
which implies that U̇ (x) ≤ − 21 U (x)(1 + T (x))−2 < 0, x ∈
D0 \f −1 (0), and hence, (10) holds with V (·) replaced by
U (·). The function U (·) now satisfies all of the conditions
of the theorem except for smoothness.
To obtain smoothness, note that since |U̇ (x)| ≤ 2 for
every x ∈ D0 , it follows that U̇ (x) satisfies a boundedness
condition in the sense of Wilson [13]. By Theorem 2.5 of
[13], we can define a smooth function W : D0 \f −1 (0) → R
satisfying
1
1
|W (x) − U (x)| < U (x)(1 + T (x))−2 < U (x)
4
2
and
1
Ẇ (x) ≤ U̇(x) + U (x)(1 + T (x))−2
4
1
≤ − U (x)(1 + T (x))−2
4
< 0, x ∈ D0 \f −1 (0).
Next, we extend W (·) to all of D0 by taking W (z) = 0 for
z ∈ f −1 (0). Now, W (·) is a continuous Lyapunov function
which is smooth on D0 \f −1 (0). Taking
−2
V (x) = W (x)e−(W (x)) ,
R EFERENCES
[1] D. S. Bernstein and S. P. Bhat, “Nonnegativity, reducibilty and
semistability of mass action kinetics,” in Proc. IEEE Conf. Decision
and Control, Phoenix, AZ, 1999, pp. 2206–2211.
[2] S. P. Bhat and D. S. Bernstein, “Nontangency-based Lyapunov tests for
convergence and stability in systems having a continuum of equilibra,”
SIAM J. Control Optim., vol. 42, pp. 1745–1775, 2003.
[3] J. A. Jacquez and C. P. Simon, “Qualitative theory of compartmental
systems,” SIAM Rev., vol. 35, pp. 43–79, 1993.
[4] W. M. Haddad, V. Chellaboina, and S. G. Nersesov, Thermodynamics:
A Dynamical Systems Approach. Princeton, NJ: Princeton Univ. Press,
2005.
[5] Q. Hui, W. M. Haddad, and S. P. Bhat, “Finite-time semistability theory with applications to consensus protocols in dynamical networks,”
in Proc. Amer. Control Conf., New York, NY, 2007, pp. 2411–2416.
[6] Q. Hui and W. M. Haddad, “Continuous and hybrid distributed control
for multiagent coordination: Consensus, flocking, and cyclic pursuit,”
in Proc. Amer. Control Conf., New York, NY, 2007, pp. 2576–2581.
[7] S. L. Campbell and N. J. Rose, “Singular perturbation of autonomous
linear systems,” SIAM J. Math. Anal., vol. 10, pp. 542–551, 1979.
[8] D. S. Bernstein and S. P. Bhat, “Lyapunov stability, semistability, and
asymptotic stability of matrix second-order systems,” ASME J. Vibr.
Acoustics, vol. 117, pp. 145–153, 1995.
[9] S. P. Bhat and D. S. Bernstein, “Arc-length-based Lyapunov tests for
convergence and stability in systems having a continuum of equilibria,”
in Proc. Amer. Control Conf., Denver, CO, 2003, pp. 2961–2966.
[10] T. Yoshizawa, Stability Theory by Liapunov’s Second Method. Tokyo,
Japan: Math. Soc. Japan, 1966.
[11] J. L. Massera, “Contributions to stability theory,” Ann. Math., vol. 64,
pp. 182–206, 1956.
[12] J. Kurzweil, “On the inversion of Lyapunov’s second theorem on
stability of motion,” Amer. Math. Soc. Transl., vol. 24, pp. 19–77,
1963, translated from Czechoslovak Math. J., vol. 81, pp. 217-259;
455-484, 1956.
[13] F. W. Wilson, “Smoothing derivatives of functions and applications,”
Trans. Amer. Math. Soc., vol. 139, pp. 413–428, 1969.
[14] Y. Lin, E. D. Sontag, and Y. Wang, “A smooth converse Lyapunov
theorem for robust stability,” SIAM J. Control Optim., vol. 34, pp.
124–160, 1996.
[15] F. H. Clarke, Y. S. Ledyaev, and R. J. Stern, “Asymptotic stability and
smooth Lyapunov functions,” J. Diff. Equat., vol. 149, pp. 69–114,
1998.
[16] N. N. Krasovskii, Stability of Motion. Stanford, CA: Stanford Univ.
Press, 1963.
[17] V. I. Zubov, Methods of A. M. Lyapunov and their Application.
Groningen, The Netherlands: Noordhoff, 1964.
[18] W. Hahn, Stability of Motion. Berlin, Germany: Springer-Verlag,
1967.
[19] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems.
Berlin, Germany: Springer-Verlag, 1970.
[20] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ:
Prentice-Hall, 2002.
[21] J. K. Hale, Ordinary Differential Equations, 2nd ed. Malabar, FL:
Krieger, 1980.
(20)
and noting that
1
1
W (x) > U (x) > dist(x, f −1 (0)), x ∈ D0 \f −1 (0),
2
2
2
so that V (·) satisfies (9) with α(r) (r/2)e−4/r , we obtain
the desired smooth Lyapunov function.
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