Jrl Syst Sci & Complexity (2009) 22: 683696

STABILIZATION OF NONLINEAR TIME-VARYING

SYSTEMS: A CONTROL LYAPUNOV FUNCTION
APPROACH
Zhongping JIANG Yuandan LIN Yuan WANG

Received: 18 June 2009

c
2009 Springer Science + Business Media, LLC

Abstract This paper presents a control Lyapunov function approach to the global stabilization
problem for general nonlinear and time-varying systems. Explicit stabilizing feedback control laws are
proposed based on the method of control Lyapunov functions and Sontags universal formula.
Key words Control Lyapunov functions (clf), global stabilization, nonlinear time-varying systems.

1 Introduction
The last two decades have witnessed tremendous progress in the field of nonlinear control.
One of the most powerful tools is the approach of control Lyapunov functions (clf)[12] which
has been employed to address various issues tied to nonlinear control systems, such as nonlinear
stabilization[3] and adaptive control[45]. However, most of the past work by the clf approach
focused on time-invariant systems. The main purpose of this paper is to develop some tools
based on control Lyapunov functions for stabilizing nonlinear time varying systems.
The problem of stabilization of time varying systems has attracted much attention; see for
instance, [610], and other work cited therein, where some recursive designs such as backstep-
ping approach were adopted. Our work will be based on the clf approach. Essentially, we will
generalize the well-known universal formula[2] to provide explicit feedback control laws that sta-
bilize a general time-varying nonlinear system, under the assumption that a control Lyapunov
function is known. It should be mentioned that, when applying the control Lyapunov function
approach to time-varying systems, new challenging issues arise, mainly due to the presence of
the time parameter t in the Lyapunov functions. In contrast to the time invariant case, a feed-
back law resulted from the universal formula may fail to stabilize the system even if a uniform
control Lyapunov function is given. In this preliminary work, we will define uniform control
Lyapunov functions and the associated concept of small control property for time varying sys-
tems. We will then provide several sufficient conditions under which the feedback laws given by
the universal formula render the closed-loop system uniformly globally asymptotically stable.
Zhongping JIANG
Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Brooklyn,
NY 11201, USA. Email: zjiang@control.poly.edu.
Yuandan LIN Yuan WANG
Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA.
Email: lin@math.fau.edu; ywang@math.fau.edu.
This work has been supported in part by National Science Foundation under Grants Nos. ECS-0093176, DMS-

0906659, and DMS-0504296, and in part by National Natural Science Foundation of China under Grant Nos.
60228003 and 60628302.
684 ZHONGPING JIANG YUANDAN LIN YUAN WANG

By the control Lyapunov function approach, we also consider a problem related to some open
questions raised in [8]. The general form of the problems proposed in [8] was that if a system
x = f (x) + g(x)u is stabilizable by a feedback law k(x), can the system x = f (x) + p(t)g(x)u
be stabilized when p() possesses some persistent excitation property? In [8], the problem was
discussed and solved for several classes of systems, which may lead to a solution to the more
general type of systems in strict feedback forms. For some related work based on the condition of
persistent excitation, please also see [7] and [11]. We will consider in this work the problem from
the clf point of view. We first show by a simple example that not every stabilizing feedback law
for a system x = f (x) + g(x)u will stabilize the system x = f (x) + p(t)g(x)u when a persistently
exciting p(t) is presented. Our main concern is then whether the feedback law derived from
the universal formula for a system is still stabilizing when p(t) multiplicatively appears in the
control channel. A natural tool in our method is the so called weakly persistently exciting
functions introduced in [12]. A weakly persistently exciting function still has the main feature
that the energy of the function over any time interval of a given length maintains at least a
certain level, but the function is allowed to take negative values. Because of the uncertainty
of the sign of such functions involved, the proofs of the results are more complicated than one
would expect, especially when it involves with manipulations of inequalities. The results we
obtained are still preliminary. On the other hand, the method by the combination of universal
formula and weakly persistently exciting functions may lead to more applications. We refer
the interested reader to some recent contributions[68,1217] for additional tools and results on
stability and stabilization for nonlinear time-varying systems.
The rest of the paper is organized as follows. In Section 2 we review the notion and properties
of uniform global asymptotic stability and Sontags universal formula. In Section 3 we present
some extension of control Lyapunov functions to deal with nonlinear time-varying systems.
Several new results on time-varying stabilization will be developed in this section. In Section 4
we close the paper with brief concluding remarks.

2 Preliminaries
Before tackling the stabilization problem for general nonlinear time-varying systems, we
first review the notion and properties of uniform global asymptotic stability that will be needed
for the development of our stabilizing control schemes. Then, in Subsection 2.2, we recall the
concept of control Lyapunov function and Sontags universal formula for time-invariant control
systems.
Throughout the paper, K is the class of continuous functions R+ R+ which is zero at
zero and strictly increasing; K is a subset of the class-K functions which are unbounded; KL
stands for the class of functions R+ R+ R+ which are of class K on the first argument and
decreases to zero on the second argument.

2.1 Uniform Global Asymptotic Stability

Consider a nonlinear time-varying system of the form:

x(t)
= f (t, x(t)), t R0 , x Rn , (1)

where f : R0 Rn Rn is locally Lipschitz. Such a system is uniformly globally asymp-

totically stable (UGAS) at the origin if there exists some KL such that for every solution
x(, t0 , x0 ) of (1) with the initial condition x(t0 ) = x0 , it holds that

|x(t + t0 , t0 , x0 )| (|x0 | , t), t 0.

 STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS 685

The following result provides an equivalent Lyapunov characterization of UGAS; see, for in-
stance, [18, Theorem 49.3] and [19, Proposition 16].
Proposition 2.1 A system as in (1) is UGAS if and only if there exists a C Lyapunov
function V : R0 Rn R0 such that
for some , K , it holds that

(||) V (t, ) (||), t 0, Rn ; (2)

for some positive definite function , it holds that

V V
(t, ) + f (t, ) (||), t 0, Rn . (3)
t

Note that to get the existence of a C function for system (1), one may relax the Lipschitz
condition of f by assuming that f (t, ) is locally Lipschitz on R0 (Rn \ {0}) and is continuous
everywhere[19].
Observe that for a function V satisfying (2), property (3) is equivalent to the existence of a
positive definite function
e() such that

V V
(t, ) + f (t, ) e
(V (t, )), t 0, Rn .
t

In recent work [12] and [16], it was shown that, for the sufficiency part, condition (3) can
be replaced by a weaker condition, as shown below.
For > 0 and > 0, let P, denote the collection of measurable, locally essentially bounded
functions p : R R0 such that
Z t+
p(s) ds , t 0. (4)
t

Proposition 2.2[16] A system as in (1) is UGAS if there exists a C 1 Lyapunov function

satisfying (2) such that for some p P, , the following holds:

V V
(t, ) + f (t, ) p(t)(||) (5)
t
for all t 0 and all Rn .
In [16], this result was stated for periodic systems and was proved by establishing equivalence
among different types of Lyapunov functions. Below we provide a more direct proof.
Suppose there is some Lyapunov function V satisfying properties (2) and (5). By Proposition
13 in [20], one sees that there exists some C 1 , proper, and positive definite function such that
(V ) 1 (V ) (V ), and hence, for the Lyapunov function W given by W := V , it holds
that
W W
(t, ) + f (t, ) p(t)W (t, ), t 0, . (6)
t
It then follows that (see also [7])
R t+t0
p(s) ds
W (t + t0 , x(t + t0 , t0 , x0 )) W (t0 , x0 )e t0
, t 0, t0 , x0 .
686 ZHONGPING JIANG YUANDAN LIN YUAN WANG

Let k denote the largest integer such that k t. By (4), one sees that
Z t0 +t Z t0 +k t 
p(s) ds p(s) ds k 1 . (7)
t0 t0
Therefore,
t
W (t + t0 , x(t + t0 , t0 , x0 )) W (t0 , x0 )e( 1) W (t0 , x0 )M et/ , (8)

where M = e . The desired UGAS property follows directly readily.

In the above discussions, one has used the linear property of W in the decay estimation (6).
For a Lyapunov function V satisfying such an exponential decay estimation, the condition of p
being nonnegative can be further relaxed as shown in [12].
For > 0 and > 0, let wP, denote the set of all measurable, locally essentially bounded
functions p such that
for some c > 0, it holds that p(t) c, a.e.;
Z t+
p(s) ds for all t 0.
t

Observe that wP is a larger class than P since a function in wP is not required to be nonnegative.
Also notice that any p P, yields a function p(t) c in wP, for any given 0 c < /
and = c . For instance, sin2 t 14 wP, with = and = /4. Since the function
takes negative values in some intervals, sin2 t 14 6 P, for any , > 0. The next result is a
variant of [12, Theorem 10].
Proposition 2.3 A system as in (1) is UGAS if there exists a C 1 Lyapunov function V
satisfying (2) such that for some p wP, , the following holds:
V V
(t, ) + f (t, ) p(t)V (t, ) (9)
t
for all t 0 and all Rn .
The proof of Proposition 2.3 follows essentially the same lines as the proof from (6) to (8),
except one needs to note that for p wP, ,
Z t0 +t Z t0 +k Z t0 +t t 
p(s) ds = p(s) ds + p(s) ds k c 1 c,
t0 t0 t0 +k

where k still has the same meaning as in (7). Hence, the decay property (9) implies that

V (t + t0 , x(t + t0 , t0 , xo )) V (t0 , xo )M1 et/ ,

where M1 = e+c .

2.2 Control Lyapunov Functions

Recall that a C function V (x) is said to be a (global) clf for a time-invariant affine-in-
control system
x = f (x) + g(x)u,
if V (x) is positive definite and proper, and satisfies the following implication

inf a() + b()u < 0, 6= 0
u
 STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS 687

with a() = V() f () and b() = V() g(). We refer the interested reader to [3] for a tutorial
on control Lyapunov functions.
In the seminal work [2], a universal formula was established for the design of feedback
stabilizers when a clf is given. More precisely, it is shown that the feedback law u = (a(), b())
given by the universal formula is of class C on Rn \ {0} and stabilizes the system, where the
function is defined by

a + a2 + b 4
, if b 6= 0,
(a, b) = b (10)

0, if b = 0.

Furthermore, if the clf function V satisfies the small control property, i.e., for any > 0,
there exists some > 0 such that whenever 0 < || < , there exists some |u| < such that
a() + b()u < 0, then the feedback given by u = (a(), b()) is almost smooth, where an
almost smooth function means a function defined on Rn which is smooth away from 0 and
continuous everywhere. In the next section, we generalize this method to general time-varying
systems.
For a system given by x(t) = F (t, x(t)) and a C 1 function V : R0 Rn Rn , we let
V (t, x(t)) denote dt
d
V (t, x(t)) along the solution x(t), that is,

V V
V (t, x(t)) = (t, x(t)) + (t, x(t))F (t, x(t)).
t x

3 Control Lyapunov Functions for Time-Varying Systems

To simplify the presentation, we only consider a single-input time-varying control system:

x(t)
= f (t, x(t)) + g(t, x(t))u(t), (11)

where f and g are smooth maps from R+ Rn to Rn . The admissible class of input functions
is composed of measurable, locally essentially bounded functions. A function : R+ Rn R
is said to be almost smooth if it is continuous everywhere and is smooth on R+ (Rn \ {0}).
Definiton 3.1 System (11) is said to be uniformly stabilizable if there exists some feedback
law k : R+ Rn R which is smooth on R+ (Rn \ {0}) such that the resulted closed-loop
system
x(t)
= f (t, x(t)) + g(t, x(t))k(t, x(t))
is UGAS at the origin.
Definiton 3.2 A C function V is said to be a clf for (11), if it satisfies property (2) and
the following:
inf {a(t, ) + b(t, )u} < 0, 6= 0, (12)
u
V V
where a(t, ) = t (t, ) + f (t, ) and b(t, ) = V
g(t, ).
Moreover, V is said to be a uniform control Lyapunov function (uniform-clf) for (11), if,
instead of (12), it satisfies a stronger property

inf a(t, ) + b(t, )u (||), (13)
u

where is a positive definite function.

688 ZHONGPING JIANG YUANDAN LIN YUAN WANG

Remark 3.3 It is of interest to note that the condition (12) is equivalent to requiring

b(t, ) = 0 a(t, ) < 0, 6= 0. (14)

Clearly, by means of Proposition 2.1, the existence of a uniform-clf is a necessary condition

for a system as in (11) to be uniformly stabilizable. Namely,
Lemma 3.4 If a system as in (11) is uniformly stabilizable, then the system admits a
uniform-clf.
It is interesting to point out that the existence of a clf does not necessarily imply that the
system can be uniformly stabilized. For instance, for the planar system
x1
x 1 = + x2 , x 2 = x2 u,
1 + t2
V (x1 , x2 ) = x21 + x22 is a clf. But the system cannot be uniformly stabilized. Indeed, for
any feedback law u = k(t, ), the x1 -component of the trajectory with the initial condition
x(t0 ) = (x1o , 0) is a solution of the one-dimensional system x 1 = x1 /(1 + t2 ), x1 (t0 ) = x1o
which is not a UGAS system.
Even if a system admits a uniform-clf, the universal formula does not necessarily yield a
feedback law that guarantees the UGAS property of the closed-loop system. Indeed, for a
general system with the feedback given by the universal formula, the following holds for the
closed-loop solutions: p
V (t, x(t)) = a2 (t, x(t)) + b4 (t, x(t)). (15)
Without knowing if the function a2 (t, ) + b4 (t, ) dominates a positive definite function of V ,
one cannot determine if the closed-loop system is uniformly asymptotically stable. For instance,
for the simple system
x+u
x = ,
1 + t2
2
the function V (x) = x2 /2 is a uniform-clf satisfying that V (x(t)) = x (t)+x(t)u(t)

1+t2
. The universal
formula leads to
2
r r
x (t) 1 2V (x(t)) 1
V (x(t)) = 1+ = 1+ .
1 + t2 1 + t2 1 + t2 1 + t2

It can be verified that the closed-loop system is not UGAS.

In this work we will investigate sufficient conditions under which the feedback laws given by
the universal formula render the closed-loop system the UGAS property. As a preparation, we
first extend the result regarding the small control property to time varying systems.
Definition 3.5 A clf V is said to satisfy the small control property (SCP) if for every > 0,
every t0 > 0, there exists some > 0 such that for all 0 < || < and all t (t0 , t0 + ),
there exists some || < such that a(t, ) + b(t, ) < 0.
Note that in Definition 3.5, the choice of is allowed to be dependent on t0 (i.e., the choice
of is not required to be uniform in t0 ). Instead of saying V satisfies the small control property,
we will sometimes say that the pair (a(t, ), b(t, )) satisfies the small control property.
Lemma 3.6 Let V be a clf for a system as in (11). If V satisfies the small control property,
then the feedback function given by the universal formula u = k(t, ) := (a(t, ), b(t, )) is
almost smooth.
The proof of Lemma 3.6 will be given in Appendix A. The next result follows directly from
Lemma 3.6, Proposition 2.2, and Equation (15).
 STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS 689

Proposition 3.7 Suppose a system as in (11) admits a clf V that satisfies the small control
property. Assume further that
p
a2 (t, ) + b4 (t, ) p(t)(||), Rn , t 0 (16)

for some p P, and some positive definite function , then the closed-loop system under the
almost smooth feedback u = (a(t, x), b(t, x)) is UGAS.
As a side remark, we would like to point out that if the property p P, is relaxed into
the following weaker property Z
p(s) ds = , (17)
0

then one will achieve a weaker stability. Namely, one has the following (see [14, Proposition
3.3]).
Proposition 3.8 Suppose a system as in (11) admits a clf V that satisfies the small control
property. Assume further that (16) holds for some positive definite functions and p satisfying
(17). Then, the closed-loop system under the almost smooth feedback u = (a(t, x), b(t, x)) is
semi-uniformly GAS (see [14]), that is, for some KL and some nonnegative function , it
holds that  
t
|x(t + t0 )| |x0 | , , t 0.
1 + (t0 )

3.1 Controller Design with p-CLF

In applications, a clf satisfying condition (14) can be hard to find. Often, the following
holds, instead of (14):

b(t, ) = 0 a(t, ) p(t)e

a(), 6= 0,

where e
a is positive definite and p P, . Such a clf is called a p-clf.
Proposition 3.9 Consider a system as in (11). Suppose that there exists a C function
V satisfying (2) and the following:

a(t, ) = p(t)a0 (t, ) a1 (t, ), b(t, ) = p(t)b0 (t, ) (18)

with the conditions

(a) a1 (t, ) 0;
(b) b0 (t, ) = 0 a0 (t, ) < 0 for all 6= 0;
(c) p P, for some > 0 and > 0; and
(d) there is some continuous, positive definite function 3 such that

|a0 (t, )| + b20 (t, ) 3 (V (t, )). (19)

Assume further that (a0 (t, ), b0 (t, )) satisfies the small control property. Then, the feedback
law by the universal formula u = (a0 (t, x), b0 (t, x)) is almost smooth, and the corresponding
closed-loop system is UGAS.
Proof As it can be directly checked, the solutions of the closed-loop system satisfy
q
V (t, x(t)) p(t) a20 (t, x(t)) + b40 (t, x(t)) p(t)3 (V (t, x(t)))/2
690 ZHONGPING JIANG YUANDAN LIN YUAN WANG

With the help of Proposition 2.2 and Lemma 3.6, Proposition 3.9 follows readily.
Remark 3.10 Suppose that the function p in property (c) of Proposition 3.9 is bounded,
that is, 0 p(t) M for some constant M < , then condition (19) can be relaxed into

|a0 (t, )| + b20 (t, ) + a1 (t, ) 3 (V (t, )).

To see this, note that with u = (a0 (t, x), b0 (t, x)), one has
q
V (t, x(t)) p(t) a20 (t, x(t)) + b40 (t, x(t)) a1 (t, )
q

p(t)
a20 (t, x(t)) + b40 (t, x(t)) + a1 (t, )
M
p(t) 3 (V (t, ))
(|a0 (t, )| + b20 (t, ) + |a1 (t, )|) p(t) .
2M 2M
The stability property then follows as in the proof of Proposition 3.9.
In Proposition 3.9, a common function p(t) P, is required for the pair (a(t, ), b(t, )).
This can be made slightly more flexible as in what follows. By using transformations on the
input variable u of the form u = q0 (t)v, a system as in (11) is changed to

x(t)
= f (t, x(t)) + q0 (t)g(t, x(t))v(t).

Proposition 3.9 can be modified as follows.

Corollary 3.11 For a system as in (11). Suppose that there exists a C function V
satisfying (2) and the following:

a(t, ) = p(t)a0 (t, ) a1 (t, ), b(t, ) = p0 (t)b0 (t, )

with the conditions:

(a) a1 (t, ) 0;
(b) there exists some smooth function q0 (t) such that p0 (t)q0 (t) = p(t);
(c) b0 (t, ) = 0 a0 (t, ) < 0 for all 6= 0;
(d) p P, for some > 0, > 0; and
(e) there is some continuous, positive definite function 3 such that (19) holds for a0 (t, )
and b0 (t, ).
Assume further that (a0 (t, ), b0 (t, )) satisfies the small control property. Then, the feedback
law by the modified universal formula u = q0 (t)(a0 (t, x), b0 (t, x)) is almost smooth, and the
corresponding closed-loop system is UGAS.
Example 3.12 The function V () = 2 /2 is a clf for the system x = u sin t as in Corollary
3.11 with p0 (t) = q0 (t) = sin t, a0 (t, ) = a1 (t, ) = 0 (Note that p0 (t) = sin t alone is not a
class P, function). A direct application of Corollary 3.11 yields the time-varying feedback
law u = x sin t for the closed-loop system to be UGAS.

3.2 Stabilization with Persistently Exciting Functions in Control Channels

Suppose a system as in (11) is uniformly stabilizable. Let p P, . An interesting question
is whether or not there is a feedback law u = k1 (t, x) that uniformly stabilizes the system

x = f (t, x) + p(t)g(t, x)u. (20)

 STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS 691

In this section, we consider the specific question that if a system as in (11) admits a clf, can
the universal formula be used to find a stabilizing feedback law for (20)?
Example 3.13 It is easily seen that u = 4x stabilizes both systems x = x + u and
x = x3 + x2 u. However, the same feedback stabilizes uniformly the system x = x + (sin2 t)u,
but not the system x = x3 + (sin2 t)x2 u. In fact, the latter system cannot be stabilized by any
linear feedback u = kx for any k > 0.
We now assume that (11) admits a clf V satisfying (2) and the following:

inf {a(t, ) + b(t, )u} < 0, 6= 0,

u

where a(t, ) = t V (t, ) + V (t, )f (t, ) and b(t, ) = V (t, )g(t, ). Assume further that
the feedback law given by the universal formula u = (a(t, x), b(t, x)) stabilizes the system (11)
uniformly, that is, for some positive definite function ,
p
a2 (t, ) + b4 (t, ) (||). (21)

Without loss of generality, because of property (2), (21) can be re-formulated as

p
a2 (t, ) + b4 (t, )
b(V (t, ))

for some positive definite function b. By Lemmas 11 and 12 in [20], one sees that, for each
> 0, there exists a proper, positive definite function which is smooth on R \ {0} and C 1
everywhere such that
(V (t, ))b
(V (t, )) (V (t, )),
and hence, with W = V , it holds that

a(t, ) + eb(t, )u} W (t, ),

inf {e (22)
u

where e
a(t, ) = t
W (t, ) + W (t, )f (t, ) and eb(t, ) =

W (t, )g(t, ). Hence, we get the
following lemma.
Lemma 3.14 Suppose a system as in (11) admits a smooth clf V satisfying (2). Assume
further that (21) holds for the function V . Then, for any > 0, there exists a C 1 clf W , smooth
on R+ (Rn \ {0}), for which (22) holds.
Consider p P, for a system as in (20). Replacing p by M p and u by v := u/M for some
M > 0 if necessary, one may assume the following condition on the function p:
Z t+
(p(s) 1) ds , t 0,
t

that is, (p 1) wP, .

We are now ready to state the following theorem, whose proof is postponed to Appendix B.
Theorem 1 Consider the problem of stabilizing a system as in (20). Assume the following:
(a) the function p satisfies that 0 p(t) M for all t 0 and some M > 0 and that
p 1 wP, ;
(b) the system (11) admits a C 1 clf V , smooth on R0 (Rn \ {0}), satisfying (2) and
p
a2 (t, ) + b4 (t, ) cV (t, ), (23)

where c = max 2 M, 2 + 2 ; and
692 ZHONGPING JIANG YUANDAN LIN YUAN WANG

(c) a(t, ) V (t, ).

Then, the feedback law given by the universal formula u = (a(t, ), b(t, )) is smooth away from
0 and it globally and uniformly stabilizes the system (20).
Furthermore, if the pair (a, b) satisfies the small control property, then the feedback law is
almost smooth.
Remark 3.15 For the conditions in Theorem 1, we note the following:
1) condition (a) amounts to requiring the function p appeared in (20) to be bounded;
2) condition (b) amounts to requiring system (11) to admit a clf satisfying (21) for some
positive definite function (which is always the case when f and g are time invariant);
3) one may always modify the clf V so that (23) holds (c.f. Lemma 3.14); and
4) condition (c) requires that for the new clf for which (23) holds, the term a(t, x) is bounded
above by V .
As one can see, condition (c) is the only real restriction required for Theorem 1. Notice
that, as suggested by Example 3.13, some conditions on the upper bound of the term a(t, )
should be imposed to guarantee the stabilizability.
In the special case when f and g are both time invariant, Theorem 1 states that under
condition (a)(c), the feedback law given by the universal formula for the system x = f (x) +
g(x)u also stabilizes the system x = f (x) + p(t)g(x)u.
One of the technical issues in the proof of Theorem 1 is that when working with a function
q() of class wP, the uncertainty of the sign of q() makes it hard to manipulate with inequalities.
In the special case when (11) admits a clf V for which a(t, ) does not change sign, the conditions
of Theorem 1 can be significantly simplified, as stated below. The proof of the following
proposition will be postponed to Appendix C.
Proposition 3.16 Let p P, for some > 0 and > 0. Assume that 0 p(t) M for
some M > 0. If system (11) admits a smooth clf V such that
for some positive definite function , (21) holds; and
a(t, ) 0 for all t, ,
then the feedback law u = (a(t, ), b(t, )) globally and uniformly stabilizes the corresponding
system (20).
Note that Proposition 3.16 means that if the term a(t, ) is nonpositive and if the persistently
excited function p() is bounded, then the system (20) can be stabilized by the universal formula.
The next result deals with the case when a(t, ) is always nonnegative. Its proof can be found
in Appendix D.
Proposition 3.17 Let p P, for some > 0 and > 0. Assume that system (11)
admits an almost smooth clf V such that
p
a2 (t, ) + b4 (t, ) V (t, ); and
0 a(t, ) cV (t, ) for some c 0.
Let M > 0 be such that (M p() c) wP, . Then, the feedback law u = M (a(t, ), b(t, ))
globally and uniformly stabilizes the system.
For the two systems in Example 3.13, let V (x) = x2 /2. It is readily seen that the first system
x = x + (sin2 t)u satisfies all conditions of Proposition 3.17; while the property a(t, ) cV (t, )
fails for the second system x = x3 + (sin2 t)x2 u.
 STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS 693

Remark 3.18 In Theorem 1 and Propositions 3.16 and 3.17, the function p in (20) is
required to be a class P, function. As in the discussion for Corollary 3.11, one may relax this
requirement slightly by converting the system to

x = f (t, x) + p1 (t)g(t, x)v (24)

with p1 (t) = p(t)q0 (t) and u = q0 (t)v. Suppose system (24) satisfies the requirements of
Theorem 1 or Proposition 3.16, then the modified feedback u = q0 (t)(a(t, ), b(t, )) globally
uniformly stabilizes the system (20).

4 Concluding Remarks
In this paper, we have proposed a control Lyapunov function approach to the global sta-
bilization problem for general nonlinear time-varying systems. Sufficient conditions are given
under which explicit feedback laws generated by the universal formula can be found to stabi-
lize globally and uniformly the system in question. Topics for future research include, among
many others, feedback stabilization of time-varying systems with restricted inputs, and adaptive
control of nonlinear time-varying systems with unknown parameters.

Appendix A Proof of Lemma 3.6

Let V be a clf for system (11) that satisfies the small control property.
The proof of Theorem 1 in [2] shows that the function (a, b) given by (10) is analytic on
the set
S := {(a, b) R2 : b 6= 0 or a < 0}.
Consequently, the feedback function k(t, ) := (a(t, ), b(t, )) is smooth on the set R+ (Rn \
{0}). Below we show that the function k(t, ) is continuous on the set R+ {0}.
Let t0 0 and > 0 be given. We will show that there exists some > 0 such that

|k(t, )| < , |t t0 | < , || < . (25)

Let B1 = {(t, ) : b(t, ) 6= 0}. Since k(t, ) = 0 whenever b(t, ) = 0, it is enough to find
such that (25) holds at the points (t, ) B1 .
By the small control property, there is some > 0 such that for every || < and every
t (t0 , t0 + ), there is some || < /3 such that

a(t, ) + b(t, ) < 0. (26)

Without loss of generality, one may also assume (by continuity and the fact that b(t0 , 0) = 0)
that > 0 is chosen so that

|b(t, )| < , |t t0 | < , || < .
3
Observe that (26) implies that
|b(t, )|
|a(t, )|
3
for all |t t0 | < and all || < for which a(t, ) 0. Hence, for every |t t0 | < and every
|| < for which (t, ) B1 and a(t, ) 0, one has
694 ZHONGPING JIANG YUANDAN LIN YUAN WANG

 a(t, ) + pa2 (t, ) + b4 (t, ) 

 
|(a(t, ), b(t, ))| =  
b(t, )
2 |a(t, )| + b2 (t, )

|b(t, )|
2
< + = .
3 3
There remains the case when a(t, ) < 0. In this case, it holds that
p
0 < a(t, ) + a2 (t, ) + b4 (t, ) b2 (t, ),

and it follows that

|(a(t, ), b(t, ))| |b(t, )| <
for all |t t0 | < , all || < for which a(t, ) < 0. The proof of Lemma 3.6 is thus completed.

Appendix B Proof of Theorem 1

Let V be a clf for (11) that satisfies all assumptions as in Theorem 1. With the feedback
law u = (a(t, ), b(t, )), it holds that
p

a(t, ) + p(t)b(t, )u = a(t, ) p(t) a(t, ) + a2 (t, ) + b4 (t, ) .

Let
a+ (t, ) = max{a(t, ), 0}, a (t, ) = max{a(t, ), 0}.
Then a(t, ) = a+ (t, ) a (t, ).
Consider the case when a(t, ) 0, that is, a(t, ) = a+ (t, ). For this case,
p

a(t, ) + p(t)b(t, )u = a+ (t, ) p(t) a+ (t, ) + (a+ )2 (t, ) + b4 (t, )
p
a(t, ) p(t) a2 (t, ) + b4 (t, )
V (t, ) p(t) cV (t, )
(p(t) 1)V (t, ).

We next consider the case when a(t, ) 0, that is, when a(t, ) = a (t, ). For this case,
consider first the case when a (t, ) b2 (t, ). It then follows that

1 p 2 cV (t, )
a (t, ) a (t, ) + b4 (t, ) M V (t, ) p(t)V (t, ).
2 2
Hence,
p

a(t, ) + p(t)b(t, )u = a (t, ) p(t) a (t, ) + (a )2 (t, ) + b4 (t, )
a (t, ) p(t)V (t, ).

Finally, we consider the case when a(t, ) = a (t, ) 0 and a (t, ) b2 (t, ). In this
case one has
1 p 2
a (t, ) a (t, ) + b4 (t, ),
2
 STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS 695

and therefore,
p

a(t, ) + p(t)b(t, )u = a (t, ) p(t) a (t, ) + (a )2 (t, ) + b4 (t, )
 1 p 2
a (t, ) p(t) 1 (a ) (t, ) + b4 (t, )
2
 1 
p(t) 1 cV (t, ) p(t)V (t, ).
2
We have thus shown that in all cases, it holds that, with u = (a(t, ), b(t, )),
V V
(t, ) + (f (t, ) + p(t)g(t, )(t, )) (p(t) 1)V (t, )
t
for the closed-loop system of (20). By Proposition 2.3, the closed-loop system is UGAS. The
proof of Theorem 1 is thus completed.

Appendix C Proof of Proposition 3.16

When a(t, ) 0, a(t, ) = a (t, ) = |a(t, )|. Without loss of generality, we assume
that M 1. With u = (a(t, ), b(t, )) given by the universal formula, the following holds:
p

a(t, ) + p(t)b(t, )(a(t, ), b(t, )) a (t, ) p(t) a (t, ) + a2 (t, ) + b4 (t, )
p(t) b4 (t, )
a (t, ) p(t) p
M a (t, ) + a2 (t, ) + b4 (t, )

p(t) b4 (t, ) 
a (t, ) + p
M a (t, ) + a2 (t, ) + b4 (t, )

p(t) (a (t, ))2 + b4 (t, )

p
M a (t, ) + a2 (t, ) + b4 (t, )
p
p(t) a2 (t, ) + b4 (t, )

M 2
(||)
p(t) .
2M
Hence, the following holds for the closed-loop system of (20):
V V (||)
(t, ) + (f (t, ) + p(t)g(t, )(t, )) p(t) .
t 2M
By Proposition 2.2, the closed-loop system is UGAS. The proof of Proposition 3.16 is thus
completed.

Appendix D Proof of Proposition 3.17

p
Suppose that 0 a(t, ) cV (t, ) and a2 (t, ) + b4 (t, ) V (t, ) for all (t, ). Let
M > 0 be a constant as stated in Proposition 3.17. With u = M (a(t, ), b(t, )), one has
p

a(t, ) + p(t)b(t, )(a(t, ), b(t, )) = a(t, ) M p(t) a(t, ) + a2 (t, ) + b4 (t, )
p
cV (t, ) M p(t) a2 (t, ) + b4 (t, )
(M p(t) c)V (t, ).
696 ZHONGPING JIANG YUANDAN LIN YUAN WANG

Thus, the following holds for the closed-loop system of (20):

V V
(t, ) + (f (t, ) + M p(t)g(t, )(t, )) (M p(t) c)V (t, ).
t

By assumption, (M p() c) wP, . One concludes that the system is ugas. The proof of
Proposition 3.17 is thus completed.

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