Lyapunov Characterizations of Input to Output Stability
Eduardo Sontag
Dept. of Mathematics
Rutgers University
New Brunswick, NJ 08903
Yuan Wangy
Dept. of Mathematics
Florida Atlantic University
Boca Raton, FL 33431
sontag@control.rutgers.edu
ywang@math.fau.edu
Abstract
This paper presents necessary and sucient characterizations of several notions of input to
output stability. Similar Lyapunov characterizations have been found to play a key role in
the analysis of the input to state stability property, and the results given here extend their
validity to the case when the output, but not necessarily the entire internal state, is being
regulated.
Keywords. Lyapunov functions, output stability, ISS, robust control
AMS subject classications. 93D05, 93D20, 93D09, 34D20
Running Head. Lyapunov Characterizations of Input to Output Stability
Submitted to SIAM Journal on Control and Optimization, January 1999.
1 Introduction
This paper concerns itself with systems with outputs of the general form
x_ (t) = f (x(t); u(t)); y(t) = h(x(t)) ;
(1)
where f : Rn Rm ! Rn and h : Rn ! Rp are both locally Lipschitz continuous, f (0; 0) = 0,
and h(0) = 0. In the paper [27] (see also [26]), the authors introduced several notions of output
stability for such systems. All these notions serve to formalize the idea of a \stable" dependence
of outputs y upon inputs (which may be thought of as disturbances, actuator or measurement
errors, or regulation signals). They dier in the precise formulation of the decay estimates and
the overshoot, or transient behavior, characteristics of the output. Among all of them, the one
of most interest is probably the one singled out for the name input to output stability, or ios,
for short.
Our main theorem in this paper provides a necessary and sucient characterization of the
ios property in terms of Lyapunov functions. In the process of obtaining this characterization,
we derive as well corresponding results for the variants of ios discussed in [27]. (The relationships between those variants, shown in [27], play a role in our proofs, but otherwise the two
papers are independent of each other.)
In the very special case when y = x, our concepts all reduce to the input to state stability
(iss) property. Much of iss control design, cf. [3, 4, 6, 7, 8, 9, 10, 14, 16, 17, 28], relies upon the
Lyapunov characterizations rst obtained in [13, 23]. Thus, it is reasonable to expect a similar
impact from the results given here for the more general case.
y
This research was supported in part by US Air Force Grant F49620-98-1-0242
This research was supported in part by NSF Grants DMS-9457826 and DMS-9403924
1
In order to review the dierent i/o stability concepts, let us make the following notational
conventions. Euclidean norms will be denoted as jxj, and kuk denotes the Lm1 -norm (possibly
innite) of an input u (i.e., a measurable and locally essentially bounded function u : I ! Rm ,
where I is a subinterval of R which contains the origin; if we do not specify the domain I of an
input u, we mean implicitely that I = R0 ). For each initial state 2 Rn and input u, we let
x(; ; u) be the unique maximal solution of the initial value problem x_ = f (x; u), x(0) = , and
write the corresponding output function h(x(t; ; u)) simply as y(; ; u). Given a system with
control-value set Rm , we often consider the same system but with controls restricted to take
values in some subset Rm ; we use M
for the set of all such controls. As usual, by a K
function we mean a function : [0; 1) ! [0; 1) that is strictly increasing and continuous, and
satises (0) = 0, by a K1 function one that is in addition unbounded, and we let KL be the
class of functions [0; 1)2 ! [0; 1) which are of class K on the rst argument and decrease to
zero on the second argument. When we state the various properties below, we always interpret
the respective estimates as holding for all inputs u and for all initial states 2 Rn .
With these conventions, we say that a system (1) is uniformly bounded input bounded state
stable, and write \ubibs" for short, if it is forward complete (for every initial state and input
u, the solution x(t; ; u) is dened for all t 0) and, for some function of class K, the following
estimate holds for all solutions:
jx(t; ; u)j maxf(jj); (kuk)g;
8t 0 :
(2)
As argued in [27], stability of internal signals is a routine constraint in regulation problems.
The following four output stability properties were discussed in [27]. A ubibs system is:
ios, or input to output stable , if there exist a KL-function and a K-function such that
jy(t; ; u)j (jj ; t) + (kuk);
8t 0
(3)
(the term (kuk) can be replaced by the norm of the restriction to past inputs (kuk[0;t]),
and the sum could be replaced by a \max" or two analogous terms);
olios, or output-Lagrange input to output stable if it is ios and, in addition, there exist
some K-functions 1 ; 2 such that
jy(t; ; u)j maxf1(jh()j); 2 (kuk)g;
siios, or state-independent input to output stable if there exist some 2 K such that
jy(t; ; u)j (jh()j ; t) + (kuk);
8t 0;
8t 0;
(4)
2 KL and some
(5)
ros, or robustly output stable (ros) if there are a smooth K1 -function and a 2 KL
such that
jy(t; ; d)j (jj ; t);
8t 0;
where we denote by y (; ; d) the output function of the system
x_ = g(x; d) := f (x; d(jyj)) ; y = h(x) ;
(6)
(7)
and we understand the estimate (6) as holding for all d 2 MB , where B = fjj 1g Rm .
2
The last concept corresponds to the preservation of output stability under output feedback
with \robustness margin" . It was shown in [27] that siios ) olios ) ios ) ros, and no
converses hold. We now introduce the associated Lyapunov concepts.
Denition 1.1 With respect to the system (1), a smooth function V : Rn ! R0 is:
an ios-Lyapunov function if there exist 1 ; 2 2 K1 such that
1 (jh( )j) V ( ) 2 (j j);
8 2 Rn ;
and there exist 2 K and 3 2 KL such that
V ( ) (jj) ) DV ( )f (; ) ?3 (V ( ); j j);
8 2 Rn ; 8 2 Rm ;
an olios-Lyapunov function if there exist 1; 2 2 K1 such that
1 (jh( )j) V ( ) 2 (jh( )j);
8 2 Rn ;
and there exist 2 K and 3 2 KL such that (9) holds;
an siios-Lyapunov function if there exist 2 K and 3 2 K such that
V ( ) () ) DV ( )f (; ) ?3 (V ( ));
8 2 Rn ; 8 2 Rm ;
and there exist 1 ; 2 2 K1 such that (10) holds;
an ros-Lyapunov function if there exist 2 K and 3 2 KL such that
jh()j (jj) ) DV ()f (; ) ?3(V (); jj); 8 2 Rn ; 8 2 Rm ;
and there exist 1 ; 2 2 K1 such that (8) holds.
(8)
(9)
(10)
(11)
(12)
2
Observe that, if an estimate (8) holds, then (12) is implied by (9), in the sense that if and 1 are as in the former, then e := ?1 1 can be used as \" for the latter. Note also that,
provided that (10) holds, condition (9) is equivalent to the existence of 2 K and 3 2 KL so
that
jh()j (jj) ) DV ()f (; ) ?3(V (); jj) :
Our main results can be summarized as follows.
Theorem 1 A ubibs system is:
1. ios if and only if it admits an ios-Lyapunov function;
2. olios if and only if it admits an olios-Lyapunov function;
3. ros if and only if it admits an ros-Lyapunov function; and
4. siios if and only if it admits an siios-Lyapunov function.
The proofs are provided in Section 4.
3
2 Remarks on Rates of Decrease
In properties (9) and (12), the decay rate of V (x(t)) depends on the state and on the value of
V (x(t)). The main role of 3 is to allow for slower convergence if V (x(t)) is very small or if
x(t) is very large. We rst note two simplications.
Remark 2.1 Inequality (9) holds for some 3 2 KL if and only if there exist K-functions 1 ; 2
such that
V ( ) (jj)
) DV ()f (; ) ? 1+1(V ((j))j)
(13)
2
for all 2 Rn and all 2 Rm . This follows from Lemma A.2, proved in the Appendix. A
similar remark applies to (12).
2
Remark 2.2 Suppose V is an ios-Lyapunov function for the system satisfying (8) with some
1 ; 2 2 K1 and (13) with some ; 1 ; 2 2 K. By the proof of Lemma 11 together with Lemma
12 in [16], one sees that there exists a C 1 K1 -function such that 0 (s)1 (s) (s) for all
s 0. Let W = V . Then W is a C 1 function satisfying the following:
(1 (jh( )j)) W ( ) (2 (j j));
and
8 2 Rn ;
(14)
) DW ()f (; ) ? 1 +W(()jj)
2
for all 2 Rn and all 2 Rm , where 1 = 2 K. This shows that if a system admits an
W ( ) 1 (jj)
ios-Lyapunov function, then it admits one satisfying inequality (14). A similar remark applies
to (12).
2
Obviously, a function which satises a decay estimate of the stronger form
V ( ) (jj)
) DV ()f (; ) ?(V ())
(15)
for some ; 2 K is in particular an ios Lyapunov function. It is thus natural to ask if there
always exists, for an ios system, a function with this stronger property. We now show, by
means of an example, that such functions do not in general exist. Consider for that purpose
the following two-dimensional single-input system:
2x + u
(16)
x_ 1 = 0; x_ 2 = ? 2 2 ; y = x2 :
1 + x1
This system is ios, because with V (x) := x22 , it holds that
V ( ) 2
) DV ()f (; ) = ?2x2 21x+2 +x2u ? 12V+(x)2 :
1
1
Namely, V is an ios-Lyapunov function for the system.
Suppose that system (16) would admit an ios-Lyapunov function W with a decay estimate
as in (15), i.e., there exist some ; 2 K such that
W ( ) (jj)
) DW ()f (; ) ?(W ()):
4
(17)
Without loss of generality, we may assume that 2 K1 . In particular, we have that
DW ( )f (; ??1 (W ( ))) ?(W ( ))
(18)
for all 2 R2 . Fix any 1 2 R, and consider the one-dimensional dierential equation
2x ? ?1 (W (1 ; x2 )) :
x_ 2 = ? 2
(19)
1 + 12
Since W (1 ; x2 (t)) ! 0 (because of (18)) and as 1 (j2 j) W (1 ; 2 ) for all (for some 1 2 K),
it follows that x2 (t) ! 0 as t ! 1. This implies that ?1 (W (1 ; 2 )) < 22 for all 1 2 R and
2 > 0. Together with (17), this implies that there exists some 2 KL such that, for every
trajectory of (16) with u(t) 0, it holds that
jx2(t)j (jx2(0)j ; t);
for all = (x1 (0); x2 (0)) such that x2 (0) > 0. This is impossible, as it can be seen that, when
u(t) 0, x2 (t) = x2 (0)e?2t=(1+(x1 (0))2 ) whose decay rate depends on both x2 (0) and x1 (0).
Observe that, if we let U (1 ; 2 ) := [(1 + 12 ) j2 j](1+12 ) , then one obtains the following
estimate:
j2j jj ) DU ()f (; ) ?U ()
(20)
for all 1 2 R, 2 6= 0, and all 2 R. (The function U is not smooth on the set where U ( ) = 0,
but using a routine smoothing argument, one may easily modify U to get a smooth Lyapunov
function.) This U is not an example of a W as here (which, in any case, we know cannot exist),
because (20) only means that U is an ros-Lyapunov function, not necessarily an ios-Lyapunov
function (since the comparison is between j2 j and jj rather than between a function of U and
jj).
Finally, we observe that property (9) in the ios-Lyapunov denition may be repharased as
follows:
V ( ) > e(jj) ) DV ( )f (; ) < 0;
8 2 Rn ; 8 2 Rm
(21)
where e(s) := (s) (for any arbitrary chosen 2 (0; 1)). This statement is obviously implied
by (9). Conversely, if V satises this property, then there is an 2 KL so that (9) holds; this
follows from Lemma A.5 given in the Appendix.
3 Uniform Stability Notions
There is a key technical result which underlies the proofs of all our converse Lyapunov theorems. It requires yet another set of denitions, which correspond to stability uniformly on all
\disturbance" inputs.
Denition 3.1 A system (1) is uniformly output stable with respect to inputs in M
, where
is a compact subset of Rm , if:
it is forward complete, and
there exists a KL-function such that
jy(t; ; u)j (jj ; t);
holds for all u and all 2 Rn .
5
8t 0
(22)
If, in addition, there exists 2 K such that
jy(t; ; u)j (jh()j); 8 t 0
(23)
holds for all trajectories of the system with u 2 M
, then the system is output-Lagrange
uniformly output stable with respect to inputs in M
. Finally, if strengthens (22) to
jy(t; ; u)j (jh()j ; t); 8 t 0
(24)
holding for all trajectories of the system with u 2 M
, then the system is state-independent
uniformly output stable with respect to inputs in M
.
Theorem 2 Let be a compact subset of Rm , and suppose that a system (1) is uniformly output
stable with respect to inputs in M
. Then the system admits a smooth Lyapunov function V
satisfying the following properties:
there exist 1; 2 2 K1 such that
1 (jh( )j) V ( ) 2 (j j);
8 2 Rn ;
(25)
there exists 3 2 KL such that
DV ( )f (; ) ?3 (V ( ); j j);
8 2 Rn ; 8 2 :
(26)
Moreover, if the system is output-Lagrange uniformly output stable with respect to inputs in
M
, then (25) can be strengthened to
1 (jh( )j) V ( ) 2 (jh( )j);
8 ;
(27)
for some 1 ; 2 2 K1 . Finally, if the system is state-independent uniformly output stable with
respect to inputs in M
, then (25) can be strengthened to (27) and also (26) can be strengthened
to:
for some 4 2 K.
DV ( )f (; ) ?4 (V ( ));
8 2 Rn ; 8 2 :
(28)
The proof of this theorem will be postponed until Section 4.6.
4 Proof of Theorem 1
We prove the various parts of the theorem after establishing a preliminary simple fact.
4.1 A Small-Gain Lemma for Output-Lagrange Stability
Lemma 4.1 For every system which satises (4), there exist a K-function and a K1-function
such that, for any and any u, if ju(t)j (jy(t; ; u)j) for almost all t 0, then
jy(t; ; u)j (jh()j); 8 t 0:
(29)
6
Proof. Assume system (1) satises (4) for some 1 ; 2 2 K. We may suppose, without loss of
generality, that 2 2 K1 and 1 (s) s. Let
(s) = 2?1 (1?1 (s=2)=2):
Below we show that (29) holds for such a choice of , and = 1 . Pick any and u such that
ju(t)j (jy(t; ; u)j). Let x(t) denote the corresponding trajectory, and y(t) = h(x(t)).
Consider the case h( ) 6= 0. It is enough to show that the supremum t1 of the times s 0 for
which (ju(t)j) jh( )j =2 a.e. on [0; s] is innite, since this together with (4) will imply that
jy(t)j 1(jh()j) for all t 0. If it were the case that t1 < 1 then, since ju(t)j jh()j =2
almost everywhere on [0; t1 ], it follows from (4) that jy(t)j 1 (jh( )j) for all t 2 [0; t1 ].
By continuity, there exists some > 0 such that jy(t)j 21 (jh( )j) for all t 2 [0; t1 + ].
Consequently,
2 (ju(t)j) 2 ((jy(t)j)) 2 ((21 (jh( )j))) jh( )j =2
for almost all t 2 [0; t1 + ]. This contradicts the denition of t1 . Hence, t1 = 1.
The case when h( ) = 0 is similar. Suppose that ju(t)j (jy(t)j) almost everywhere,
but there is some t1 > 0 and some " > 0 such that jy(t1 )j ". Then t2 = inf ft : jy(t)j 1?1 ("=2)g 2 (0; t1 ), and jy(t2 )j = 1?1 ("=2). Using the rst part of the proof with the initial
time t1 , we obtain that jy(t)j 1 (jy(t2 )j) "=2 for all t t2 , contradicting the assumption
that jy(t1 )j ". It follows that y(t) 0.
4.2 Proof of Theorem 1, Part 1.
Necessity. Consider an olios system (1). By Lemma 4.1, there exist 1 2 K1 and 2 K such
that, for any and any u such that ju(t)j 1 (jy(t; ; u)j) a.e., (29) holds. Since the system
is olios, and in particular, ios, and as shown in [27], any ios system is necessarily also ros,
there exists some smooth K1 -function 2 such that the system
x_ = f (x; d2 (jyj)); y = h(x);
(30)
where d 2 MB , is forward complete, and there exists some 2 KL such that, for all trajectories
x2 (t; ; u) with the output functions y2 (t; ; u), it holds that
y2 (t; ; d)
8 t 0; 8 2 Rn ; 8 d 2 MB :
Let 3 (s) = minf1 (s); 2 (s)g, and let () be any smooth K1 -function so that (s) 3 (s)
(jj ; t);
for all s. Then, for the system
x_ = f (x; d(jyj)); y = h(x);
(31)
where d 2 MB , it holds that
y (t; ; d) (j j ; t) and
y (t; ; d) (jh( )j);
8 t 0:
Observe that the rst inequality together with the ubibs property implies that system (31) is
forward complete, since on any nite interval, one has
jx (t; ; d)j maxf(jj); (kdk kyk)g maxf(jj); ( (jj ; 0))g
(where we write x (; ; d) for the state trajectory associated to the system). Hence, system (31)
is uos. Applying Theorem 2, one sees that there exists some smooth function V such that
7
there exist 1 ; 2 2 K1 such that
1 (jh( )j) V ( ) 2 (jh( )j);
8 ;
(32)
DV ( )f (; (jh( )j)) ?3 (V ( ); j j)
(33)
there exist some 3 2 KL such that
for all 2 Rn and all j j 1.
It then follows that
DV ( )f (; ) ?3 (V ( ); j j)
whenever jj (jh( )j), or equivalently, whenever jh( )j ?1 (jj). Let = ?2 1 ?1 . Then
one has:
V ( ) (jj) ) DV ( )f (; ) ?3 (V ( ); j j)
for all and all . Hence, V is an olios-Lyapunov function for the system.
Suciency. Let V be an olios-Lyapunov function for system (1). Let 1 ; 2
2 K1 such
that (10) holds. By (9), and arguing as in Remark 2.1, one also knows that there exists some
1 and 2 2 K1 such that
(V ( ))
V ( ) (jj) ) DV ( )f (; ) ? 1
(34)
1 + 2 (j j)
for all and .
Let 2 KL be as in Lemma A.4 for the function 1 . Pick any initial state and any u.
Let x(t) and y(t) denote the ensuing trajectory and output function respectively. If for some
t1 0, V (x(t1 )) (kuk), then V (x(t)) (kuk) for all t t1 . (Proof: pick any " > 0. If
t1 := inf ft > t1 j V (x(t)) > (kuk) + "g is nite, then V (x(t)) > (kuk) for all t in some left
neighborhood of t2 , so DV (x(t))=dt < 0 and V (x(t)) > V (x(t2 )) for such t, contradicting its
minimality. As " was arbitrary, the claim follows.) Now let
ft 0 : V (x(t)) (kuk)g
with the understanding that et = 1 if V (x(t)) > (kuk) for all t 0. Then
V (x(t)) (kuk);
8 t et;
e
t = inf
and on [0; et), it holds that
(V (x(t)))
d
V (x(t)) ? 1
dt
1 + 2 (jx(t)j) :
Since the system is ubibs, there exists some such that (2) holds. Hence,
1 (V (x(t)))
d
V (x(t)) ?
dt
1 + maxfe2 (j j); e2 (kuk)g
for all t 2 [0; et), where e2 = 2 . It then follows Lemma A.4 that
t
V (x(t)) V ( );
1 + maxfe2 (j j); e2 (kuk)g
8
(35)
for all t 2 [0; et).
Let v0 (s) = maxjjs V ( ). Then v0 is nonincreasing, v0 (0) = 0, and V ( ) v0 (j j). Note
then that
t
V ( );
1 + max
f
e2 (j j); e2 (kuk)g
t
t
max V (); 1 + e (jj) ; v0 (kuk); 1 + e (kuk)
2
2
max V (); 1 + et (jj) ; (v0(kuk); 0)
2
(consider two cases: j j kuk and j j kuk). This shows that
t
e
V (x(t)) max V ( );
1 + e2 (j j) ; 0 (kuk)
for all t 2 [0; et), where e0 (s) = (v0 (s); 0). Combining this with (35), one sees that
V (x(t)) max V ( );
t
1 + e2 (j j) ; e(kuk)
(36)
for all t 0, where e(s) = e(s) + (s). Using the fact that jh( )j ?1 1 (V ( )), we conclude
that
t
e
jy(t)j max jh()j ; 1 + e (jj) ; (kuk)
(37)
2
for all t 0, where e(s; r) = ?1 1 ( (2 (s); r)), and (s) = ?1 1 (e(s)).
Observe that the proof used in the suciency part of Theorem 1 in fact shows that if a system
admits an olios-Lyapunov function, then it satises an estimate of type (37). Combining this
with the necessity part, one concludes as follows:
Lemma 4.2 A ubibs system is olios if and only if there exist 2 KL; 2 K, and 2 K such
that
jy(t; ; u)j jh()j ; 1 + t(jj) + (kuk);
8 t 0;
holds for all trajectories of the system.
2
4.3 Proof of Theorem 1, Part 2.
Necessity. Consider an ios system (1). By Theorem 1 in [27], there exist some locally Lipschitz
map h0 and 2 K1 with the property that h0 ( ) (jh( )j) such that the system
x_ = f (x; u); y = h0 (x)
(38)
is olios. By Part 1 of this theorem, system (38) admits an olios-Lyapunov function V . This
means that there exist 1 ; 2 ; 2 K1 and 3 2 KL such that
1 (jh0 ( )j) V ( ) 2 (jh0 ( )j);
and
8 2 Rn ;
V ( ) (jj) =) DV ( )f (; ) ?3 (V ( ); j j):
9
To show that V is an ios-Lyapunov function, it only remains to show that V ( ) e1 (jh( )j)
for some e1 2 K1. But this follows immediately from the fact that jh( )j ?1 (h0 ( )). So
one can let e1 := 1 . Hence, V is indeed an ios-Lyapunov function for system (1).
Suciency. Let V be an ios-Lyapunov function for system (1). From the proof of Part 1 of
Theorem 1 (suciency), one can see that if V satises (34) for some ; 1 ; 2 2 KL, then there
exist e 2 KL; e2 ; e 2 K1 such that (36) holds. This means that the system
x_ = f (x; u); y = V (x)
is olios. Since V (x) 1 (jh( )j) for some 1 2 K1 , it follows that system (1) is ios.
4.4 Proof of Theorem 1, Part 3.
Necessity. Since the system (1) is ros, there is a smooth K1 -function such that (6) holds
for the corresponding system (7). As remarked in the necessity proof of Theorem 1, part 1,
one sees that system (7) is also forward complete, and hence, uos. By Theorem 2, system (7)
admits a smooth Lyapunov function V satisfying (25) and
8 2 Rn ; 8 jj 1;
DV ( )f (; (jyj)) ? 3 (V ( ); j j);
for some 3 2 KL. This is equivalent to
jyj ?1 (j j) ) DV ()f (; ) ? 3 (V (); jj);
8 2 R n ; 8 j j 2 R m :
Hence, one concludes that V is an ros-Lyapunov function for system (1).
Suciency. Let V be an ros-Lyapunov function. As in Remark 2.1, there exist ; 1 ; 2 2 K1
such that
(V ( ))
DV ( )f (; ) ? 1
1 + (j j)
2
whenever jh( )j (jj). Let = ?1 . Without loss of generality, one may assume that is
smooth. (Otherwise, one can always replace by a smooth K1 -function that is majorized by
.) It then follows that
(V ( ))
DV ( )f (; (jh( )j)) ? 1
1 + (j j)
for all 2
system
Rn
2
and all j j 1. This implies that for any trajectory x (t) = x (t; ; d) of the
where d 2 MB , it holds that
x_ = f (x; d(jyj)); y = h(x);
1 (V (x (t)))
d
V (x (t)) ?
dt
1 + 2 (x (t))
(39)
for all t 0. It follows immediately that V (x (t)) V ( ) for all t 0. Since V ( ) 1 (jh( )j)
for some 1 2 K1 , it follows that, for some 2 K1 ,
y
(jj); 8 t 0:
Since the system is ubibs, there exists some 0 2 K such that
x (t; ; d)) maxf0 (j j); 0 (kud k)g;
8 t 0;
(t)
10
(40)
where ud (t) = d(t)(jy(t)j). Combining this with (40), it follows that
x
(t; ; d) e(j j);
8 t 0;
where e(s) = maxf0 (s); 0 (((s)))g. Substituting this back into (39), one has
(V (x (t)))
d
V (x (t)) ? 1
dt
1 + 3 (j j) ;
8 t 0;
where 3 (s) = 2 (e(s)). Again, by Lemma A.4, one knows that there exists some 2 KL
(which depends only upon 1 ) such that
t
V (x (t)) V ( );
1 + 3 (j j) ;
8 t 0:
Together with the fact that jh( )j ?1 1 (V ( )), this yields
y
e
(t; ; d) (j j ; t);
8 t 0;
where e(s; r) = ?1 1 [ (2 (s); t=(1 + 3 (s)))] is in KL, and 2 is any K1 -function such that
V ( ) 2 (j j) for all . This shows that the system is ros.
4.5 Proof of Theorem 1, Part 4.
Necessity. Assume that a ubibs system (1) admits an estimate (5) for some 2 KL and some
2 K. Then it admits an estimate of type (4) with 1 (s) = 2 (s; 0) and 2 (s) = 2 (s). By
following the proof of Lemma 4.1, one can show that there exists some 2 K1 such that for
any and u, if (ju(t)j) (jy(t; ; u)j for almost all t 0, it holds that (ju(t)j) jh(2)j .
Again, without loss of generality, one may assume that is smooth. One then can show that
for the system
x_ (t) = f (x(t); d(t)(jy(t)j)); y(t) = h(x(t));
there exists e 2 KL so that, for all trajectories x (t; ; d), it holds that
y
(t; ; d)
e(jh()j ; t)
for all t 0. Applying the last part of Theorem 2 one sees that there exists V satisfying (27)
for some 1 ; 2 2 K1 and
DV ( )f (; (jy( )j)) ?3 (V ( ))
for all and all j j 1. This is equivalent to the existence of 2 K1 such that
V ( ) (jj) ) DV ( )f (; u) ?3 (V ( )):
(41)
Suciency. It is routine to show that if there is a smooth function V satisfying (27) and (41),
then the system admits an estimate of type (5).
11
4.6 Proof of Theorem 2
Consider the system
x_ (t) = f (x(t); u(t)); y = h(x(t));
(42)
where the input u takes values in a compact subset of Rm . Assume that the system is ubibs
and there exists some 2 KL such that (22) holds for all trajectories of (42). Let ! : Rn ! R0
be dened by
!( ) := sup fjy(t; ; u)j : t 0; u 2 M
g :
It then holds that
(43)
8 2 Rn ;
(44)
where 0 (s) = (s; 0). Moverover, if there exists some 2 K such that (23) holds for all
jh()j !() 0 (jj);
trajectories, then the above can be strengthened to
jh()j !() (jh()j); 8 2 Rn :
Observe that, for any 2 Rn , u 2 M
, and t1 0,
!(x(t1 ; ; u)) sup jy(t1 + t; ; v)j (j j ; t1 ):
t0;v2M
(45)
(46)
Also ! decreases along trajectories, i.e.,
!(x(t; ; u)) !( );
8t 0; 2 Rn ; u 2 M
:
(47)
Dene
D := f : y(t; ; u) = 0; 8 t 0; 8 u 2 M
g:
Then !( ) = 0 if and only if 2 D. For 2= D, it holds that
jy(t; ; u)j ;
!( ) =
sup
0tt ; u2M
(48)
where t = Tjj (!( )=2), and Tr (s) is dened as in Lemma A.1 associated with the function .
Lemma 4.3 The function !() is locally Lipschitz on Rn n D and continuous everywhere.
Proof. First notice that
lim !( ) !(0 );
8 0 2 R n ;
(49)
that is, !( ) is lower semi-continuous on Rn . Indeed, pick 0 and let c := !(0 ). Take any " > 0.
Then there are some u0 and t0 so that jy(t0 ; 0 ; u0 )j c ? "=2. By continuity of y(t0 ; ; u0 ), there
is some neighborhood Ue0 of 0 so that jy(t0 ; ; u0 )j c ? " for all 2 Ue0 . Thus !( ) c ? " for
all 2 Ue0 , and this establishes (49).
Fix any 0 2 Rn n D, and let c0 = !(0 )=2. Then there exists a bounded neighborhood U0
of 0 such that
!( ) c0 ;
8 2 U0 :
Let s0 be such that j j s0 for all 2 U0 . Then
!0
!( ) = sup fjy(t; ; u)j : t 2 [0; t1 ]; u 2 M
g ;
12
8 2 U0 ;
where t1 = Ts0 (c0 =2). By [13, Proposition 5.5], one knows that x(t; ; u) is locally Lipschitz in
uniformly on u 2 M
and on t 2 [0; t1 ], and therefore, so is y(t; ; u). Let C be a constant
such that
jy(t; ; u) ? y(t; ; u)j C j ? j ; 8 ; 2 U0; 8 0 t t1; 8 u 2 M
:
For any " > 0 and any 2 U0 , there exist some t;" 2 [0; t1 ] and some u;" such that
!( ) jy(t;"; ; u;")j + ":
It then follows that, for any ; 2 U0 , for any " > 0,
!( ) ? !() jy(t;"; ; u;")j + " ? jy(t; "; ; u;")j C j ? j + ":
Consequently,
By symmetry,
!( ) ? !() C j ? j ;
8 ; 2 U0 :
!() ? !( ) C j ? j ;
8 ; 2 U0 :
This proves that ! is locally Lipschitz on Rn n D.
We now show that ! is continuous on D. Fix 0 2 D. One would like to show that
lim !( ) = 0:
!0
(50)
Assume that this does not hold. Then there exists some "0 > 0 and a sequence fk g with
k ! 0 such that !(k ) > "0 for all k. Without loss of generality, one may assume that
jk j s1;
8 k;
for some s1 0. It then follows that
!(k ) = sup fjy(t; k ; u)j : t 2 [0; t2 ]; u 2 M
g ;
where t2 = Ts1 ("0 =2). Hence, for each k, there exists some uk 2 M
and some k 2 [0; t2 ] such
that
jy(k ; k ; uk )j !(k ) ? "0 =2 "0 =2:
Again, by the locally Lipschitz continuity of the trajectories, one knows that there is some
C1 > 0 such that
jy(t; k ; u) ? y(t; 0 ; u)j C1 jk ? 0j ;
Hence,
8 k 0; 8 0 t t2; 8 u 2 M
:
jy(k ; 0; uk )j "0 =4
for k large enough, contradicting the fact that y(t; 0 ; u) 0 for all u 2 M
. This shows
that (50) holds if 0 2 D.
Next, we pick any smooth and bounded function k : R0 ! R>0 whose derivative is everywhere positive, and dene W : Rn ! R0 by
W ( ) := sup f!(x(t; ; u))k(t) : t 0; u 2 M
g :
13
(51)
Corresponding to k there are two positive real numbers c1 < c2 such that k(t) 2 [c1 ; c2 ] for all
t 0, and so
c1 !( ) W ( ) c2 !( );
8 2 Rn ;
which implies that
c1 jh( )j W ( ) c2 0 (j j);
8 2 Rn :
(52)
Note, for future reference, that it is always possible to nd a bounded, positive, and decreasing continuous function () with (t) ! 0 as t ! 1, such that
k0 (t) (t) for all t 0 :
(53)
By (46), one knows that !(x(t; ; u)) ! 0 as t ! 1. It follows that there is some 0
such that
W ( ) = sup f!(x(t; ; u))k(t) : u 2 M
; 0 t g :
(54)
Furthermore, one can get the following estimate, where fTr g is a family of functions associated
to as in Lemma A.4.
Lemma 4.4 For any 62 D with jj r,
W ( ) = sup f!(x(t; ; u))k(t) : u 2 M
; 0 t g ;
where = Tr ( 2cc12 !( )).
Proof. If the statement is not true, then for any " > 0, there exists some t" > Tr ( 2cc12 !( ))
and some u" 2 M
such that
W ( ) !(x(t" ; ; u" ))k(t" ) + ":
This implies the following:
!( )
1 W ( ) 1 !(x(t ; ; u ))k(t ) + "
"
"
"
c
c
c
1
1
"
c2
!(x(t" ; ; u" )) +
c1
c1
!( ) "
= 2 +c :
1
1
c2 c1
"
!( ) +
c1 2c2
c1
Taking the limit as " ! 0 results in a contradiction.
Lemma 4.5 The function W () is locally Lipschitz on Rn n D and continuous everywhere.
Proof. Fix 0 62 D. Let K0 be a compact neighborhood of 0 such that K0 \ D = ;. Since ! is
continuous, it follows that there is some r0 > 0 such that !( ) > r0 for all 2 K0 , and hence,
W ( ) > r1 := c1 r0 for all 2 K0 . Let
T0 = Ts0
r1
8c2 ;
where s0 > 0 is such that j j s0 for all 2 K0 . Let C > 0 be such that
jy(t; ; u) ? y(t; ; u)j C j ? j ; 8 t 2 [0; T0 ]; 8 ; 2 K0 ; 8 u 2 M
:
14
Let
r
K1 = K0 \ : j ? 0 j 1
16Cc2 :
Fix any " 2 (0; r1 =4). Then, for any 2 K1 , there exist t;" 2 [0; T0 ] and u;" 2 M
such that
W ( ) !(x(t;" ; ; u;"))k(t;" ) + ":
Claim: For any ; 2 K1 , !(x(t;" ; ; u;" )) 8rc12 .
Proof. First we note that for any 2 K1 K0 ,
W ( ) ? " W ( )
2c r2 ;
!(x(t;"; ; u;" )) c2
2
r
1
where r2 := 2c2 . Thus, for each 2 K1 , there exists some v 2 M
and some > 0 such that
jy( ; x(t;"; ; u;"); v )j !(x(t;"; ; u;")) ? r2=2 r2 =2:
Observe that
y( ; x(t;" ; ; u;"); v ) = y( + t;"; ; v ;");
where v;" is the concatenation of u;" and v , i.e.,
t);
if 0 t < t;";
v;"(t) = vu;"(t(?
t;"); if t t;":
Noticing that jy(t; ; u)j r2 =2 for all t Ts0 (r2 =4), one concludes that + t;" < Ts0 (r2 =4) =
T0 . Note also that for any 2 K1 ,
jy( ; x(t;"; ; u;" ); v )j = jy( + t;"; ; v;")j
jy( + t;"; ; v;")j ? jy( + t;"; ; v;") ? y( + t;"; ; v;")j
r22 ? C j ? j
1 = r1 ? r1 = r1 :
r22 ? 2C 16rCc
2 4c2 8c2 8c2
This implies that !(x(t;"; ; u;" )) 8rc12 for all ; 2 K1 , as claimed.
According to [13, Proposition 5.1], there is some compact set K2 such that x(t; ; u) 2 K2
for all 0 t T0 , all 2 K1 and all u 2 M
. Let
K3 = K2 \ f : !( ) r1 =8c2 g:
Applying Lemma 4.3, one knows that there is some C1 > 0 such that
j!(1) ? !(2)j C1 j1 ? 2j ; 8 1; 2 2 K3:
Since for all ; 2 K1 and all 0 < " < r1 =4, x(t;"; ; u;" ) 2 K3 , we have:
j!(x(t;"; ; u;")) ? !(x(t;"; ; u;"))j C1 jx(t;"; ; u;") ? x(t;"; ; u;" )j ;
for all ; 2 K1 , all " 2 (0; r1 =4). Hence,
W ( ) ? W () !(x(t;" ; ; u;"))k(t;" ) ? !(x(t;"; ; u;" ))k(t;" ) + "
c2 j!(x(t;"; ; u;")) ? !(x(t;"; ; u;"))j + "
c2 C1 jx(t;"; ; u;") ? x(t;"; ; u;")j + "
c2 C1C2 j ? j + ";
15
where C2 > 0 is such a constant that jx(t; ; u) ? x(t; ; u)j C2 j ? j for all ; 2 K3 , all
t 2 [0; T0 ], all u 2 M
. Note that the above holds for any " 2 (0; r1 =4), and thus,
W ( ) ? W () C3 j ? j
for all ; 2 K1 , where C3 = c2 C1 C2 . By symmetry, one proves that
W () ? W ( ) C3 j ? j
for all ; 2 K1 .
To prove the continuity of W on D, it is enough to notice that for any 2 D, W ( ) = 0 and
jW () ? W ()j c2 !() ! 0;
as ! :
The proof of Lemma 4.5 is thus concluded.
Below we show that W is decreasing along trajectories. Pick any 62 D. Let 0 > 0 be such
that
!(x(t; ; v)) !( )=2; 8 t 2 [0; 0 ]; 8 v 2 ;
where v denotes the constant function v(t) v. (Observe that such a 0 exists because ! is
continuous.) Pick any 2 [0; 0 ], and let v = x(; ; v). For any " > 0, there exists some tv;"
and uv;" 2 M
such that
W (v )
!(x(tv;"; v ; uv;"))k(tv;" ) + "
= !(x(tv;" + ; ; uv;"))k(tv;" + ) 1 ? k(tv;"k(+t ) +? k)(tv;") + "
v;"
k(tv;" + ) ? k(tv;" )
+ ";
W () 1 ?
c2
(55)
where uv;" denotes the concatenation of v and uv;". Still for the xed and , and for any
r > j j, dene
c1
r
e
(56)
T; := max Tr
2c2 !(x(; ; v)) :
ve2
Notice that x(; ; ve) is jointly continuous as a function of (; ; ve). Since ! and Tr are both
r is continuous as a function of ,
continuous, this maximum is well dened and, moreover, T;
so in particular
To be xed! { It should be
it was
lim T r
!0+ ;
= Tr 2cc1 !( ) :
2
(57)
lim T r = Tr cc1 !( ) :
!0+ ;
2
r
Claim: tv;" + T; for all v 2 and for all " 2 (0; c41 !( )).
?
Proof. Assume that this is not true. Then there is some v 2 and some " 2 0; c41 !( )
r , and in particular,
such that tv;" + > T;
tv;" + Tr
c1
2c2 !(x(; ; v)) ;
16
from which it follows that
c
c
!(x(tv;" ; v ; uv;")) = !(x(tv;" + ; ; uv;" )) 1 !(x(; ; v)) = 1 !(v );
2c2
2c2
for some input function uv;" (which we can take to be the concatenation of v and uv;"; note
that the inequality follows from (46) and the denition of the functions Tr ).
By the denition of W , one has
1 W ( ) 1 !(t ; ; u )k(t ) + "
!( ) v
v
c1
c1 v;" v v;"
"
c2
!(tv;" + ; ; uv;") +
c1
c1
1 !( ) + " ;
2 v c1
v;"
c1
which is impossible, since " < c41 !( ) c21 !(v ). This proves the claim.
From (55), we have, for any v 2 D and for any " small enough,
W ( )
W (x(; ; v)) ? W ( ) ?
(t + c) + "
v;"
c2
for some c 2 (0; 1), where we used the mean value theorem in order to estimate the change in
k, and where is a function as in (53). Using the monotonicity of (), and the above claim,
one concludes
W (x(; ; v)) ? W ( )
r +"
? Wc() T;
?
2
for all " small enough. Letting " ! 0, one obtains:
W ( ) ? r T; ; 8 v 2 :
W (x(; ; v)) ? W ( ) ?
c
2
Thus one concludes that for any v 2 and any > 0,
W (x(; ; v)) ? W ( )
W ( )
r ):
?
(T;
c
2
Since W is locally Lipschitz on Rn n D, it is dierentiable almost everywhere on Rn n D, and
hence, for any v 2 , any r > j j, and any at which W is dierentiable,
W (x(; ; v)) ? W ( )
? lim W () (T r )
DW ( )f (; v) = lim
!0+
c2
!
0+ ;
r = ? W ( ) Tr c1 !( )
= ? Wc( ) lim+ T;
c2
c2
!0
2
? Wc() Tr cc12 W () = ?e3 (W (); r);
2
2
(58)
where e3 (s; r) = cs2 (Tr (c3 s)) with c3 = c1 =c22 . Since (58) holds for all r > j j, it follows that
DW ( )f (; v) ?e3 (W ( ); 2 j j);
for all v 2 and for almost all 2 Rn n D.
17
(59)
Since Tr (s) is dened for all r 0 and s > 0, one sees that e3 is dened on R>0 R0 .
Extend e3 to R0 R0 by letting e3 (0; r) := 0 for all r 0. By the continuity property of
and Tr (), one sees that e3 (; r) is continuous for each r. (The continuity at s = 0 follows
from e3 (s; r) = s (Tr (c3 s))=c2 s (0)=c2 for all s > 0.) Furthermore, since (Tr (c3 s)) is
nondecreasing in s, it follows that e3 (s; r) is of class K in s. Let 3 (s; r) = e3 (s; 2r)=(1 + r).
This function tends to zero as r ! 1, because e3 (s; r) is nonincreasing in r; thus 3 (s; r) is of
class KL. Moreover,
DW ( )f (; v) ? 3 (V ( ); j j);
8 2 Rn n D; 8 v 2 :
By Corollary A.3, there exists a continuous KL-function b3 such that
DW ( )f (; v) ?b3 (V ( ); j j);
8 2 Rn n D; 8 v 2 :
(60)
To complete the proof, we follow the strategy used in [13] to nd a smooth approximation
of W . First of all, by Theorem B.1 in [13], applied on Rn n D, there is a continuous function
W1 that is smooth on Rn n D such that
(61)
jW1() ? W ()j W2() ; 8 2 Rn n D ;
and
DW1 ( )f (; v) ?b3 (W ( ); j j)=2;
8 2 Rn n D ; 8 v 2 :
(62)
We extend W1 to all of Rn by letting W1 0 on D; thus, the approximation (61) holds on all
of Rn . (Note that W and b3 (V ( ); j j) are both continuous, so the result in [13] can indeed be
applied.)
Next, we appeal to Lemma 4.3 in [13]. This shows that there exists some 2 K1 with
0 (s) > 0 for all s > 0 such that W1 is smooth everywhere. Let V = W1 . It follows
from (52) and (61) that
1 (jh( )j) V ( ) 2 (j j);
8 2 Rn ;
where 1 (s) = (c1 s=2), 2 (s) = (2c2 0 (s)), and it follows from (61) and (62) that
DV ( )f (; ) ?0 (W1 ( ))b3 (W ( ); j j)=2 ?3 (V ( ); j j)
(63)
for all 2 Rn n D, all 2 , where
3 (s; r) =
0 (?1 (s))b3 (?1 (V ( ))=2; r)
:
2
Since V has local (actually, global) minima at all points in D, it follows that DV ( ) 0 on D,
so we know that the estimate (63) holds also on all of Rn .
Finally, observe that if there exists 2 K such that (23) holds for all trajectories of the
system, then (45) holds for all , which, in turn, implies that
c1 jh( )j W ( ) c2 (jh( )j);
8 2 Rn :
(64)
8 2 Rn ;
(65)
This results in the desired inequality:
1 (jh( )j) V ( ) 1 (jh( )j);
18
where 1 (s) = (2c2 (s)). This shows that if (23) holds for some 2 K, then property (25)
can be strengthened to property (27).
Finally, suppose that, in the above proof, one strengthens (22) to (24). Associated to the
function there are, as before, functions fTr g. Since also we have an estimate as in (22), there
are functions fTr g associated to a as in (22); without loss of generality, we will assume that
the same Tr 's work for both. Thus, we know that, provided t Tr (s), jy(t; ; u)j s whenever
jh()j r or jj r. The claim stated after Equation(57) holds now for all r > jh()j (instead
of merely if r > j j), because (46) can be strenghtened to
!(x(t1 ; ; u))
(jh()j ; t1) :
We now repeat the above proof to get a function W ( ) satisfying (64), and corresponding
to (60), one has now also
W ( )
;
DW ( )f (; v) ?b3 (W ( ); jh( )j) ?b3 W ( );
c1
for all 2 Rn n D and all v 2 . Therefore, on Rn n D,
DW ( )f (; v) ?4 (W ( ));
where e4 (s) = e3 (s; s=c1 ) is a continuous positive denite function. Using the same smoothing
argument as earlier, we can show that there is a smooth function V such that (65) holds for
some 1 ; 2 2 K1 , and (63) can be strengthened to
DV ( )f (; v) ?b4 (V ( ))
(66)
for all 2 Rn , all v 2 , where b4 () is some continuous positive denite function.
Now we modify the function V to get V1 so that V1 satises inequalities of type (64) and (66)
with b4 replaced by a K1 function 5 . For this purpose, let 0 () be a smooth K1 -function
such that 0 (s)b4 (s) 1 for s 1, and let
Rs
1 (s) = e 0 0 (s1 ) ds1 ? 1:
Dene V1 ( ) = 1 (V ( )). It holds that
b1 (jh( )j) V1 ( ) b2 (jh( j);
8 2 Rn ;
where b1 (s) = 1 (1 (s)), and b2 (s) = 1 (2 (s)), and
DV1 ( )f (; v) = ?(V1 ( ) + 1)0 (V ( ))b4 (V ( )) ?5 (V1 ( ));
for all 2 Rn , all v 2 , where 5 is any K1 function with the property that
5 (1 (s))
(1 (s) + 1)0 (s)b4 (s)
for all s 0 (such a K1 -function exists because (s + 1)0 (s)b4 (s) s for all s 1). Using V1
as a Lyapunov function, this completes the proof.
19
A Some Facts Regarding KL Functions
The following simple observation is proved in [27] and will be needed here too.
Lemma A.1 For any KL-function , there exists a family of mappings fTr gr0 such that
onto
for each xed r > 0, Tr : R>0 ?!
R>0 is continuous and strictly decreasing, and T0 (s) 0;
for each xed s > 0, Tr (s) is strictly increasing as r increases, and is such that (r; Tr (s)) <
s, and consequently, (r; t) < s for all t Tr (s).
Lemma A.2 For any KL function , there exist two K functions 1 and 2 so that
(s; t)
for all s 0 and all t 0.
1 +1(s)(t)
2
(67)
Proof. We assume that b := sups (s; 0) < 1 (otherwise, we rst nd a 0 with that
property, and prove the result for 0 ). We dene, for all s 0 and t 0,
e(s; t) :=
Z
t+1
t
(s; ) d :
Note that e is again of class KL, and e(s; t) (s; t) for all s; t. Let
(t) := sup e(s; t) :
s0
This is nite everywhere, since it is bounded by b. Moreover, it is a continuous function, because
e(t) :=
Z
t+1
t
( ) d
where is the decreasing function (not necessarily strictly) dened by (t) := sups0 (s; t).
We will write from now on \e(1; t)" instead of \e(t)". Finally, we let
(x) := maxfx; 0g
for all x 2 R, and introduce the following function:
1
2
e
; (y) ? (?x) ? (?y)
c : R ! R : (x; y) 7! ? ln (x)
where we understand e( 01 ; t) as e(t). As in [1], we let N denote the class of all functions
k : R ! R that are nondecreasing, continuous, and unbounded below. Note that c is of class N
on each variable separately. (Continuity follows from the continuity of each of e(1; ), e(s; )
for each s 0, and e(; t) for each t 0 as well as continuity of . The nondecreasing property
is clear, using that e(; t) for each t 0 and are nondecreasing, and that e(1; ) and e(s; )
for each s 0 are nonincreasing. Unbounded below follows from the fact that for x ! ?1 we
e
have c(x; y0 ) = a + x, where
a = (1; (y0 )) ? (?y0 ) and for y ! ?1 we have c(x0 ; y ) = a + y ,
1
where a = ? ln e (x0 ) ; 0 ? (?x0 ).
20
By Proposition 3.4 in [1], there is some k 2 N such that c(x; y) k(x) + k(y) for all x; y.
So, we can write, after using that e: (1=x; y) e?k(x) e?k(y) for all x; y > 0. Equivalently,
1 (s)
(s; t) 1 + 2 (t)
for all s; t > 0, when we dene
1 (s) := e?k(1=s)?k(0)
for all s > 0 and
2 (t) := ek(t)?k(0) ? 1
for all t 0. Observe that both of these functions are continuous, nonincreasing, and nonnegative. Moreover, 2 (0) = 0, so 2 is in K. From the inequality
[1 + 2 (0)] (s; 0) 1 (s)
for all s > 0, and the fact that (0; 0) = 0, we conclude that lims!0+ 1 (s) = 0, so we may
extend 1 by dening 1 (0) = 0, and thus 1 is in K as well.
As 1 and 2 in Lemma A.2 are continuous, we have, in particular:
Corollary A.3 For any KL-function , there is a (jointly) continuous KL-function 1 such
that (s; r) 1 (s; r) for all (s; r) 2 R0 R0 .
2
The following is a generalization of the comparison lemma given in [13]. It plays a role in
the proofs of suciency, which are the easier parts of the theorems.
Lemma A.4 For any K-function , there exists a KL function such that if y() is any locally
absolutely continuous function dened on some interval [0; T ] with y(t) 0, and if y() satises
the dierential inequality
y_ (t) ?c (y(t)) for almost all t 2 [0; T ]
(68)
for some c 0 with y(0) = y0 0, then it holds that
y(t) (y0 ; ct)
for all t 2 [0; T ].
Proof. First of all, by Lemma 4.4 in [13], for each 2 K, there exists 2 KL such that for any
locally absolutely continuous function z (t) 0, if it satises the inequality
z_ (t) ?(z (t))
on [0; T ], it holds that z (t) (z (0); t) for all t. (The statement in that reference applies to z
dened on all of [0; 1), but exactly the same proof works for a nite interval.)
Let y(t) be a function as in the statement of the lemma for some c > 0; T > 0. Let ye(t) =
y(t=c). Then ye is again locally absolutely continuous and nonnegative on [0; cT ]. Moreover, ye
satises the inequality
d
ye(t) ?(ye(t)):
dt
21
Hence,
ye(t) (ye(0); t)
for all t 2 [0; cT ]. This then implies that
y(t) (y(0); ct)
for all t 2 [0; T ].
Finally, we have this fact, mentioned when discussing decrease conditions:
Lemma A.5 Let V : Rn ! R be a C 1 positive denition function with the following property:
for some K function , it holds that
V ( ) (jj) and V ( ) 6= 0 ) DV ( )f (; ) < 0 :
Then, there is a function 2 KL so that:
V ( ) (jj) ) DV ( )f (; ) ?(V ( ); j j)
for all 2 Rn ; 2 Rm .
Proof. Without loss of generality, we assume that 2 K1 . Dene the set, for each s; t 0:
R(s; t) := f(x; u) : j j t; V ( ) s; jj ?1 (V ( ))g :
These sets are compact (possibly empty), for each s and t. Note the following properties:
s > s0 ) R(s; t) R(s0 ; t)
t > t0 ) R(s; t0 ) R(s; t) :
Now let
0 (s; t) = min ?DV ( )f (; )
(;)2R(s;t)
(with the convention that 0 (s; t) = +1 if R(s; t) = ;). Then, 0 (s; t) is nonincreasing in t and
nondecreasing in s. Moreover, (s; t) > 0 whenever s > 0 (by the hypothesis of the lemma).
Next let:
b(s; t) := minf0 (s; t); sg :
This function has the same monotonicity properties as 0 , it satises 0 (s; t) b(s; t) for all
s; t, and is nite-valued. It also satises b(s; t) 6= 0 for s > 0. Now pick
e(s; t) :=
Z
s
s?1
b(; t) d
(let b(s; t) := 0 for s < 0). This function still has the same monotonicity properties, satises
e(s; t) > 0 for s > 0, and it is continuous in s. It may not be strictly increasing in s, nor need
it converge to zero as t ! 0, so we obtain nally a KL function by dening:
se(s; t)
(s; t) :=
(1 + s)(1 + t) :
This satises the desired properties by construction, because
V ( ) (jj) ) DV ( )f (; ) ?(V ( ); jj)
and 0 b e pointwise.
22
References
[1] D. Angeli, E.D. Sontag, and Y. Wang, A characterization of integral input to state
stability, submitted. (Preliminary version in Remarks on integral input to state stability,
Proc. IEEE Conf. Decision and Control, Tampa, Dec. 1998, IEEE Publications, 1998, to
appear.)
[2] A. Bacciotti, \External stability of nonlinear systems," Proce. of IFAC Non-Linear
Control Systems Design Symposium, (NOLCOS '95), Tahoe City, CA, 1995.
[3] S. Battilotti, Robust stabilization of nonlinear systems with pointwise norm bounded
uncertainties: A control Lyapunov approach, IEEE Trans. Autom. Control, to appear.
[4] P. D. Christofides and A. R. Teel, Singular perturbations and input-to-state stability,
IEEE Trans. Automat. Control, 41(1996): 1645-1650.
[5] L. Hatvani, On the asymptotic stability by nondecrescent Ljapunov function, Nonlinear
Analysis, Theory, Methods and Appl, 8 (1984), pp. 67{77.
[6] A. Isidori, Global almost disturbance decoupling with stability for non minimum-phase
single-input single-output nonlinear systems, Systems & Control Letters, 28 (1996),
pp. 115{122.
[7] Z.-P. Jiang, A. Teel, and L. Praly, Small-gain theorem for ISS systems and applications, Mathematics of Control, Signals, and Systems, 7 (1994), pp. 95{120.
[8] H. K. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, second ed.,
1996.
[9] M. Krstic and H. Deng, Stabilization of Uncertain Nonlinear Systems , Springer-Verlag,
London, 1998.
[10] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive
Control Design, John Wiley & Sons, New York, 1995.
[11] V. Lakshmikantham, S. Leela, and A. A. Martyuk, Practical Stability of Nonlinear
Systems, World Scientic, New Jersey, 1990.
[12] Y. Lin, Lyapunov Function Techniques for Stabilization, PhD thesis, Rutgers, The State
University of New Jersey, New Brunswick, New Jersey, 1992.
[13] Y. Lin, E. D. Sontag, and Y. Wang, A smooth converse Lyapunov theorem for robust
stability , SIAM Journal on Control and Optimization, 34 (1996), pp. 124{160.
[14] W. M. Lu, A class of globally stabilizing controllers for nonlinear systems , Systems &
Control Letters, 25 (1995), pp. 13{19.
[15] W. M. Lu, A state-space approach to parameterization of stabilizing controllers for nonlinear systems , IEEE Trans. on Automatic Control, 40 (1995), pp. 1576{1588.
[16] L. Praly, and Y. Wang, Stabilization in spite of matched unmodelled dynamics and an
equivalent denition of input-to-state stability, Math. of Control, Signals, and Systems 9
(1996): 1-33.
23
[17] R. Sepulchre, M. Jankovic, and P.V. Kokotovic, Integrator forwarding: a new
recursive nonlinear robust design, Automatica 33(1997): 979-984.
[18] E. D. Sontag, Smooth stabilization implies coprime factorization , IEEE Transactions on
Automatic Control, AC-34 (1989), pp. 435{443.
[19]
, Some connections between stabilization and factorization , in Proc. IEEE Conf.
Decision and Control, Tampa, Dec. 1989, IEEE Publications, 1989, pp. 990{995.
[20]
, On the input-to-state stability property , European Journal of Control, 1 (1995),
pp. 24{36.
, Comments on integral variants of iss , Systems & Control Letters, 34(1998): 93-100.
[21]
[22]
, Mathematical Control Theory, Deterministic Finite Dimensional Systems , Second
Edition, Springer-Verlag, New York, 1998.
[23] E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property ,
Systems & Control Letters, 24 (1995), pp. 351{359.
, Output-to-state stability and detectability of nonlinear systems, Systems & Control
[24]
Letters, 29 (1997), pp. 279{290.
[25]
, New characterizations of the input to state stability property , IEEE Transactions on
Automatic Control, 41 (1996), pp. 1283{1294.
[26]
, A notion of input to output stability, Proc. European Control Conf., Brussels, July
1997, Paper WE-E A2, CD-ROM le ECC958.pdf, 6 pages.
, Notions of input to output stability , submitted.
[27]
[28] Tsinias, J., Input to state stability properties of nonlinear systems and applications to
bounded feedback stabilization using saturation, ESAIM Control Optim. Calc. Var. 2(1997):
57-85.
[29] V. I. Vorotnikov, Stability and stabilization of motion: Research approaches, results,
distinctive characteristics , Automation and Remote Control, 54 (1993): 339{397.
[30] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic
Solutions, Springer-Verlag, New York, 1975.
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