Set-Valued Analysis 7: 375–405, 1999.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
375
Smooth Lyapunov Functions for Discontinuous
Stable Systems
LIONEL ROSIER
Laboratoire d’Analyse Numérique et EDP, Université Paris 11, Bât. 425, 91405 Orsay Cedex,
France. e-mail: [email protected]
(Received: 7 July 1999)
Abstract. It has been proved that a differential system dx/dt = f (t, x) with a discontinuous righthand side admits some continuous weak Lyapunov function if and only if it is robustly stable. This
paper focuses on the smoothness of such a Lyapunov function. An example of an (asymptotically)
stable system for which there does not exist any (even weak) Lyapunov functions of class C 1 is given.
In the more general context of differential inclusions, the existence of a weak Lyapunov function of
class C 1 (or C ∞ ) is shown to be equivalent to the robust stability of some perturbed system obtained
in introducing measurement error with respect to x and t. This condition is proved to be satisfied by
most of the robustly stable systems encountered in the literature. Analogous results are given for the
Lagrange stability. As an application to the study of the links between internal and external stability
for control systems, an extension of a result by Bacciotti and Beccari is obtained by means of a
smooth Lyapunov function associated with a robustly Lagrange stable system.
Mathematics Subject Classifications (1991): 34A60, 34D20, 93D05, 93D25.
Key words: converse first Lyapunov theorem, smooth Lyapunov function, Lagrange stability,
bounded input bounded state stability, differential inclusion, Filippov’s solution.
1. Introduction
Let us consider the ODE
ẋ = f (t, x),
(1.1)
where t ∈ R+ is time, x ∈ Rn is the state variable, ẋ = dx/dt and f is some field
defined on R+ × Rn . Let Q be an open subset of Rn and let V = V (t, x) be a
function defined on R+ × Q.
DEFINITION 1. V is said to be a weak Lyapunov function for (1.1) on R+ × Q
if
!
"
!
"
t1 ! t2 ⇒ V t1 , x(t1 ) " V t2 , x(t2 )
for each pair of nonnegative numbers t1 , t2 and each solution x(t): [t1 , t2 ] → Q
of (1.1). If, moreover, Q = {x : &x& < h} for some h > 0 (resp. Q = Rn \ {0})
376
LIONEL ROSIER
and if there exist two increasing continuous maps a: R+ → R+ and b: R+ → R+
such that
(I) a(0) = 0, a(r) > 0 for r > 0,
(II) b(0) = 0 (resp. a(r) → +∞ as r → +∞),
(III) a(&x&) ! V (t, x) ! b(&x&) for t " 0, x ∈ Q,
then V is said to be a weak Lyapunov function in the small (resp. in the large).
A well-known version of the first Lyapunov theorem asserts that if there exists
a weak Lyapunov function in the small for (1.1), then the origin is a (uniformly)
stable equilibrium position for (1.1). The question of whether such a theorem is
invertible, with some (hopefully) smooth Lyapunov function, constitutes an important issue in control theory. (Indeed, many applications of stabilization theory
are based on the so-called Jurdjevic–Quinn method for the design of the feedback,
which, in turn, rests on the knowledge of a smooth weak Lyapunov function for
the unforced associated system.) The first positive answers to this question were
given independently by Krasovski [12], Kurzweil [13] and Yoshizawa [23]. In these
papers, rather strong assumptions are made as far as the regularity of the field f is
concerned. (For instance, f is assumed to be C 1 in [13], or continuous and locally
Lipschitz continuous w.r.t. x in [23].) These assumptions guarantee the uniqueness
of the solution of (1.1) issuing from x0 at t = t0 for any pair (t0 , x0 ), and also
the smoothness of the Lyapunov function which is constructed. It turns out that
the situation is more involved when f is only assumed to be continuous. Indeed
Kurzweil and Vrkoč [15] have shown that a stable system (1.1) with a continuous
r.h.s. may fail to have a continuous weak Lyapunov function V = V (t, x). In fact
the first Lyapunov theorem may be inverted provided that the stability concept is
slightly strengthened. Let us denote by x(t; t0 , x0 ) any solution of (1.1) such that
x(t0 ) = x0 .
DEFINITION 2. Assume that x = 0 is an equilibrium position for (1.1). We say
that the origin is robustly stable for (1.1) if there exist a sequence (G−i )i=0,1,2,...
of open sets in R+ × Rn and two sequences (a−i )i=0,1,2,... , (b−i )i=0,1,2,... of real
numbers such that
(i) 0 < b−i−1 < a−i ! b−i for each i = 0, 1, 2, . . . and b−i → 0 as i → +∞;
(ii) R+ × {x : &x& < a−i } ⊂ G−i ⊂ R+ × {x : &x& < b−i } for each i =
0, 1, 2, . . .;
(iii) for each i = 0, 1, 2, . . ., each initial pair (t0 , x0 ) ∈ G−i and each solution
x(·; t0 , x0 ), one has (t, x(t; t0 , x0 )) ∈ G−i for each t " t0 .
Obviously the robust stability implies the usual stability and it may be proved
that these two concepts agree when the continuous field f is also assumed to
be locally Lipschitz continuous w.r.t. x. The main result in [15] asserts that for
any continuous field f = f (t, x) such that f (t, 0) = 0 for any t, the existence
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
377
of a continuous weak Lyapunov function in the small V = V (t, x) for (1.1) is
equivalent to the robust stability of the origin. This result is extended in [6] to
+
n
n
discontinuous systems, i.e. f ∈ L∞
loc (R × R , R ) and solutions are taken in
Filippov’s sense [11]. The aim of this paper is to investigate the existence of a
smooth Lyapunov function for a discontinuous robustly stable system.
In the (somewhat different) context of asymptotic stability, significant progress
have recently been made by Clarke et al. in the construction of smooth Lyapunov
functions for discontinuous (autonomous) systems. Indeed, it follows from [8,
Thm. 1.2] that a globally asymptotically stable system ẋ = f (x) (with f ∈
n
n
L∞
loc (R , R ) and solutions of (1.1) taken in Filippov’s sense) admits a (strong)
Lyapunov function of class C ∞ . Notice that such a result cannot be generalized to
time-dependent systems, as is shown by the following result (to be proved in this
paper).
PROPOSITION 1. There exists a bounded Borel map λ: R → R such that the
origin is globally asymptotically stable (hence, robustly stable) for the system
ẋ = λ(t)x
(1.2)
and such that there does not exist any (even weak) Lyapunov function in the large
(or in the small) of class C 1 .
Hence, the local Lipschitz continuity of a (strong) Lyapunov function for a discontinuous (time-dependent) asymptotically stable system (1.1), asserted by [18,
Thm. 2], cannot be improved in general. Notice that the counterexample given in
Proposition 1 proves to be rather pathological, since the restriction of the function
λ to the complement of any zero measure set is nowhere continuous. If we intend
to construct a smooth Lyapunov function, some additional assumption has to be
made as far as the regularity of f w.r.t. time is concerned.
+
n
m
DEFINITION 3. Let f ∈ L∞
loc (R × R , R ), m, n " 1. The function f is said
to be essentially continuous w.r.t. time (e.c.t. in short) if there exists a (Borel) set
N ⊂ R+ × Rn of measure zero such that for each pair (t0 , x0 ) ∈ R+ × Rn and for
each ε > 0, there exists δ > 0 such that for each pair (t, x) ∈ R+ × Rn

|t − t0 | + &x − x0 & < δ 
(t, x) )∈ N
⇒ &f (t, x) − f (t0 , x)& < ε.
(1.3)

(t0 , x) )∈ N
Roughly speaking, the function f is e.c.t. if the restriction of f to the complement
of some set of measure zero is continuous w.r.t. time, in a (locally) uniform way
w.r.t. x. Such a property is fulfilled when, for instance, f takes the form
f (t, x) := h(u(t), v(x)),
n
q
where u: R+ → Rp is a (locally) Riemann integrable map, v ∈ L∞
loc(R , R )
0
p+q
m
and h ∈ C (R , R ), m, n, p, q " 1. Indeed, a bounded function defined on a
378
LIONEL ROSIER
segment of R is Riemann integrable iff the set of points at which the function fails
to be continuous is of measure zero [19]. Notice that every field f of the form
f (t, x) := f0 (x) +
p
&
ui (t) fi (x),
i=1
n
n
where for each i " 0 fi ∈ L∞
loc (R , R ) and for each i " 1 ui is a piecewise
continuous (or piecewise monotonous) function, is concerned.
The following result, which asserts the existence of a smooth weak Lyapunov
function for a discontinuous (but e.c.t.) robustly stable system, is one of the main
results in this paper.
+
n
n
THEOREM 1. Let f ∈ L∞
loc (R × R , R ) be e.c.t. If the origin is an equilibrium
position which is robustly stable for (1.1) (with Filippov’s solutions), then there
exists a weak Lyapunov function in the small for (1.1) which is of class C ∞ .
We recall that a Filippov solution x(t) of (1.1) on some interval I ⊂ R+ is a
map x: I → Rn which is absolutely continuous on any [a, b] ⊂ I and such that
for a.e. t ∈ I,
where
Kx f (t, x) :=
ẋ(t) ∈ Kx f (t, x(t)),
' '
δ>0 µ(N)=0
co{f (t, B(x, δ) \ N)}.
(1.4)
(1.5)
In (1.5), B(x, δ), co and µ denote, respectively, the closed ball of center x and
radius δ, the convex closure and the usual Lebesgue measure. Notice that Filippov’s
system (1.4) is a special instance of a differential inclusion
ẋ ∈ F (t, x)
(1.6)
in which the multivalued map F assumes compact convex values and is upper
semicontinuous w.r.t. x. Actually, Theorem 1 will follow from a more general
result which applies to differential inclusions (Theorem 4, see below). Moreover, a
smoothing procedure for multivalued maps (inspired from the one in [8]) appears
as a key step in the proofs of Theorems 1, 2 and 4, and we emphasize that this
procedure cannot be achieved within the framework of ODE. Consequently, the
exposition in the following sections will be developed within the framework of
differential inclusion theory.
The same ideas may be applied to the problem of boundedness of solutions, the
so-called Lagrange stability. Recall that system (1.1) is said to be Lagrange stable
if, for each R > 0, there exists S > 0 such that for each pair (t0 , x0 ) with t0 " 0,
for each solution x(·; t0 , x0 ) defined on some interval [t0 , t1 ] one has
&x0 & < R ⇒ ∀t ∈ [t0 , t1 ] &x(t; t0 , x0 )& < S.
(1.7)
As for the stability, this concept has to be slightly strengthened in order to be
characterized by means of some continuous weak Lyapunov function.
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
379
DEFINITION 4. Let f (t, x) (resp. F (t, x)) be defined on R+ × Rn . We say that
(1.1) (resp. (1.6)) is robustly Lagrange stable if there exist a sequence {Gi }i=0,1,2,...
of open sets in R+ × Rn and two sequences {ai }i=0,1,2,... , {bi }i=0,1,2,... of real numbers such that
(i+ ) 0 < ai ! bi < ai+1 for each i = 0, 1, 2, . . . and ai → +∞ as i → +∞;
(ii+ ) R+ × {x : &x& < ai } ⊂ Gi ⊂ R+ × {x : &x& < bi } for each i = 0, 1, 2, . . .;
(iii+ ) for each i = 0, 1, 2, . . ., each initial pair (t0 , x0 ) ∈ Gi and each solution
x(.; t0 , x0 ) one has (t, x(t; t0 , x0 )) ∈ Gi for each t " t0 .
Obviously a result analogous to Theorem 1 still holds for the robust Lagrange
stability, but only the version for differential inclusions (Theorem 3, see below)
will be reported here.
The paper is organized as follows: The main results (existence of smooth Lyapunov functions for robustly stable or Lagrange stable differential inclusions) are
presented in Section 2. The proofs are given in Sections 3 and 4. Section 5 contains
an application to control theory, namely an extension of a result by Bacciotti and
Beccari which asserts that an affine Lagrange stable system is UBIBS stabilizable.
2. The Main Results
In what follows, ,·, ·- denotes the usual scalar product in Rn , &·& the corresponding
Euclidean norm, dist(x, S) the Euclidean distance from a point x to a set S and
h(S1 , S2 ) the Hausdorff distance between two closed sets S1 and S2 , defined by
(
)
h(S1 , S2 ) := max sup dist(x1 , S2 ), sup dist(x2 , S1 ) .
x1 ∈S1
x2 ∈S2
We shall adopt the notation B (resp. N) for a closed ball in Rnx (resp., a zero
measure set in R+
t ), and the notation B (resp. N ) for a closed ball (resp., a zero
measure set) in the extended space Rn+1
(t,x).
Let F (t, x) be a multivalued map. For the sake of brevity we shall denote by
(H) the following hypothesis on F : There exists a zero measure set N0 ⊂ R+ such
that
(H1) for each (t, x) ∈ (R+ \ N0 ) × Rn , F (t, x) is a nonempty convex compact
subset of Rn ,
(H2) for each R > 0, there exists M > 0 such that
(t ∈ [0, R] \ N0 and &x& ! R) ⇒ F (t, x) ⊂ B(0, M),
(H3) for every t ∈ R+ \ N0 , F (t, x), as a multivalued map of x, is upper semicontinuous,
(H4) for every x ∈ Rn , F (t, x), as a multivalued map of t, is measurable.
380
LIONEL ROSIER
By a solution x(t) of (1.6) on some interval I ⊂ R+ we mean a map x: I → Rn
which is absolutely continuous on any [a, b] ⊂ I and such that ẋ(t) ∈ F (t, x(t))
for a.e. t ∈ I . It is well known (see [1, 9]) that any Cauchy problem
ẋ ∈ F (t, x),
x(t0 ) = x0 ,
admits a (local) solution provided that (H) holds true, and that (H) is fulfilled by
+
n
n
Filippov’s multivalued map F (t, x) := Kx f (t, x) for any f ∈ L∞
loc (R × R , R ).
Proposition 1 is obtained as a direct consequence of the following result, which
claims that the existence of a smooth Lyapunov function in the large is actually
equivalent to the robust Lagrange stability of the perturbed system we get when
introducing measurement error w.r.t. the full state (t, x). (Compare with [16], in
which the existence of a smooth uniform Control Lyapunov Function is shown to
be equivalent to the existence of a stabilizing feedback law which is robust w.r.t.
measurement error and external disturbances.)
THEOREM 2. Let F (t, x) be a multivalued map such that (H) holds and let N ⊃
N0 be a zero measure set in R+ . Consider the following assertions:
(A1): there exists a weak Lyapunov function in the large V of class C 1 for (1.6)
such that
∀t ∈ R+ \ N, ∀x ∈ Rn \ {0}, ∀v ∈ F (t, x)
∂V
(t, x) + ,∇x V (t, x), v- ! 0;
∂t
(2.1)
(A2): There exists a continuous map δ: R+ × Rn → R+ such that δ(t, x) = 0 ⇔
x = 0 and the system
ẋ ∈ G(t, x)
* + ! !
"
",
co F B (t, x), δ(t, x) ∩ (R+ \ N) × Rn
if x )= 0;
:=
(2.2)
F (t, 0)
if x = 0,
is also robustly Lagrange stable. Then (A1) ⇔ (A2). Moreover, the Lyapunov
function V in (A1) may as well be taken of class C ∞ .
Remark 1. Consider the following multivalued map (defined on R+ × R)
*
[−x, 1], if x = t ∈ Q+ ;
F (t, x) :=
{−x},
otherwise.
Obviously (H) is satisfied (with N0 = ∅), the origin is globally asymptotically
stable for (1.6) and (2.1) holds true with N := Q+ and V (t, x) := x 2 . We claim
that there does not exist any (smooth) weak Lyapunov function V such that (2.1)
holds true for every t " 0 (that is with N = ∅): Indeed, x(t) = t is a trajectory of
(2.2) for any function δ as in (A2) if N = ∅.
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
381
Since condition (A2) is not always easy to be checked in practice, we introduce
a strengthened version of (H) which ensures the existence of smooth Lyapunov
functions for robustly stable (or Lagrange stable) multivalued systems:
Let (H+ ) denote the following assumption for the multivalued map F (t, x):
There exists a zero measure set N0 ⊂ R+ such that (H1), (H2) hold true together
with
(H3+ ): The multivalued map F |(R+ \N0 )×Rn is upper semicontinuous, i.e. for each
pair (t0 , x0 ) ∈ (R+ \ N0 ) × Rn , for each ε > 0, there exists δ > 0 such that
for any (t, x) ∈ (R+ \ N0 ) × Rn ,
|t − t0 | + &x − x0 & < δ ⇒ F (t, x) ⊂ F (t0 , x0 ) + B(0, ε).
It is easy to see that (H+ ) ⇒ (H). Moreover, it follows from the next proposition
that (H+ ) holds true when F (t, x) = Kx f (t, x) for a wide class of discontinuous
fields f .
+
n
n
+
PROPOSITION 2. Let f ∈ L∞
loc (R × R , R ). If f is e.c.t., then (H ) is fulfilled
by F (t, x) := Kx f (t, x).
We are now in position to state the main result of the paper.
THEOREM 3. Let F (t, x) be a multivalued map such that (H+ ) holds. If system
(1.6) is robustly Lagrange stable, then there exists a weak Lyapunov function in the
large V : R+ × Rn \ {0} → R+ which is of class C ∞ and such that for almost every
t " 0 and every x )= 0,
∀v ∈ F (t, x)
∂V
(t, x) + ,∇x V (t, x), v- ! 0.
∂t
Of course, a result analogous to Theorem 3 also holds true for a robustly stable
differential inclusion, provided that (H+ ) is fulfilled.
THEOREM 4. Let F (t, x) be a multivalued map such that (H+ ) holds. If the origin
is robustly stable for (1.6), then there exists a weak Lyapunov function in the small
V : R+ × {x : &x& < h} → R+ which is of class C ∞ and such that for almost
every t " 0 and every x with &x& < h,
∀v ∈ F (t, x)
∂V
(t, x) + ,∇x V (t, x), v- ! 0.
∂t
Theorem 1 is therefore a direct consequence of Theorem 4 and Proposition 2.
Remark 2. The converse of the second Lyapunov theorem is not investigated in
this paper, but we believe that, for time-dependent asymptotically stable differential inclusions, (H+ ) is a relevant assumption to ensure the existence of a smooth
(strong) Lyapunov function.
382
LIONEL ROSIER
3. Proofs of Theorem 2 and Proposition 1
Before going to the proof of Theorem 2, let us remark that for any function δ(t, x)
as in (A2),
∀t )∈ N, ∀x ∈ Rn ,
F (t, x) ⊂ G(t, x).
Therefore any solution of
ẋ ∈ F (t, x),
x(t0 ) = x0 ,
is also a solution of
ẋ ∈ G(t, x),
x(t0 ) = x0 ,
(3.1)
and we do not need to check whether assumption (H) is fulfilled by G in order to
claim the (local) existence of solutions for the Cauchy problem (3.1).
Proof of Theorem 2. Let us first deal with (A1) ⇒ (A2). Assume that (A1) holds.
Let a(·), b(·) be as in Definition 1. We first use a trick by Yoshizawa [24], which
allows us to strengthen the inequality (2.1). We substitute
Ṽ (t, x) :=
1+t
V (t, x)
1 + 2t
for V (t, x), and we observe that
1
2
a(&x&) ! Ṽ (t, x) ! b(&x&)
but this time
∂ Ṽ
(t, x) + ,∇x Ṽ (t, x), v∂t
.
1
1 + t ∂V
=−
V
(t,
x)
+
V
(t,
x),
v(t,
x)
+
,∇
x
(1 + 2t)2
1 + 2t ∂t
1
!−
a(&x&)
(1 + 2t)2
(3.2)
for every t ∈ R+ \ N, every x ∈ Rn \ {0} and every v ∈ F (t, x). To prove (A2),
it is clearly sufficient (thanks to the first Lyapunov theorem [6, Thm. 2.1] and to a
standard compactness argument) to check that for any (t0 , x0 ) ∈ R+ × (Rn \ {0}),
there exists δ > 0 s.t. for each pair (t, x) ∈ B((t0 , x0 ), δ) ∩ R+ × (Rn \ {0})
"
!
∀v ∈ F B((t, x), δ) ∩ (R+ \ N) × Rn
∂ Ṽ
(t, x) + ,∇x Ṽ (t, x), v- ! 0.
∂t
By (H2), there exists M > 0 such that
&(t − t0 , x − x0 )& ! 1 and
t )∈ N0 ⇒ ∀ v ∈ F (t, x)
&v& ! M.
(3.3)
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
383
Since Ṽ is C 1 on R+ × (Rn \ {0}) there exists δ ∈ (0, min(1/2, &x0 &/2)) such
that for all pairs (t, x), (t¯, x̄) in B((t0 , x0 ), 2δ) ∩ R+ × Rn
/
/
/ 0
0
/ ∂ Ṽ
∂ Ṽ
/ 0
0
/
/ ∂t (t, x) − ∂t (t¯, x̄)/ + ∇x Ṽ (t, x) − ∇x Ṽ (t¯, x̄) · M
1 a(&x0 &)
!
(3.4)
2 (1 + 2t0 )2
and
a(&x̄&)
1 a(&x0 &)
"
.
2
(1 + 2t¯)
2 (1 + 2t0 )2
(3.5)
Then let
(t, x) ∈ B((t0 , x0 ), δ) ∩ R+ × (Rn \ {0})
and
"
!
v ∈ F B((t, x), δ) ∩ (R+ \ N) × Rn .
This means that v ∈ F (t¯, x̄) for some pair (t¯, x̄) ∈ B((t, x), δ) ∩ (R+ \ N) × Rn .
Hence &(t¯ − t0 , x̄ − x0 )& ! 2δ < min(1, &x0 &), t¯ )∈ N and x̄ )= 0. We infer from
(3.2), (3.3), (3.4) and (3.5) that
∂ Ṽ
(t, x) + ,∇x Ṽ (t, x), v∂t
( ∂ Ṽ
)
!
(t¯, x̄) + ,∇x Ṽ (t¯, x̄), v- +
∂t
/ ∂ Ṽ
/ 0
0
∂ Ṽ
/
/
+/
(t, x) −
(t¯, x̄)/ + 0∇x Ṽ (t, x) − ∇x Ṽ (t¯, x̄)0 · M
∂t
∂t
1 a(&x0 &)
a(&x̄&)
+
! 0.
!−
(1 + 2t¯)2 2 (1 + 2t0 )2
!
We now turn to the
Proof of (A2) ⇒ (A1). We assume that (A2) holds. Without loss of generality,
we may also assume that the function δ is Lipschitz continuous with a Lipschitz
constant equal to 1, that is for all pairs (t1 , x1 ) and (t2 , x2 ) in R+ × Rn
|δ(t1 , x1 ) − δ(t2 , x2 )| ! &(t1 − t2 , x1 − x2 )&.
Notice that for any t " 0 and any x ∈ Rn
|δ(t, x)| = |δ(t, x) − δ(t, 0)| ! &x&.
384
LIONEL ROSIER
In order to get a Lyapunov function which is smooth w.r.t. (t, x) up to t = 0, we
are led to work on the open set
U := (−1, +∞) × (Rn \ {0}).
Set δ(t, x) = δ(0, x) for t ∈ (−1, 0), x ∈ Rn and
.1
*
t + 1 δ(t, x)
(3.6)
,
W (t, x) := (s, y) ∈ U; &(s − t, y − x)& < min
2
3
for (t, x) ∈ U.
CLAIM 1. The family
!
"
W (t, x) (t,x)∈((−1,+∞)\N)×(Rn\{0})
constitutes an open covering of U.
Indeed, for each pair (t, x) in N × (Rn \ {0}), there exists t˜ ∈ (−1, +∞) \ N
such that
.
t +1
δ(t, x)
t + 1 δ(t, x)
, t˜ + 1 >
|t − t˜| < min
,
and δ(t˜, x) >
.
4
6
2
2
((−1, +∞) \ N is dense in (−1, +∞) since µ(N) = 0.) It follows that (t, x) ∈
!
W(t˜,x) .
Set N = N × Rn . (N is a zero measure set in Rn+1 .) Let (ψi )i!1 denote
some C ∞ partition of unity such that the carriers of the ψi ’s form a refinement of
(W (t, x))(t,x)∈U\N . Setting, for every i " 1,
Si := {(t, x) ∈ Rn+1 ; ψi (t, x) )= 0},
this means that (i) each ψi is a nonnegative function of class C ∞ on Rn+1 ; (ii) for
each i " 1, there2exists (ti , xi ) ∈ U \ N such that Si ⊂ W (ti , xi ); (iii) for each
pair (t, x) ∈ U, ∞
i=1 ψi (t, x) = 1 and there exist a positive number ρ > 0 and a
finite subset I of N∗ such that
!
"
B (t, x), ρ ∩ Si )= ∅ ⇒ i ∈ I.
By (H2) there exists a continuous function M: R+ → R+ such that
!
"
(t ∈ R+ \ N0 , &x& ! 1) ⇒ F (t, x) ⊂ B 0, M(t) .
We regularize the multivalued map F as follows. Set for t " 0 and x ∈ Rn
.
* - 
2
δ(t
,
x
)

i
i

,
ψi (t, x) co F B (ti , xi ),


3
 i!1
.1
FL (t, x) :=
(3.7)
∩ (R+ \ N) × Rn
if x )= 0,




"
 !
B 0, M(t)
if x = 0,
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
385
and FL (t, x) = FL (0, x) for −1 < t < 0 and x ∈ Rn . It is easy to see that (1)
FL (t, x) is a (nonempty) convex compact set in Rn for any (t, x) ∈ (−1, +∞)×Rn ,
(2) the multivalued map FL is locally Lipschitz continuous for the Hausdorff metric
on U (since on compact subsets of R+ × (Rn \ {0}) the sum in (3.7) is finite) and
(3) the assumption (H) is also satisfied by the multivalued map FL |R+ ×Rn . (Thanks
to (3.6), we see that
ψi (t, x) )= 0 ⇒ &(t − ti , x − xi )& <
δ(ti , xi )
&xi &
!
.
3
3
Hence
&xi & < 32 &x&
and
|t − ti | < 12 &x&.
This is useful when proving that for each t ∈ R+ \ N0 the multivalued map x 4→
FL (t, x) be upper semicontinuous at x = 0.)
CLAIM 2.
∀t ∈ R+ \ N, ∀x ∈ Rn
F (t, x) ⊂ FL (t, x).
(3.8)
The result is obvious for x = 0, so let x )= 0 and t ∈ R+ \ N. Let i " 1 be such
that ψi (t, x) > 0. It follows from (3.6) that &(t − ti , x − xi )& < 13 δ(ti , xi ) hence
- .
.
"
! +
δ(ti , xi )
n
F (t, x) ⊂ F B (ti , xi ),
∩ R \N ×R
3
and then F (t, x) ⊂ FL (t, x).
CLAIM 3.
!
∀(t, x) ∈ R+ × (Rn \ {0}) FL (t, x) ⊂ G(t, x).
To prove this claim, let x )= 0, t ∈ R+ , and i " 1 be such that ψi (t, x) > 0.
Since δ is a Lipschitz continuous function with a Lipschitz constant equal to 1, we
may write
δ(ti , xi ) − δ(t, x) ! &(t − ti , x − xi )& < 13 δ(ti , xi ),
hence
"
!
"
!
"
!
B (ti , xi ), 13 δ(ti , xi ) ⊂ B (t, x), 23 δ(ti , xi ) ⊂ B (t, x), δ(t, x)
from which we infer that FL (t, x) ⊂ G(t, x).
As a consequence, the system
ẋ ∈ FL (t, x)
(t " 0, x ∈ Rn )
!
(3.9)
386
LIONEL ROSIER
is also (robustly) Lagrange stable. By [6, Thm. 2.2], there exist continuous func7 is locally Lipschitz
7: R+ × (Rn \ {0}) → R+ , a, b: R+ → R+ such that V
tions V
continuous, a and b are increasing, a(0) = 0, a(r) > 0 for r > 0, a(r) → ∞ as
7(t, x) ! b(&x&) for each pair (t, x) ∈ R+ × (Rn \ {0}), and
r → ∞, a(&x&) ! V
for any nonvanishing solution x(t) of (3.9) on some interval [t1 , t2 ] ⊂ R+ we have
7(t2 , x(t2 )) ! V
7(t1 , x(t1 )). (In other words Ṽ is a locally Lipschitz continuous
V
weak Lyapunov function in the large for (3.9).) We now turn to the
CLAIM 4. For a.e. (t0 , x0 ) ∈ R+ × (Rn \ {0})
∀v ∈ FL (t0 , x0 )
7
∂V
7(t0 , x0 ), v- ! 0.
(t0 , x0 ) + ,∇x V
∂t
(3.10)
7 is differenRecall that it follows from Rademacher’s theorem (see [10]) that V
tiable at (t0 , x0 ) for almost every (t0 , x0 ). Therefore, it is sufficient to prove that for
every pair (t0 , x0 ) ∈ R+ × (Rn \ {0}),
∀v ∈ FL (t0 , x0 )
lim sup
h→0+
Ṽ (t0 + h, x0 + hv) − Ṽ (t0 , x0 )
! 0.
h
Notice first that for each v ∈ FL (t0 , x0 ) there exists a solution x(t) of (3.9) such
that x(t0 ) = x0 and ẋ(t0 ) = v. Indeed, we may infer from the local Lipschitz
continuity of FL that the map
g(s, y) := πFL (s,y)(v)
(3.11)
(where πFL (s,y) denotes the projection on the convex compact set FL (s, y)) is continuous, hence, by Peano’s theorem, there exists a solution x(t) on some interval
[t0 , t0 + ε] of the (classical) Cauchy problem
ẋ = g(t, x) (∈ FL (t, x)),
x(t0 ) = x0 .
Obviously ẋ(t0 ) = g(t0 , x0 ) = v. Now there exists L > 0 such that for any (t1 , x1 ),
(t2 , x2 ) in some neighborhood of (t0 , x0 )
7(t1 , x1 ) − V
7(t2 , x2 )| ! L &(t1 − t2 , x1 − x2 )&.
|V
It follows that for h > 0 small enough
7(t0 , x0 )
7(t0 + h, x0 + hv) − V
V
h
7
7(t0 + h, x(t0 + h))
V (t0 + h, x0 + hv) − V
=
+
h
7(t0 + h, x(t0 + h)) − V
7(t0 , x0 )
V
+
h
0 x(t + h) − x
0
0 0
0
0
! L0
− v 0,
h
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
387
hence
lim sup
h→0+
7(t0 + h, x0 + hv) − V
7(t0 , x0 )
V
! 0.
h
!
Set for t " 0 and x ∈ Rn \ {0}
VL (t, x) :=
WL (t, x) :=
1+t 7
V (t, x),
1 + 2t
1
a(&x&).
(1 + 2t)2
We infer from (3.10) that for a.e. (t0 , x0 ) ∈ R+ × (Rn \ {0})
∂VL
(t0 , x0 ) + ,∇x VL (t0 , x0 ), v- ! −WL (t0 , x0 ).
∂t
∀v ∈ FL (t0 , x0 )
(3.12)
Before applying a smoothing procedure, we are led to extend VL and WL on a
neighborhood ( of R+ × (Rn \ {0}) in such a way that (3.12) remains valid for a.e.
(t0 , x0 ) ∈ (. Since VL (0, ·) is locally Lipschitz continuous on Rn \ {0} and FL is
locally bounded on U, there exists a continuous map l: Rn \ {0} → (0, +∞) such
that
for a.e. x ∈ Rn \ {0},
&∇x VL (0, x)& ! l(x)
and
∀(t, x) ∈ (− 21 , 0) × (Rn \ {0}), ∀v ∈ FL (t, x),
&v& ! l(x).
Let d: Rn \ {0} → (0, +∞) denote some function of class C ∞ such that
∀x ∈ Rn \ {0} d(x) > l(x)2 + a(&x&).
For (t, x) ∈ (−1, 0) × (Rn \ {0}) we now set
VL (t, x) = VL (0, x) − td(x),
WL (t, x) = WL (|t|, x).
VL is obviously locally Lipschitz continuous on U. Set
+
O := (t, x) ∈ (− 12 , 0) × (Rn \ {0}); VL (t, x) > 12 a(&x&) and
"
,
!
−d(x) + l(x) l(x) + |t| · &∇x d(x)& < −WL (t, x) ,
and ( := R+ × (Rn \ {0}) ∪ O. It is easy to see that ( is an open neighborhood of
R+ × (Rn \ {0}). On the other hand,
∀(t, x) ∈ (,
1
2
a(&x&) ! VL (t, x) ! b(&x&),
WL (t, x) > 0.
388
LIONEL ROSIER
CLAIM 5. For a.e. (t, x) ∈ (,
∂VL
(t, x) + ,∇x VL (t, x), v- ! −WL (t, x).
∀v ∈ FL (t, x),
∂t
Since the result is already proved for a.e. (t, x) ∈ R+ × (Rn \ {0}) (see (3.12)),
it is sufficient to observe that for a.e. (t, x) ∈ O and every v ∈ FL (t, x)
∂VL
(t, x) + ,∇x VL (t, x), v∂t
= −d(x) + ,∇x VL (0, x) − t ∇x d(x), v! −d(x) + l(x)(l(x) + |t| · &∇x d(x)&)
! −WL (t, x).
!
Regularization of VL on compact subsets of ( is performed by means of convolution with mollifiers:
LEMMA 1. Let φ, ψ: ( → (0, +∞) be continuous functions such that φ is
locally Lipschitz continuous w.r.t. (t, x) on ( and
for a.e. (t0 , x0 ) ∈ (, ∀v ∈ FL (t0 , x0 ),
∂φ
(t0 , x0 ) + ,∇x φ(t0 , x0 ), v- ! −ψ(t0 , x0 ).
∂t
(3.13)
Let S ⊂ ( be any compact set and let ε > 0. Then there exists a function φ̃ of class
C ∞ with compact support in ( such that
&φ − φ̃&L∞ (S) < ε
(3.14)
and
∀(t0 , x0 ) ∈ S, ∀v ∈ FL (t0 , x0 ),
∂ φ̃
(t0 , x0 ) + ,∇x φ̃(t0 , x0 ), v- ! − 12 ψ(t0 , x0 ).
∂t
(3.15)
∞
n+1
Proof of Lemma 1. Let
8 ρ ∈ C (R , R) be a nonnegative function such that
supp(ρ) ⊂ B(0, 1) and Rn+1 ρ(t, x) dt dx = 1. Set for δ > 0,
.
1
t x
ρδ (t, x) = n+1 ρ
,
δ
δ δ
and
φδ (t, x) := φ ∗ ρδ (t, x)
9
=
φ(t − s, x − y)ρδ (s, y) ds dy
n+1
9R
φ(t − δ s̄, x − δ ȳ)ρ(s̄, ȳ) ds̄ dȳ,
=
&(s̄,ȳ)&"1
ψδ (t, x) := ψ ∗ ρδ (t, x).
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
389
Fix some open neighborhood (+ of S such that (+ ⊂⊂ (, i.e. (+ is a compact
subset of (. Then, for δ > 0 small enough, the functions φδ and ψδ are defined and
of class C ∞ on (+ , and φδ (t, x) → φ(t, x) (resp. ψδ (t, x) → ψ(t, x)) uniformly
on S as δ → 0. Let θ be a function of class C ∞ with compact support in (+ and
such that θ(t, x) = 1 for every (t, x) in a neighborhood of S. Setting φ̃ := θφδ we
see that the function φ̃ is of class C ∞ on Rn+1 with compact support in (+ and that
(3.14) holds true provided that δ is small enough. To complete the proof of Lemma
1, we now show that there exists δ0 > 0 such that for any δ ∈ (0, δ0 )
∀(t0 , x0 ) ∈ S, ∀v ∈ FL (t0 , x0 ),
∂φδ
(t0 , x0 ) + ,∇x φδ (t0 , x0 ), v- ! − 12 ψ(t0 , x0 ).
∂t
Let δ1 > 0 be such that S + B(0, δ1 ) ⊂ ( and L > 0 be such that for any pairs
(t1 , x1 ), (t2 , x2 ) in S + B(0, δ1 )
!
"
h FL (t1 , x1 ), FL (t2 , x2 ) + |φ(t1 , x1 ) − φ(t2 , x2 )|
(3.16)
! L &(t1 − t2 , x1 − x2 )&.
It follows that for a.e. (t, x) ∈ S + B(0, δ1 ), φ is differentiable at (t, x) and
0
0.
0
0 ∂φ
0
0
(3.17)
0 ∂t , ∇x φ (t, x)0 ! L.
Let δ ∈ (0, δ1 ), (t0 , x0 ) ∈ S and v ∈ FL (t0 , x0 ). Applying Lebesgue’s theorem, we
infer from (3.16) that
∂φδ
(t0 , x0 ) + ,∇x φδ (t0 , x0 ), v∂t
9
φ(t0 − δ s̄ + η, x0 − δ ȳ + ηv) − φ(t0 − δ s̄, x0 − δ ȳ)
ρ(s̄, ȳ) ds̄ dȳ
= lim
η→0 &(s̄,ȳ)&"1
η
9
∂φ
(t0 − δ s̄, x0 − δ ȳ)+
=
∂t
&(s̄,ȳ)&"1
.
+ ,∇x φ(t0 − δ s̄, x0 − δ ȳ), v- ρ(s̄, ȳ)ds̄ dȳ.
(3.18)
g also denoting the map defined by (3.11), we may write (thanks to (3.13), (3.17)
and (3.18))
∂φδ
(t0 , x0 ) + ,∇x φδ (t0 , x0 ), v∂t 9
.
∂φ
=
+ ,∇x φ, g- (t0 − δ s̄, x0 − δ ȳ)ρ(s̄, ȳ) ds̄ dȳ +
∂t
&(s̄,ȳ)&"1
9
+
,∇x φ(t0 − δ s̄, x0 − δ ȳ), v −
&(s̄,ȳ)&"1
− g(t0 − δ s̄, x0 − δ ȳ)-ρ(s̄, ȳ) ds̄ dȳ,
9
&v − g(t0 − δ s̄, x0 − δ ȳ)&ρ(s̄, ȳ) ds̄ dȳ.
! −ψδ (t0 , x0 ) + L
&(s̄,ȳ)&"1
(3.19)
390
LIONEL ROSIER
Now observe that for &(s̄, ȳ)& ! 1,
&v − g(t0 − δ s̄, x0 − δ ȳ)&
!
"
= dist v, FL (t0 − δ s̄, x0 − δ ȳ)
"
!
! h FL (t0 , x0 ), FL (t0 − δ s̄, x0 − δ ȳ)
! Lδ,
owing to (3.16). Pick any δ0 ∈ (0, δ1 ) such that
L2 δ0 <
1
4
min ψ(t, x)
(3.20)
(t,x)∈S
and such that, for any 0 < δ < δ0 ,
max |ψ(t, x) − ψδ (t, x)| <
(t,x)∈S
1
4
min ψ(t, x).
(t,x)∈S
(3.21)
Then we infer from (3.19), (3.20) and (3.21) that, for 0 < δ < δ0 ,
∂φδ
(t0 , x0 ) + ,∇x φδ (t0 , x0 ), v- ! −ψδ (t0 , x0 ) + 14 ψ(t0 , x0 ) ! − 12 ψ(t0 , x0 )·
∂t
The proof of Lemma 1 is complete.
!
For i " 1, set
+
,
Ji = j " 1, Si ∩ Sj )= ∅ .
(Recall that Si is the support of ψi and that Si ⊂ U.) It follows from the compactness of Si and the local finiteness of the covering of U by the Sj , j " 1, that Ji is
finite. Applying Lemma 1, we see that for each i " 1 there exists a function Vi of
class C ∞ with compact support in ( and such that, if we set
!
"
Si+ := Si ∩ R+ × Rn ,
*
!
"
+
1
+
+
εi = 2 min min(t,x)∈Si VL (t, x), min(t,x)∈Si WL (t, x) (> 0), if Si )= ∅,
1, otherwise,
*
/ ∂ψi
/
+
/
/
+
qi = max(t,x)∈Si , v∈FL(t,x) ∂t (t, x) + ,∇x ψi (t, x), v- , if Si )= ∅,
0, otherwise,
we have
∀(t, x) ∈ Si+
and
|Vi (t, x) − VL (t, x)| <
minj ∈Ji εj
2i (1 + qi )
∀(t, x) ∈ Si+ , ∀v ∈ FL (t, x),
∂Vi
(t, x) + ,∇x Vi (t, x), v- ! − 12 WL (t, x).
∂t
(3.22)
(3.23)
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
391
(Notice that if Si+ = ∅, (3.22) and (3.23) are fulfilled by any function Vi of class
C ∞ with compact support in (.) Set
V =
+∞
&
(3.24)
ψi Vi .
i=1
It is clear that V is of class C ∞ on (. Moreover, for each (t, x) ∈ R+ × (Rn \ {0}),
|V (t, x) − VL (t, x)|
+∞
&
ψi (t, x)|Vi (t, x) − VL (t, x)|
!
!
i=1
+∞
&
i=1
1
VL (t, x) =
2i+1
VL (t, x)
,
2
hence
1
4
a(&x&) ! V (t, x) ! 32 b(&x&).
Let (t, x) ∈ R+ × (Rn \ {0}) and v ∈ FL (t, x). We aim to prove that ∂V /∂t(t, x)+
,∇x V (t, x), v- ! 0. By (3.24)
.
.
& - ∂ψi
& - ∂Vi
∂V
+ ,∇x V , v- =
+ ,∇x ψi , v- Vi +
+ ,∇x Vi , v- ,
∂t
∂t
∂t
i
i
& - ∂ψi
. & ..
∂Vi
+ ,∇x ψi , v- Vi − VL +
+ ,∇x Vi , vψi
∂t
∂t
i
i
&
!
"
minj ∈Ji εj &
+
qi i
ψi · − 12 WL (t, x) .
!
2 (1 + qi )
i
{i; (t,x)∈Si }
Let i0 " 1 be such that (t, x) ∈ Si0 and εi0 = min{i; (t,x)∈Si } εi . If i " 1 is such that
(t, x) ∈ Si , then i0 ∈ Ji and εi0 " minj ∈Ji εj . It follows that
&
{i; (t,x)∈Si }
qi
& εi
minj ∈Ji εj
0
! εi0 .
!
2i (1 + qi ) {i; (t,x)∈S } 2i
i
On the other hand,
&
&
εi0 =
ψi (t, x)εi0 !
ψi (t, x)εi ,
{i; (t,x)∈Si }
i
hence
.
& ∂V
1
ψi εi − WL (t, x) ! 0.
+ ,∇x V , v- !
∂t
2
i
(3.25)
392
LIONEL ROSIER
Now (2.1) follows from (3.8) and (3.25). The proof of Theorem 2 is complete. !
The end of this section is devoted to the
Proof of Proposition 1. Let B ⊂ R be a Borel set such that
0 < µ(B ∩ I ) < µ(I )
for every nondegenerate finite interval I ⊂ R. (Proving the existence of B is a
classical exercise in measure theory, see [20, ex. 8, p. 59].) Set
" !
"
!
F = {0} ∪ B ∩ [0, 12 ] ∪ 12 + [0, 12 ] \ B .
Obviously, µ(F ) = 12 and also
!
"
0 < µ F ∩ (a, b) < b − a
for any 0 ! a < b ! 1. Finally, set E =
;
1
if t ∈ E,
λ(t) =
−2 otherwise.
:
n∈Z (F
+ n) and
8t
λ is a bounded 1-periodic Borel function. Let ,(t) = 0 λ(τ ) dτ . Since , is
absolutely continuous on any [a, b] ⊂ R, there exists a zero measure Borel set
N ⊂ R such that for any t )∈ N, , is differentiable at time t and ,+ (t) = λ(t).
Obviously, for any (t0 , x0 ) ∈ R2 , the (unique) solution in Carathéodory’s sense of
the Cauchy problem
x(t0 ) = x0
ẋ = λ(t)x,
(3.26)
is given by x(t) = e,(t )−,(t0) x0 . If s is a real number, let [s] denote its integral part.
Since λ is periodic, we get for any t " t0
,(t) − ,(t0 )
9
9 t0 +[t −t0 ]
λ(τ ) dτ +
=
t0
hence
9
1
t
λ(τ ) dτ
t0 +[t −t0 ]
9 t
λ(τ ) dτ +
dτ
! [t − t0 ]
0
t0 +[t −t0 ]
!
"
! [t − t0 ] µ(F ) − 2µ([0, 1] \ F ) + 1,
,(t) − ,(t0 ) ! − 12 [t − t0 ] + 1,
and
" 1
!
|x(t)| ! e|x0 | e− 2 [t −t0 ] .
(3.27)
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
393
We infer from (3.27) that the origin is globally asymptotically stable (hence,
robustly stable and Lagrange stable) for system (1.2). To complete the proof of
Proposition 1, we argue by contradiction: We assume that there exists a weak Lyapunov function in the large V = V (t, x), which is of class C 1 . We shall infer from
this assumption that the system ẋ = x is (robustly) Lagrange stable, a property
which is obviously false. Let x(t) denote the solution of (3.26). If t0 )∈ N and
x0 )= 0, then x(·) is differentiable at t0 and ẋ(t0 ) = λ(t0 )x0 . Hence
d[V (t, x(t))]
∂V
∂V
(t0 , x0 ) + λ(t0 ) x0
(t0 , x0 ) =
|t =t0 ! 0
∂t
∂x
dt
(since V (t, x(t)) is a nonincreasing function), i.e. (A1) holds true (here N0 = ∅
and F (t, x) = {λ(t)x}). It follows from Theorem 2 that there exists a nonnegative
continuous function δ(t, x) such that δ(t, x) = 0 ⇔ x = 0, and such that the
system ẋ ∈ G(t, x) (where G(t, x) is defined in (2.2)) is also (robustly) Lagrange
stable. We now define a (Borel) function s = s(t, x) as follows: If x = 0 and t " 0,
we set s(t, 0) = 0. If n ! t < n + 1, 2k ! |x| < 2k+1 with n ∈ N, k ∈ Z we first
pick p ∈ N∗ such that
+
,
1
< min δ(t¯, x̄); n ! t¯ ! n + 1 and 2k ! |x̄| ! 2k+1 .
p
Next for each i ∈ {1, . . . , p} we choose
.
i
i−1
,n +
\ N.
ti ∈ E ∩ n +
p
p
(This is possible, since
..
i −1
i
µ E∩ n+
,n +
>0
p
p
whereas µ(N) = 0.) Then we set s(t, x) = ti (∈ E \ N) for
.
<
i
i −1
,n +
, 1 ! i ! p and 2k ! |x| < 2k+1 .
t ∈ n+
p
p
Observe that |s(t, x) − t| < 1/p < δ(t, x) and that λ(s(t, x)) = 1. Obviously,
0 ∈ G(t, 0) and for x )= 0
"
! !
"
x = λ(s(t, x))x ∈ F B (t, x), δ(t, x) ∩ (R+ \ N) × R ⊂ G(t, x).
It follows that the system ẋ = x, whose trajectories are also solutions of ẋ ∈
G(t, x), should be Lagrange stable, a property which is clearly false. A slight modification of previous reasoning shows that there does not exist any weak Lyapunov
!
function of class C 1 in the small. The proof of Proposition 1 is complete.
394
LIONEL ROSIER
4. Proofs of Proposition 2, Theorem 3 and Theorem 4
+
n
n
Proof of Proposition 2. Since f ∈ L∞
loc (R × R , R ), we may assume without
loss of generality (by extending the zero measure (Borel) set N in Definition 3)
that
∀R > 0, ∃M > 0,
(0 ! t ! R, &x& ! R
Set
*
9
N = t " 0;
Rn
and
(t, x) )∈ N ⇒ &f (t, x)& ! M).
(4.1)
1
χN (t, x) dx > 0 .
(χN denotes the characteristic function of the set N .) N is a zero measure (Borel)
set in R+ , and it is clear that
0 ! t ! R, &x& < R
and
t )∈ N ⇒ Kx f (t, x) ⊂ B(0, M).
Now let t0 )∈ N, x0 ∈ Rn and ε > 0. We aim to show that there exists δ + > 0 s.t.
"
!
|t − t0 | + &x − x0 & < δ + and t ∈ R+ \ N
⇒ Kx f (t, x) ⊂ Kx f (t0 , x0 ) + B(0, ε).
(4.2)
Let δ > 0 be such that (1.3) holds true with ε/2 instead of ε. Pick any
(t, x) ∈ (R+ \ N) × Rn
δ
s.t. |t − t0 | + &x − x0 & < .
2
Since t )∈ N and t0 )∈ N, for almost every y ∈ B(x, δ/2), we have (t, y), (t0 , y) )∈
N , hence (by (1.3)) &f (t, y) − f (t0 , y)& < ε/2. We infer from [17, Thm. 1] that
!
"
Kx f (t, x) ⊂ Kx f (t0 , x) + Kx f (t, x) − f (t0 , x)
.
ε
⊂ Kx f (t0 , x) + B 0,
.
(4.3)
2
Since the multivalued map x 4→ Kx f (t0 , x) is upper semicontinuous, there exists
0 < δ + < δ/2 such that
.
ε
+
(4.4)
&x − x0 & < δ ⇒ Kx f (t0 , x) ⊂ Kx f (t0 , x0 ) + B 0,
2
Then (4.2) follows from (4.3) and (4.4).
We now turn to the
!
Proof of Theorem 3. By Theorem 2 it is sufficient to prove that (A2) holds, that is
there exists a nonnegative continuous function δ(t, x) such that δ(t, x) = 0 ⇔ x =
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
395
0 and the system (2.2) is robustly Lagrange stable. Since it proves to be difficult to
(directly) construct a sequence {Gi }i!1 of open sets (as in Definition 4) for the flow
of (2.2), we are led to define together with δ(t, x) a (locally Lipschitz) continuous
weak Lyapunov function in the large V = V (t, x) for (2.2), which ensures the
robust Lagrange stability of (2.2). The construction of V being essentially the same
as that given in the proof of [6, Thm. 2.2] we limit ourselves to giving the main
steps, the reader being referred to [6] for the details of the proof.
Let us first remark that for any function δ(t, x) as in (A2), the multivalued map
G defined by (2.2) (with N = N0 ) satisfies (H+ ), as well. Indeed (H1) and (H2) are
obvious and (H3+ ) follows easily from the upper semicontinuity of the multivalued
maps F |(R+ \N0 )×Rn and (t, x) 4→ B((t, x), δ(t, x)) ∩ (R+ \ N0 ) × Rn .
The following compactness lemma, which may be seen as a synthesis of [6,
Lemma 4.1] and [8, Lemma 2.3], is the first step in the proof of Theorem 3.
LEMMA 2. Assume that (H+ ) holds true for the multivalued map F (t, x). Let R >
j
j
0, T > 0 be given. Let (t1 )j =0,1,2,... , (t2 )j =0,1,2,... and (δ j )j =0,1,2,... be sequences of
j j
numbers such that [t1 , t2 ] ⊂ [0, T ], δ j → 0+ as j → +∞ and let (y j )j =0,1,2,...
j j
be a sequence of absolutely continuous functions y j : [t1 , t2 ] → B(0, R). Assume
that
9 tj
2
!
+ ! !
"
","
dist ẏ j (τ ), co F B (τ, y j (τ )), δ j ∩ (R+ \ N0 ) × Rn dτ = 0.
lim
j →+∞ t j
1
Then there exist t1 , t2 ∈ [0, T ], a function z: [t1 , t2 ] → B(0, R) and a sequence
j
(jl ) such that z(·) is a solution of (1.6) on [t1 , t2 ], liml→∞ jl = +∞, t1 l → t1 ,
j
t2 l → t2 , and
j
lim y jl (t1 l ) = z(t1 ),
l→∞
j
lim y jl (t2 l ) = z(t2 ).
l→∞
Proof of Lemma 2. Without loss of generality, we may assume that δ j < 1 for
j " 0. Let M > 0 be such that
t ∈ [0, T + 1] \ N0 ,
&y& ! R + 1 ⇒ F (t, y) ⊂ B(0, M).
For j " 0 and t ∈ [0, T ], we set
 j j
j
 y (t1 ) if 0 ! t ! t1 ,
j
j
7
y (t) = y j (t) if t1 ! t ! t2j ,
 j j
j
y (t2 ) if t2 ! t ! T ,
and we let p j (t) denote the projection of ỹ˙j (t) on co{F (B((t, ỹ j (t)), δ j ) ∩ (R+ \
N) × Rn )}. Then the (almost everywhere defined) map p j is measurable (thanks to
[2, Cor. 8.2.13]) and it satisfies &p j (t)& ! M and
!
+ !
","
dist ỹ˙j (t), co F B((t, ỹ j (t)), δ j ) ∩ (R+ \ N) × Rn
= &ỹ˙j (t) − p j (t)&
396
LIONEL ROSIER
for each j " 0 and every t ∈ [0, T ] such that the derivative ỹ˙j (t) exists.
Arguing as in the proof of [6, Lemma 4.1], we see that there exist two numbers
t1 , t2 ∈ [0, T ], an increasing sequence jl → ∞ and an absolutely continuous map
j
z(·): [0, T ] → B(0, R) such that ỹ jl (t) → z(t) uniformly on [0, T ], t1 l → t1 ,
jl
j
j
t2 → t2 , ỹ jl (t1 l ) → z(t1 ), ỹ jl (t2 l ) → z(t2 ) and (p jl ) converges to ż weakly in
L2 ([0, T ], Rn ) as l → +∞.
It remains to prove that ż(t) ∈ F (t, z(t)) for a.e. t ∈ (t1 , t2 ). We assume t1 < t2 ,
otherwise there is nothing to prove. Let us consider the functional
9 t2
!
"
dist w(t), F (t, z(t)) dt
J (w) =
t1
for w ∈ L2 ([t1 , t2 ], Rn ). The same argumentation as in the proof of [6, Lemma 4.1]
shows that the functional J is well-defined, convex and continuous (for the strong
topology), hence lower semicontinuous w.r.t. the weak topology of L2 ([t1 , t2 ], Rn ).
Since p jl / ż in L2 ([t1 , t2 ], Rn ), we get
J (ż) ! lim inf J (p jl ).
l→∞
To prove that J (ż) = 0 (which means that ż(t) ∈ F (t, z(t)) for a.e. t ∈ (t1 , t2 )), we
are led to show that J (p jl ) → 0 as l → ∞. Since dist(p jl (t), F (t, z(t))) ! 2M
a.e., it is sufficient (by Lebesgue’s theorem) to prove that for a.e. t ∈ (t1 , t2 ):
"
!
dist p jl (t), F (t, z(t)) → 0 as l → ∞.
Let t0 ∈ (t1 , t2 ) \ N0 be such that ỹ˙jl (t0 ) exists for all l " 0 and let ε > 0. By (H3+ )
there exists δ > 0 such that
&(t − t0 , x − z(t0 ))& ! δ and t )∈ N0
⇒ F (t, x) ⊂ F (t0 , z(t0 )) + B(0, ε).
(4.5)
Let L be a positive integer such that
l " L ⇒ δ jl <
δ
2
and
δ
&ỹ jl (t0 ) − z(t0 )& < .
2
(4.6)
It follows from (4.6) that for any l " L
!
"
!
"
B (t0 , ỹ jl (t0 )), δ jl ⊂ B (t0 , z(t0 )), δ ,
hence, by (4.5),
+ !
",
co F B((t0 , ỹ jl (t0 )), δ jl ) ∩ (R+ \ N0 ) × Rn
⊂ F (t0 , z(t0 )) + B(0, ε).
We infer from the definition of p jl (t0 ) and (4.7) that for l " L,
dist(p jl (t0 ), F (t0 , z(t0 ))) ! ε.
(4.7)
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
397
The proof of Lemma 2 is complete.
!
Since system (1.6) is robustly Lagrange stable, there exist positive numbers
a0 , a1 , . . . , b0 , b1 , . . . and open sets G0 , G1 , . . . in R+ × Rn such that conditions
(i+ ), (ii+ ) and (iii+ ) in Definition 4 are fulfilled for the flow of (1.6).
The following two results are easy consequences of Lemma 2. The first one
(Proposition 3) is, word by word, [6, Proposition 4.1], and the proof of the second
one (Proposition 4) does not differ from the proof of [6, Proposition 4.2].
PROPOSITION 3. Let i and k be fixed nonnegative integers. Let K ⊂ ([k, k +
1] × Rn ) ∩ Gi+1 be a (nonempty) compact set. Finally, let
C(K) = {(t2 , x2 ) ∈ [k, k + 1] × Rn : there exist (t1 , x1 ) ∈ K and a solution
x(·) : [t1 , t2 ] → Rn of (1.6) s.t. x(t1 ) = x1 , x(t2 ) = x2 }.
Then C(K) is a compact subset of Gi+1 .
PROPOSITION 4. Let i, k, K, C(K) be as in Proposition 3, and let
!
"
α = dist C(K), R+ × Rn \ Gi+1 .
(α > 0 by Proposition 3.) Then there exists a number ψ > 0 which enjoys the
following property: for every y(·): [t1 , t2 ] ⊂ [k, k + 1] → Rn such that
(1) y(·) is absolutely continuous,
(2) (t
8 1t2, y(t1 )) ∈ K,
(3) t1 dist(ẏ(τ ), co{F (B((τ, y(τ )), ψ) ∩ (R+ \ N) × Rn )}) dτ < ψ, we have
!
" α
min dist (t, y(t)), R+ × Rn \ Gi+1 " .
t ∈[t1 ,t2 ]
2
Let i " 0 be fixed. We define two sequences (αik )k!0 , (ψik )k!0 of positive
numbers and a sequence (Kik )k!0 of compact subsets of R+ × Rn by induction
on k " 0. For k = 0, set Ki0 = [0, 1] × Rn ∩ Gi , αi0 = α, ψi0 = ψ, where α and ψ
are given in Proposition 4 (applied with K = Ki0 , k = 0).
Then assume that Kil , αil , ψil have been defined for 0 ! l < k. Set
!
"
Kik = [k, k + 1] × Rn ∩ Gi
1
*
" αik−1
!
+
n
,
∪ (t, x); t = k and dist (t, x), R × R \ Gi+1 "
2
αik = α and ψik = ψ, where α and ψ are given in Proposition 4 (applied with
K = Kik ).
We now take a function ψ: R+ × Rn → (0, +∞) in such a way that
(1) ψ(·, ·) is of class C 1 ;
(2) ψ(t, x) > 1/ψik for t ∈ [k, k + 1] and (t, x) ∈ Gi+1 \ Gi (i " 0);
398
LIONEL ROSIER
(3) ψ(t, x) > 1 for t ∈ R+ and x ∈ Rn .
We also pick a function δ: R+ × Rn → R+ such that
(1) δ(·, ·) is continuous;
(2) δ(t, x) = 0 ⇔ x = 0;
(3) δ(t, x) < ψik for t ∈ [k, k + 1] and (t, x) ∈ Gi+1 \ Gi (i " 0).
From now on G denotes the multivalued map (defined by (2.2)) associated with
this function δ (with N = N0 ). We finally define, for each pair (t0 , x0 ) ∈ R+ × Rn ,
the class of functions
*
F (t0 , x0 ) = y(·): [t0 , +∞) → Rn such that y(·) is absolutely continuous
on every segment [t0 , b], y(t0 ) = x0 , and
1
9 +∞
!
"
ψ(t, y(t)) dist ẏ(t), G(t, y(t)) dt < 1 ·
t0
The proof of the following result is the same as the one of [6, Proposition 4.3].
PROPOSITION 5. Let i " 1 be a fixed integer, and let (t0 , x0 ) ∈ Gi . For each
y ∈ F (t0 , x0 ) and each t " t0 we have (t, y(t)) ∈ Gi+1 .
We are now ready to define a (locally Lipschitz) continuous weak Lyapunov
function in the large for G. For (t0 , x0 ) ∈ R+ × Rn we set
V (t0 , x0 )
:=
sup
y∈F (t0 ,x0 )
*-
1−
9
+∞
t0
.
1
!
"
ψ(t, y(t)) dist ẏ(t), G(t, y(t)) dt · sup &y(t)& .
t !t0
The same reasoning as in the proofs of [6, Propositions 4.4, 4.5 and 4.6] shows
that V is a (locally Lipschitz) continuous weak Lyapunov function in the large for
system (2.2), and therefore we infer from the first Lyapunov theorem [6, Thm. 2.1]
that (2.2) is robustly Lagrange stable. Next it follows from Theorem 2 that (A1)
holds true for F . The proof of Theorem 3 is complete.
!
Proof of Theorem 4. The proof is almost identical to the proof of Theorem 3,
the main difference being that the function V is now required to be smooth up
to x = 0. Let V be a continuous weak Lyapunov function such that (I), (II) and
(III) in Definition 1 hold true (with Q = {x : &x& < h}) for some continuous
functions a, b: [0, h) → R+ , V |R+ ×{x: 0<&x&<h} is of class C ∞ and (2.1) holds for
(t, x) ∈ (R+ \ N) × {x : 0 < &x& < h}. By [14, Thm 6] there exists a function
ν ∈ C ∞ (R) such that ν(r) = 0 for r ! 0, ν(r) > 0 and ν + (r) > 0 for r > 0,
ν(r) → +∞ as r → +∞ and
ν ◦ V ∈ C ∞ (R+ × {x : &x& < h}),
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
399
∂ α (ν ◦ V )(t, 0) = 0 ∀α ∈ Nn+1 , ∀t " 0.
Notice that for every (t, x) ∈ R+ × {x : &x& < h}
ν(a(&x&)) ! ν(V (t, x)) ! ν(b(&x&))
(4.8)
and for every (t, x) ∈ (R+ \ N) × {x : &x& < h}
∀v ∈ F (t, x)
∂(ν ◦ V )
(t, x) + ,∇x (ν ◦ V )(t, x), v- ! 0.
∂t
(4.9)
We infer from (4.8) and (4.9) that the assertions in Theorem 4 hold true when ν ◦V ,
ν ◦ a and ν ◦ b are substituted for V , a and b, respectively.
!
5. Some Application to Control Systems
Let us now consider the following control system:
ẋ = f (t, x, u),
(5.1)
where t " 0 is time, x ∈ Rn is the state variable and u ∈ Rm is the control
variable. We aim to investigate the links between the (robust) Lagrange stability of
the unforced system
ẋ = f (t, x, 0)
(5.2)
and the uniform bounded input bounded state (UBIBS) stability of (5.1), defined as
follows:

t0 " 0

&x0 & < R
∀R > 0, ∃M > 0 s.t.

&u&L∞ (t0 ,+∞) < R
⇒ ∀t " t0 &x(t; t0 , x0 , u)& < M
for any solution x(·; t0 , x0 , u) of (5.1) such that x(t0 ) = x0 . Obviously the UBIBS
stability of (5.1) implies the Lagrange stability of (5.2), but the converse is false
[4]. We shall say that (5.1) is UBIBS stabilizable if there exists some feedback law
u = k(t, x)+v such that the system ẋ = f (t, x, k(t, x)+v) is UBIBS stable (w.r.t.
the new control variable v). In practice, we look for maps k with the same regularity
as f . It is well known that in general the (even robust) Lagrange stability of (5.2)
does not imply the UBIBS stabilizability of (5.1) [4]. However,
dealing with
2when
m
(discontinuous) affine time-independent systems ẋ = f0 (x) + i=1 ui fi (x), it has
been proved by Bacciotti and Beccari [5] that the existence of a time-independent
weak Lyapunov function V = V (x) of class C 1 such that ,∇V (x), f0 (x)- ! 0
for each x ∈ Rn guarantees the UBIBS stabilizability of (5.1). Notice that the
existence of a time-independent continuous weak Lyapunov function V = V (x) (in
400
LIONEL ROSIER
the large) is equivalent [3, Thm. 9] to the so-called absolute boundedness (or absolute Lagrange stability), a concept which is stronger than the (robust) Lagrange
stability, since it accounts not only for the behaviour of trajectories, but also for
the behaviour of their prolongations. On the other hand, even for a smooth field f0 ,
the existence of a (time-independent) continuous weak Lyapunov function does
not imply the existence of a (time-independent) weak Lyapunov function of class
C 1 [7].
A time-dependent version of the Bacciotti–Beccari result in which we merely
assume the (robust) Lagrange stability of the unforced system is as follows.
THEOREM 5. Consider a system
ẋ = f0 (t, x) +
m
&
ui fi (t, x)
(5.3)
i=1
+
n
n
where for each i ∈ {0, . . . , m} the field fi ∈ L∞
loc (R × R , R ) and it is e.c.t.
Assume that the unforced system ẋ = f0 (t, x) is (robustly) Lagrange stable. Then
the system (5.3) is UBIBS stabilizable by means of a feedback law u = k(t, x) + v
+
n
m
such that k ∈ L∞
loc (R × R , R ) and k is e.c.t. as well. Moreover, if for each i " 0
r
fi is of class C for some r ∈ N ∪ {∞}, then k is also of class C r .
Notice that a preliminary version of this result was announced in [6].
Proof. It follows from Theorem 3 (applied with F (t, x) = Kx f0 (t, x)) that
there exist a set N ⊂ R+ of measure zero and three functions V = V (t, x), a =
a(r) and b = b(r) such that (I), (II), (III) and (A1) hold true. Without loss of
generality, we may assume that V and a|(0,+∞) are of class C ∞ and also (thanks to
Yoshizawa’s trick, see (3.2)) that
∀t ∈ R+ \ N, ∀x ∈ Rn \ {0}, ∀v ∈ Kx f0 (t, x),
∂V
1
(t, x) + ,∇x V (t, x), v- ! −
a(&x&).
∂t
(1 + 2t)2
Let ψ ∈ C ∞ (R) be such that ψ(r) = 0 for r ! 12 , ψ(r) = 1 for r " 1. Define
the components ki (1 ! i ! m) of k = k(t, x) as follows: ki (t, 0) = 0 for t " 0
and
ki (t, x)
"
!
= −&x&ψ(&x&)
sgn ,∇x V (t, x), fi (t, x)- × .
(1 + 2t)2
×ψ |,∇x V (t, x), fi (t, x)-| · &x& ·
a(&x&)
for t " 0 and x ∈ Rn \ {0}.
Notice that if fi is continuous, (,∇x V (t0 , x0 ), fi (t0 , x0 )- = 0 or x0 = 0) ⇒
ki ≡ 0 in a neighborhood of (t0 , x0 ). It follows that k is of class C r , r ∈ N ∪ {∞},
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
401
if the fi ’s are of class C r . When the fi ’s are merely assumed to be measurable,
locally (essentially) bounded and e.c.t., it is clear that k is a measurable locally
bounded function. We now prove that k is e.c.t. Let N ⊂ R+ × Rn be a zero
measure (Borel) set such that (4.1) is satisfied by fi (for 0 ! i ! m) and such that
for any (t0 , x0 ) ∈ R+ × Rn , for any ε > 0 there exists δ > 0 s.t.

|t − t0 | + &x − x0 & < δ 
(t, x) )∈ N

(t0 , x) )∈ N
⇒ ∀i ∈ {0, . . . , m}, &fi (t, x) − fi (t0 , x)& < ε.
(5.4)
Fix i ∈ {1, . . . , m}, a pair (t0 , x0 ) ∈ R+ × Rn and ε > 0. We are done if the
following claim is proved.
CLAIM 6. There exists δ̃ > 0 s.t.

|t − t0 | + &x − x0 & < δ̃ 
⇒ |ki (t, x) − ki (t0 , x)| < ε.
(t, x) )∈ N

(t0 , x) )∈ N
If x0 = 0, then (&x& < 12 , t " 0) ⇒ ki (t, x) = 0 and δ̃ = 12 is convenient.
Assume now x0 )= 0. Set Q := [0, t0 + 1] × B(x0 , &x0 &/2). There exists M > 0 s.t.
∀(t, x) ∈ Q \ N , &fi (t, x)& ! M. By an uniform continuity argument we infer
from (5.4) that there exists δ̃ > 0 such that for each pair (t, x) ∈ Q,

|t − t0 | + &x − x0 & < δ̃ 
⇒
(5.5)
(t, x) )∈ N

(t0 , x) )∈ N
|,∇x V (t, x), fi (t, x)- − ,∇x V (t0 , x), fi (t0 , x)-|
1
a(&x&)
1
<
,
(5.6)
2
20 (1 + 2t0 ) &x&
/ .
/
(1 + 2t)2
&x&ψ(&x&) //ψ |,∇x V (t, x), fi (t, x)-| &x&
a(&x&)
./
(5.7)
(1 + 2t0 )2 //
<
ε
−ψ |,∇x V (t0 , x), fi (t0 , x)-| &x&
a(&x&) /
and also
1
2
<
·
(1 + 2t0 )2
(1 + 2t)2
(5.8)
Now let (t, x) ∈ Q be such that the l.h.s. of (5.5) holds. If
|,∇x V (t0 , x), fi (t0 , x)-| <
1
a(&x&)
,
10 (1 + 2t0 )2 &x&
(5.9)
402
LIONEL ROSIER
then, by (5.6) and (5.8),
|,∇x V (t, x), fi (t, x)-| <
4
2
a(&x&)
a(&x&)
<
,
2
10 (1 + 2t0 ) &x&
10 (1 + 2t)2 &x&
hence ki (t, x) = ki (t0 , x) = 0 and we are done. If, instead, (5.9) does not hold, we
infer from (5.6) that
sgn(,∇x V (t, x), fi (t, x)-) = sgn(,∇x V (t0 , x), fi (t0 , x)-).
Hence, thanks to (5.7),
|ki (t, x) − ki (t0 , x)| < ε.
!
The proof of Claim 6 is complete.
We now proceed to the UBIBS stability.
CLAIM 7. The UBIBS stability holds true if for every R > 1, every x ∈ Rn and
every v ∈ Rm
>
= m
&
∂V
|vi | < R < &x& ⇒
(t, x) + ,∇x V (t, x), w- ! 0
∂t
i=1
for every=t ∈ R+ \ N and every
(5.10)
>
m
&
(ki (t, x) + vi )fi (t, x) .
w ∈ Kx f0 (t, x) +
i=1
(Notice that in the expression Kx (. . .) the vi ’s are reviewed as parameters.)
Indeed if (5.10) holds true, if a control map v(t) and an initial state x0 are such that
2
m
i=1 |vi (t)| < R for a.e. t " t0 and &x0 & < R, then for every solution x(t) in
Filippov’s sense of
ẋ = f0 (t, x) +
m
&
!
i=1
"
ki (t, x) + vi (t) fi (t, x),
x(t0 ) = x0 ,
we have &x(t)& ! a (b(R)) for every t " t0 : Otherwise there would exist times
t1 < t2 such that &x(t1 )& = R, &x(t)& > R for t1 < t ! t2 and a(&x(t2 )&) > b(R).
Then, thanks to (5.10), V̇ (t) ! 0 for a.e. t ∈ (t1 , t2 ), hence
−1
a(&x(t2 )&) ! V (t2 , x(t2 )) ! V (t1 , x(t1 )) ! b(R),
a contradiction.
!
2
(k
(t,
x)
Now it follows from [17, Thm. 1] that any w 2
∈ Kx (f0 (t, x) + m
i=1 i
+vi )fi (t, x)) may be decomposed as w = w0 + m
i=1 wi with w0 ∈ Kx f0 (t, x)
and wi ∈ Kx ((ki (t, x) + vi )fi (t, x)) for 1 ! i ! m. Fix R > 1. Since for each pair
(t, x) ∈ (R+ \ N) × (Rn \ {0}) and each w0 ∈ Kx f0 (t, x), we have
∂V
a(&x&)
(t, x) + ,∇x V (t, x), w0 - ! −
,
∂t
(1 + 2t)2
403
SMOOTH LYAPUNOV FUNCTIONS FOR DISCONTINUOUS STABLE SYSTEMS
n
in order to get (5.10) it is sufficient
2m to prove that for each t " 0, each x ∈ R \
m
B(0, R) and each v ∈ R s.t. i=1 |vi | < R we have
!
"
∀i = 1, . . . , m, ∀wi ∈ Kx (ki (t, x) + vi )fi (t, x) ,
a(&x&) |vi |
.
(5.11)
,∇x V (t, x), wi - !
(1 + 2t)2 &x&
Fix i ∈ {1, . . . , m} and wi ∈ Kx ((ki (t, x) +2vi )fi (t, x)). By [17, Thm. 1] and
j j
j
Carathéodory’s theorem, we may write wi = n+1
j =1 α ω , where α " 0 for each
2n+1 j
j " 1, j =1 α = 1 and for j " 1
j
j
ωj = lim (ki (t, xk ) + vi )fi (t, xk )
k→∞
j
for some sequence xk → x (as k → +∞), and we may assume that for every
j
j
k " 1, &xk & > R > 1, as well. Therefore ψ(&xk &) = 1 for each k " 1. We get
,∇x V (t, x), wi - = lim
k→∞
n+1
&
j =1
j
j
j
α j ,∇x V (t, xk ), fi (t, xk )-(ki (t, xk ) + vi ).
CLAIM 8. For any k
j
"
a(&xk &) |vi |
j
j !
j
,∇x V (t, xk ), fi (t, xk )- ki (t, xk ) + vi !
.
(1 + 2t)2 &xkj &
(5.12)
Notice first that
j
j
j
,∇x V (t, xk ), fi (t, xk )-ki (t, xk )
j
j
j
= −&x-k & |,∇x V (t, xk ), fi (t, xk )-|×
j
j
j (1
|,∇x V (t, xk ), fi (t, xk )-| &xk &
×ψ
! 0.
If
j
|,∇x V (t, xk ), fi (t, xk )-| !
j
j
a(&xk &)
j
(1 + 2t)2 &xk &
then (5.12) follows from (5.13). Otherwise
j
j
j
|,∇x V (t, xk ), fi (t, xk )-|
>
a(&xk &)
j
(1 + 2t)2 &xk &
+ 2t)2
j
a(&xk &)
.
(5.13)
404
LIONEL ROSIER
and we get
"
j
j !
j
,∇x V (t, xk ), fi (t, xk )- ki (t, xk ) + vi
j
j
j
= |,∇x V (t, xk ), fi (t, xk )-|(−&xk & ± vi )
! 0 (since
j
!
j
|vi | < R < &xk &)
a(&xk &) |vi |
,
(1 + 2t)2 &xkj &
!
as required.
We now infer from (5.12) that
,∇x V (t, x), wi - ! lim
k→∞
n+1
&
j =1
j
αj
a(&x&) |vi |
a(&xk &) |vi |
=
,
j
2
(1 + 2t) &xk &
(1 + 2t)2 &x&
i.e. (5.11) holds true. This completes the proof of Theorem 5.
Acknowledgement
The author would like to thank Professor A. Bacciotti for useful comments.
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