Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 3 (2007), 077, 21 pages
Global Stability of Dynamic Systems of High Order
Mohammed BENALILI and Azzedine LANSARI
Department of Mathematics, B.P. 119, Faculty of Sciences,
University Abou-bekr BelKaı̈d, Tlemcen, Algeria
E-mail: m benalili@mail.univ-tlemcen.dz, a lansari@mail.univ-tlemcen.dz
Received December 18, 2006, in final form June 04, 2007; Published online July 15, 2007
Original article is available at http://www.emis.de/journals/SIGMA/2007/077/
Abstract. This paper deals with global asymptotic stability of prolongations of flows
induced by specific vector fields and their prolongations. The method used is based on
various estimates of the flows.
Key words: global stability; vector fields; prolongations of flows
2000 Mathematics Subject Classification: 37C10; 34D23
1
Introduction
Global stability of dynamic systems is a vast domain in ordinary differential equations and it
is one of its main topics. Many works have been done in this context, we list some of them:
[3, 4, 5, 6, 7, 8]. However, little is known in the stability of high order (see [10] and [2]).
In this paper, we are concerned with the global asymptotic stability of prolongations of flows
generated by some specific vector fields and their perturbations. The method used is based on
various estimates of the flows and their prolongations. To justify the study of the dynamic of
prolongations of flows, we consider the Lie algebra χ(Rn ) of vector fields on Rn endowed with
the weak topology, which is the topology of the uniform convergence of vector fields and all
their derivatives on a compact sets. The Lie bracket is a fundamental operation not only in
differential geometry but in many fields of mathematics, such as dynamic and control theory.
The invertibility of this latter is of many uses i.e. given any vector fields X, Z find a vector
field Y such that [X, Y ] = Z. In the case of vector fields X defined in a neighborhood of
a point a with X(a) 6= 0 we have a positive answer: since in this case the vector field X is
locally of the form ∂x∂ 1 and the solution is given by
Z
x1
Y (x1 , . . . , xn ) =
Z(t, x2 , . . . , xn )dt,
−r
where kxk = max |xi | < r. In the case of singular vector fields, i.e. X(a) = 0 little is known.
1≤i≤n
Consider a singular vector field X defined in a neighborhood U of the origin 0 with X(0) = 0
and let φt be the flow generated by X. Suppose that X is complete and consider a vector field Y
defined on an open set V ⊃ φt (U ) for all t ∈ R. The transportation of a vector field Y along
the flow φt is defined as
(φt )∗ Y (x) = (Dφt · Y ) ◦ φ−t (x)
and the derivative with respect to t is given as follows
d
(φt )∗ Y = [(φt )∗ X, (φt )∗ Y ] .
dt
2
M. Benalili and A. Lansari
Put Yt = −
Rt
0 (φs )∗ Zds,
[X,Yt ] = −
then
d (φt )∗
dt t=0
Z
t
Z
(φs )∗ Zds = −
0
0
t
d
(φs )∗ Zds = Z − (φt )∗ Z.
ds
R +∞
So if (φt )∗ Z converges to 0 and the integral Y = − 0 (φs )∗ Zds is convergent in the weak
topology, then Y is a solution of our equation.
As applications of the right invertibility of the bracket operation on germs of vector fields at
a singular point we refer the reader to the papers by the authors [1, 2] (see also [10]).
2
Generalities
First we recall some definitions on global asymptotic stability as introduced in [9]. Let k·k be
the Euclidean norm on Rn , K ⊂ Rn is a compact set and f any smooth function on Rn , we put
α
kf kK
r = sup max kD f (x)k .
(1)
x∈K |α|≤r
Definition 1. A point a ∈ Rn is said globally asymptotically stable (in brief G.A.S.) of the
flow φt if
i) a is an asymptotically stable (in brief A.S.) equilibrium of the flow φt ;
ii) for any compact set K ⊂ Rn and any ε > 0 there exists TK > 0 such that for any
t ≥ TK we have kφt (x) − ak ≤ for all x ∈ K.
Definition 2. The point a ∈ Rn is said globally asymptotically stable of order r (1 ≤ r ≤ ∞)
for the flow φt if
i) a is a G.A.S. point for the flow φt ;
ii) for any compact set K ⊂ Rn and
∀ > 0, ∃ TK > 0 such that ∀ t ≥ TK ⇒ kφt − aIkK
r ≤ ,
where I denotes the identity map.
A vector field X will be called semi-complete if the X-flow φt = exp(tX) is defined for all
t ≥ 0.
First we quote the following proposition which characterizes the uniform asymptotic stability,
for a proof see the book of W. Hahn [5].
Let (φ)t denote a flow defined on Rn .
Proposition 1. The origin 0 in Rn is G.A.S. point for the flow φt if for any ball B(0, ρ),
centered at 0 and of radius ρ > 0, there exist t0 ≥ 0 and functions a, b such that
kφt (x)k ≤ a(kxk)b(t)
(2)
with a a continuous function on B(0, ρ) monotonously increasing such that a(0) = 0 and b is
a continuous function defined for any t ≥ t0 monotonously decreasing such that lim b(t) = 0.
t→+∞
3
Estimates of prolongations of f lows
We start with some perturbations of linear vector fields.
Global Stability of Dynamic Systems of High Order
3.1
3
Perturbation of linear vector f ields
Consider the following linear vector field
X1 =
n
X
αi xi
i=1
∂
,
∂xi
where the coefficients αi ∈ [a, b] ⊂ R and are not all 0.
The X1 -flow, ψt1 = exp(tX1 ) is then
ψt1 (x) = xeαt = x1 eα1 t , . . . , xn eαn t
∀t∈R
(3)
and its estimates are given by
kxk eat ≤ kψt1 (x)k ≤ kxk ebt .
(4)
Consider now a perturbation of the vector field X1 of the form Y1 = X1 + Z1 , where Z1 is
a smooth vector field globally Lipschitzian on Rn . The explicit form of the Y1 -flow is then
ψt1 (x) = xeAt +
Z
t
Z1 ψs1 (x) ds,
(5)
0

α1 · · ·
 .. . .
where A =  .
.
0 ···

0
.. .
. 
αn
Lemma 1. If the perturbation Z1 fulfills
kZ1 (x)k ≤ c0
∀ x ∈ Rn
(6)
then the vector field Y1 is complete and the Y1 -flow satisfies the estimates
kxk −
c0 bt c0
c0 bt c0
e +
≤ kψt1 (x)k ≤ kxk +
e − .
a
a
b
b
Proof . Clearly the Y1 -flow ψt1 is bounded for any t ∈ [0, T ] with T < +∞ and any x ∈ Rn .
The same is true if we replace t by −t. Then ψt1 is complete.
Consider now the equation
1d ψt1 (x)2 = ψt1 (x), αψt1 (x) + Z1 ψt1 (x) .
2 dt
(7)
Letting y = kψt1 (x)k, we deduce
ay 2 − c0 y ≤
1d 2
y ≤ by 2 + c0 y,
2 dt
y(0) = kxk
and by integrating we obtain
c0 bt c0
c0 bt c0
kxk −
e +
≤ y ≤ kxk +
e − .
a
a
b
b
Let B(0, 1) be the open unit ball centered at the origin 0.
4
M. Benalili and A. Lansari
Lemma 2. If the perturbation Z1 fulfills the estimates
kZ1 (x)k ≤ c00 kxk1+m
kZ1 (x)k ≤
c000 kxk
∀ x ∈ B (0, 1) and any integer m ≥ 1,
for every x ∈ Rn \ B (0, 1) ,
then Y1 is complete and the Y1 -flow fulfills the following estimates for ant t ≥ 0
kxk ea0 t ≤ ψt1 (x) ≤ kxk eb0 t ,
1
(x) ≤ kxk e−a0 t
kxk e−b0 t ≤ ψ−t
(8)
(9)
with c0 = max {c00 , c000 }, a0 = a − c0 and b0 = b + c0 .
Proof . Taking account of the explicit form of the flow (5) and the estimates (8), we deduce
1+m
that Y1 is complete. If x ∈ B (0, 1) then kZ1 (x)k ≤ c00 kxk
≤ c00 kxk, letting c0 = max {c00 , c000 }
then kZ1 (x)k ≤ c0 kxk for any x ∈ Rn . If we put y = ψt1 (x) the equation (7) leads to
(a − c0 )y ≤
d
y ≤ (b + c0 )y,
dt
y(0) = kxk
and putting b0 = b + c0 , a0 = a − c0 , we deduce the following estimates
kxk ea0 t ≤ y ≤ kxk eb0 t
for any t ≥ 0.
The same is also true in the on Rn \ B (0, 1).
Lemma 3. Suppose that all the coefficients αi are negative, a ≤ αi ≤ b < 0.
If the perturbation Z1 fulfills the estimates
kZ1 (x)k ≤ c0 kxk1+m
for any x ∈ Rn and any integer m ≥ 1,
(10)
then the vector field Y1 is semi-complete and the Y1 -flow satisfies the estimates for any t ≥ 0
− 1
c0
m
kxk eat 1 − kxkm (1 − eamt )
a
− 1
c0
m
≤ ψt1 (x) ≤ kxkebt 1 − kxkm (1 − ebmt )
.
b
(11)
Proof . By the relation (5) and
the estimates (10), we deduce that the vector field Y1 is semicomplete. Letting y = ψt1 (x) and taking into account the equation (7) and the estimates (10)
we deduce that
ay − c0 y 1+m ≤
d
y ≤ by + c0 y 1+m ,
dt
y(0) = kxk
and by integration we have
− 1
− 1
c0
c0
m
m
kxk eat 1 − kxkm (1 − eamt )
≤ y ≤ kxk ebt 1 − kxkm (1 − ebmt )
.
a
b
Example 1. Let the vector field
X3 =
n X
i
αi xi + βi x1+m
i
i=1
∂
∂xi
such that all the coefficients fulfilling
a ≤ αi ≤ b < 0,
a0 ≤ βi ≤ b0 ≤ 0
Global Stability of Dynamic Systems of High Order
5
and all the exponents mi are even positive integers with 0 < m00 ≤ mi ≤ m0 . The associated
flow φ3t = exp(tX3 ) is the solution of the dynamic system
d
φt (x) = X3 ◦ φt (x),
dt
φ0 (x) = x
or in coordinates
d
i
(φt (x))i = αi (φt (x))i + βi (φt (x))1+m
,
i
dt
φ0 (x) = x.
This latter is a Bernoulli type equation and its solution is given by
φ3t (x) i
αi t
= xi e
βi i
1 + xm
1 − eαi mi t
i
αi
−1
mi
.
(12)
The X3 -flow φ3t = exp (tX3 ) then has the explicit form
φ3t (x)
αt
= xe
β
1 + xm 1 − eαmt
α
−1
m
and the following estimates are true, ∀ t ≥ 0
kxk eat ≤ φ3t (x) ≤ kxk ebt .
3.2
(13)
Estimation of the kth prolongation of the Y1 -f low
Denote by η11 (t, x, ν) = Dψt1 (x)ν, where ν ∈ Rn , the first derivative with respect to x of the
Y1 -flow, solution of the dynamic system
d 1
η (t, x, ν) = (Dy X1 + Dy Z1 ) η11 (t, x, ν),
dt 1
η11 (0, x, ν) = ν
with y = ψt1 (x).
Lemma 4. If the perturbation Z1 fulfills the estimate
kDZ1 (x)k ≤ c1
for any x ∈ Rn ,
(14)
then the derivative of the Y1 -flow is complete and has the following estimates, for any t ≥ 0
1
ea1 t ≤ Dψt1 (x) ≤ eb1 t ,
e−b1 t ≤ Dψ−t
(x) ≤ e−a1 t
(15)
with a1 = a − c1 and b1 = b + c1 .
Proof . Consider as in previous lemmas the following equation
1d η11 (t, x, ν)2 = η11 (t, x, ν), (α + DZ1 ) η11 (t, x, ν)
2 dt
and put z = η11 (t, x, ν), so
(a − c1 )z 2 ≤
1d 2
z ≤ (b + c1 )z 2 ,
2 dt
z(0) = kνk
(16)
(17)
and then
kνk ea1 t ≤ z ≤ kνk eb1 t
for any t ≥ 0 and ν ∈ Rn .
6
M. Benalili and A. Lansari
Lemma 5. If the perturbation Z1 fulfils the estimates
l
D Z1 (x) ≤ c0 kxk1−l+m for any x ∈ B (0, 1) and all integers m ≥ 1,
l
l
D Z1 (x) ≤ c00 kxk1−l ∀ x ∈ Rn \ B (0, 1)
l
with l = 0, 1, then the first derivative of the Y1 -flow is complete and is estimated by, for any
t≥0
ea1 t ≤ kDψt1 (x)k ≤ eb1 t ,
1
e−b1 t ≤ kDψ−t
(x)k ≤ e−a1 t
(18)
with cl = max {c0l , c00l }, al = a − cl and bl = b + cl , l = 0, 1.
Proof . For any x ∈ B (0, 1) we have kDl Z1 (x)k ≤ c0l kxk1−l+m ≤ c0l kxk1−l and letting cl =
max {c0l , c00l }, we get for any x ∈ Rn kDl Z1 (x)k ≤ cl kxk1−l . By the same arguments as in
previous lemmas we get the estimates (18).
Lemma 6. Suppose that all the coefficients αi are negative, a ≤ αi ≤ b < 0.
If the perturbation Z1 fulfills the estimates
kZ1 (x)k ≤ c0 kxk1+m ,
kDZ1 (x)k ≤ c1 kxkm
for all x ∈ Rn and any integers m ≥ 1.
Then the estimates of the first derivation of the Y1 -flow are as follows, for any t ≥ 0
− c1
− c1
c0
c0
mc0
mc0
m
m
bt
1
amt
bmt
1 − kxk (1 − e )
≤ Dψt (x) ≤ e 1 − kxk (1 − e )
.
e
a
b
Proof . Letting y = ψt1 (x) and z = η11 (t, x, ν) in equation (16), we get
at
(a − c1 y m )z 2 ≤
1d 2
z ≤ (b + c1 y m )z 2 ,
2 dt
z(0) = kνk
and taking into account the estimates given by the relation (11), we obtain
−1
−1
c0
c0
kxkm emat 1 − kxkm (1 − eamt )
≤ y m ≤ kxkm embt 1 − kxkm (1 − ebmt )
a
b
consequently
kxkm emas ds
kνk exp at − c1
m
c0
ams )
0 1 − a kxk (1 − e
Z t
kxkm embs ds
≤ z ≤ kνk exp bt + c1
m
c0
bms )
0 1 − b kxk (1 − e
Z
t
which has the solution
− c1
c0
mc0
m
at
amt
kνk e
1 − kxk (1 − e )
a
− c1
c0
mc0
m
bt
bmt
≤ z ≤ kνk e 1 − kxk (1 − e )
b
for ν ∈ Rn .
Example 2. We consider the same vector field as in Example 1. Denote by ξ 13 (t, x, ν) =
Dφ3t (x)ν, ∀ ν ∈ Rn , the first derivation of the X3 -flow. In coordinates, we have for any i, j =
1, . . . , n,
φ3t (x) i
αi t
= xi e
βi i
1 + xm
1 − eαi mi t
i
αi
−1
mi
Global Stability of Dynamic Systems of High Order
7
so we deduce that
1
−1− mi i
∂
βi mi
3
αi t
αi mi t
1 + xi 1 − e
φ (x) i = e
δj
∂xj t
αi
and by the estimates (13) we get
eat ≤ Dφ3t (x) ≤ ebt .
The second derivative is
1
−2− mi
∂2
βi mi
βi −1+mi αi t
3
αi mi t
αi mi t
1 + xi 1 − e
.
φ (x) i = −(1 + mi ) xi
e
1−e
αi
αi
∂x2i t
Consequently, for l = 1, 2 and any x ∈ B (0, ρ) with ρ > 0 arbitrary fixed, there are constants
Ml > 0 such that
l 3 D φt (x) ≤ Ml ebt .
3.3
Perturbation of a nonlinear vector f ield
Consider the nonlinear vector field
X2 =
n
X
i
βi x1+m
i
i=1
∂
∂xi
with all mi > 0 and all βi ≤ 0.
The explicit form of the X2 -flow is then given by
−1
φ2t (x) = x(1 − mβtxm ) m
(19)
for any t ≥ 0 in the sense
−1
i mi
φ2t (x) i = xi (1 − mi βi txm
,
i )
1 ≤ i ≤ n.
Lemma 7. If the following assumptions are true
i) all the coefficients β i are non positive, −a0 ≤ βi ≤ −b0 ≤ 0
ii) all the exponents mi are even positive integers; 0 < m0 ≤mi ≤ m00 .
Then the vector field X2 is semi-complete and the X2 -flow satisfies the estimates
0
−1
kxk 1 + b m0 t kxk
m0 m0
≤
kφ2t (x)k
0
≤ kxk 1 + a
0
m00 t kxkm0
−10
m0
for any t ≥ 0.
(20)
Proof . Clearly the flow φ2t = exp(tX2 ) given by (19) is semi-complete i.e. defined for all t ≥ 0.
Consider the equation
D
1+m E
1d 2
kφt (x)k2 = φ2t (x), β φ2t (x)
2 dt
and put y = φ2t (x), then
b0 y 2+m0 ≤
1d 2
0
y ≤ a0 y 2+m0 ,
2 dt
and we get the estimates given in (20).
y(0) = kxk
8
3.4
M. Benalili and A. Lansari
Estimation of the kth order derivation of the X2 -f low
Let ξ 12 (t, x, ν) = Dφ2t (x)ν, ∀ ν ∈ Rn be the first derivation of the X2 -flow.
By formula (19), we get in coordinates
1
∂
1 if i = j,
j
mi −1− mi j
2
δi with δi =
φ (x) i = (1 − mi βi txi )
0 if i 6= j,
∂xj t
where i, j = 1, . . . , n.
Consequently
1 + b0 mt kxkm0
−1−
1
m0
0
≤ kDφ2t (x)k ≤ 1 + a0 m00 t kxkm0
−1− m10
0
.
(21)
To get the estimates of the second derivative, we put
i
wi = 1 − mi βi txm
i ,
so
d
wi = mi (wi − 1)x−1
i
dxi
and
∂
−1− m1
i.
φ2t (x) i = wi
∂xi
Consequently
∂2
− m1
− m1
i
i
wi−2 − wi−1 = x−1
φ2t (x) i = (1 + mi )x−1
i wi
i wi
2
∂xi
a21
a2
+ 22
wi wi
,
where a21 and a22 are real constants. Let ρ > 0 be any arbitrary and fixed real number, then for
any x ∈ B(0, ρ) and any t ≥ t0 > 0 and l = 1, 2 there is Ml > 0 such that
1
l 2 D φt (x) ≤ Ml t−1− m00 .
Suppose that for l = 1, . . . , k − 1, with fixed k, there exist constants alj and Ml > 0 such that
l
l
1 X a
∂l
j
1−l − mi
2
φ
(x)
=
x
w
,
i
i
j
i
∂xli t
w
j=1 i
where alj are real constants and
1
l 2 D φt (x) ≤ Ml t−1− m00
∀ t > 0.
For the estimates of the k th derivative, we compute
k
k
1 X a
∂k
j
1−k − mi
2
φ
(x)
=
x
w
,
i
t
i
j
i
k
∂xi
j=1 wi
k−1 k−1
d 2−k − m1 X aj
∂k
2
i
φt (x) i =
x wi
j
dxi i
∂xki
j=1 wi
=
x1−k
wi
i
− m1
i
k−1
X
ak−1
j
j=1
wij
where akj are real constants.
So we resume
(1 − k − jmi ) +
ak−1
j
wij+1
!
(1 + jmi )
1
=
−
xi1−k wi mi
k
X
akj
j=1
wij
,
Global Stability of Dynamic Systems of High Order
9
Proposition 2. Suppose that
i) all the coefficients satisfy β i ≤ 0, −a0 ≤ βi ≤ −b0 ,
ii) the exponents mi are even natural numbers such that 0 < m0 ≤ mi ≤ m00 .
Let ρ > 0 be any arbitrary fixed real number. For any x ∈ B(0, ρ), for any t ≥ t0 > 0 and
∀ k ≥ 1 there exist a constant Mk > 0 such that
1
k 2 D φt (x) ≤ Mk t−1− m00 .
3.5
(22)
Estimates of the Y2 -f low
Let
Y2 =
n
X
i=1
∂
i
βi x1+m
+ Z2i (x)
i
∂xi
the perturbation of the nonlinear vector field X2 and denote by ψt2 = exp(tY2 ) the solution of
the dynamic system
d 2
ψ (x) = Y2 ◦ ψt2 (x),
dt t
ψ02 (x) = x.
In coordinates we have, i = 1, . . . , n,
∂
1+mi
(t, x) + Z2i ψt2 (x) ,
ψ2,i (t, x) = βi ψ2,i
∂t
ψ2,i (0, x) = xi .
Putting
−mi
yi (t) = ψ2,i
(t, x)
and
ψt2 (x)
=y
−1
m
(t) =
−1
m1
−1
mn
y1 (t), . . . , yn
(t)
we get
−1−mi
yi0 (t) = −mi ψ2,i
(t, x)
∂
ψ2,i (t, x).
∂t
The Cauchy problem reads as
1+ m1
yi0 (t) = −mi βi − mi (yi (t))
−1
i
i
yi (0) = x−m
i
Z2i (y m (t)),
and has the following solution
i
yi (t) = x−m
− mi βi t − mi
i
Z
t
1
1+ m
i
yi (s)
−1
Z2i (y m (s)ds,
0
i.e.
ψ2,i (t, x) = xi 1 −
i
mi βi txm
i
−
i
mi xm
i
Z
t
−1−mi
ψi (s, x)
− m1
Z2i (ψs2 (x))ds
i
,
0
so we have the explicit form of the Y2 -flow
ψt2 (x)
m
m
Z
= x 1 − mβtx − mx
0
Now we will estimate the Y2 -flow.
t
− m1
ψs2 (x)−1−m Z2 (ψs2 (x))ds
.
(23)
10
M. Benalili and A. Lansari
Lemma 8. Suppose that
i) all the coefficients satisfy β i ≤ 0, −a0 ≤ βi ≤ −b0 ;
ii) the exponents mi are even natural numbers with 0 < m0 ≤ mi ≤ m00 ;
iii)
kZ2i (x)k ≤ c00 |xi |2+mi
kZ2i (x)k ≤
if x ∈ B (0, 1) ,
c000 |xi |1+mi
if x ∈ Rn \ B (0, 1)
with c0 = max {c00 , c000 }, b0 = b0 − c0 > 0, a0 = a0 + c0 .
Then
1) the vector field Y2 is semi-complete;
2) the Y2 -flow has the estimates
−1
0
kxk (1 + a0 m0 t kxkm0 ) m0 ≤ kψt2 (x)k ≤ kxk 1 + b0 m00 t kxkm0
−1
m
0
0
;
(24)
3) let ρ > 0 and t0 > 0 be fixed, then for any x ∈ B (0, ρ) and any t ≥ t0 > 0 there is
a constant M0 > 0 such that
kψt2 (x)k ≤ M0 kxk t
− m10
0
.
(25)
Proof . Let x ∈ B (0, 1), by assumption we have kZ2i (x)k ≤ c00 |xi |2+mi ≤ c00 |xi |1+mi , put
c0 = max {c00 , c000 } then for any x ∈ Rn we deduce kZ2i (x)k ≤ c0 |xi |1+mi . Now taking account of
the relation (23) we deduce that for any t ∈ [0, T ]
2 1
ψt (x) ≤ kxk 1 + mt kxkm (b0 − c0 ) − m ≤ kxk
hence the vector Y2 is semi-complete, i.e. defined for all t ≥ 0.
Consider the equation
D
1+m
E
1d
k ψt2 (x) i k2 = ψt2 (x) i , βi ψt2 (x) i i + Z2i ψt2 (x)
2 dt
we get yi = k ψt2 (x) i k and yi (0) = |xi |, so we deduce
1d 2
y ≤ (βi + c0 )yi2+mi ≤ −(b0 − c0 )yi2+mi
2 dt i
and
1d 2
y ≥ (βi − c0 )yi2+mi ≥ −(a0 + c0 )yi2+mi .
2 dt i
We put b0 = b0 − c0 and a0 = a0 + c0 , the solutions are estimated as
− m1
(|xi |−mi + a0 mi t)
i
− 1
≤ k ψt2 (x) i k ≤ (|xi |−mi + b0 mi t) mi .
Hence, we have the estimate (25).
(26)
Now, we estimate the first derivation of the Y2 -flow. Let η 12 (t, x, ν) = Dψt2 (x)ν, ∀ ν ∈ Rn
the solution of the dynamic system
d 1
η (t, x, ν) = (Dy X2 + Dy Z2 ) η21 (t, x, ν),
dt 2
with y = ψt2 (x).
η21 (0, x, ν) = ν
Global Stability of Dynamic Systems of High Order
11
Lemma 9. Suppose that
i) the coefficients are such that β i ≤ 0, −a0 ≤ βi ≤ −b0 ;
ii) the coefficients mi are even natural numbers, 0 < m0 ≤ mi ≤ m00 ;
iii)
kDl Z2i (x)k ≤ c0l |xi |2−l+mi if x ∈ B (0, 1) ,
kDl Z2i (x)k ≤ c00l |xi |1−l+mi if x ∈ Rn \ B (0, 1)
with l = 0, 1;
iv)
a0 = a0 + c0 ,
b0 = b0 − c0 > 0
and
a1 = a0 (1 + m0 ) + c1 ,
b1 = b0 (1 + m0 ) − c1 > 0
with cl = max {c0l , c00l }.
Then the first derivation of the Y2 -flow has the following estimates, for any t > 0
1 + b0 m0 t kxkm0
−
a1
b0 m0
0
≤ kDψt2 (x)k ≤ 1 + a0 m00 t kxkm0
1
− a bm
0
0
0
.
(27)
Let ρ > 0 be arbitrary and fixed for any x ∈ B(0, ρ), and any t ≥ t0 > 0 there is a constant
M1 > 0 such that
kDψt2 (x)k ≤ M1 t
b1
0
0 m0
−a
.
(28)
Proof . Let x ∈ B (0, 1), for l = 0, 1 we have
kDl Z2i (x)k ≤ c0l |xi |2−l+mi ≤ c0l |xi |1−l+mi .
Let cl = max {c0l , c00l } then for x ∈ Rn one has
kDl Z2i (x)k ≤ cl |xi |1−l+mi .
Consider the equation
1d 1
kη2 (t, x, ν)k2 = η21 (t, x, ν), (Dy X2 + Dy Z2 ) η21 (t, x, ν)
2 dt
and put z(t) = kη21 (t, x, ν)k with z(0) = kνk, then
1d 2
z ≤ sup ((1 + mi )βi + c1 )k ψt2 (x) i kmi z 2 ≤ z 2 sup −b1 k ψt2 (x) i kmi
2 dt
i=1,...,n
i=1,...,n
and
1d 2
z ≥ inf ((1 + mi )βi − c1 )k ψt2 (x) i kmi z 2 ≥ z 2 inf −a1 k ψt2 (x) i kmi .
i=1,...,n
i=1,...,n
2 dt
The solutions fulfill the following estimates
Z t
m
2
kνk exp inf
−a1
k ψs (x) i k i ds
i=1,...,n
0
Z t
m
2
i
≤ z(t) ≤ kνk exp sup
−b1
k ψs (x) i k ds
i=1,...,n
0
12
M. Benalili and A. Lansari
with, by (26)
m
|xi |mi
|xi |mi
2
i
≤
.
mi ≤ k ψt (x) i k
1 + a0 mi t |xi |
1 + b0 mi t |xi |mi
So we deduce
|xi |mi
−a1
kνk exp inf
mi ds
i=1,...,n
0 1 + b0 mi s |xi |
Z t
|xi |mi
−b1
≤ z(t) ≤ kνk exp sup
mi ds .
i=1,...,n
0 1 + a0 mi s |xi |
t
Z
Consequently the solutions satisfy
a1
0 mi
−b
kνk inf (1 + b0 mi t |xi |mi )
i=1,...,n
b1
0 mi
−a
≤ z(t) ≤ kνk sup (1 + a0 mi t |xi |mi )
.
i=1,...,n
Then there are constants m0 > 0 and m00 > 0 such that
a1
0 m0
−b
kνk (1 + b0 m0 t kxkm0 )
≤ kDψt2 (x)νk
b1
0 −
0
≤ kνk 1 + a0 m00 t kxkm0 a0 m0 ∀ ν ∈ Rn and for any t > 0.
Hence, we have the estimate (28).
3.6
Perturbation of binomial vector f ields
Let
Y3 =
n
X
i=1
∂
i
αi xi + βi x1+m
+ Z3i (x)
i
∂xi
with a ≤ αi ≤ b < 0, a0 ≤ βi ≤ b0 ≤ 0 and 0 < m0 ≤ mi ≤ m00 , be the perturbation of
the binomial vector field X3 and let ψt3 = exp(tY3 ) be the Y3 -flow which is the solution of the
dynamic system
d
ψt (x) = Y3 ◦ ψt (x),
dt
ψ0 (x) = x
and in coordinates, we get
∂
1+mi
ψ3,i (t, x) = αi ψ3,i (t, x) + βi ψ3,i
(t, x) + Z3,i ψt3 (x) ,
∂t
ψi (0, x) = xi
which is a Bernoulli type equation and by the same method as in the proof of previous lemmas
and with putting
−mi
yi (t) = ψ3,i
(t, x)
and
−1
ψt3 (x) = y m (t) =
−1
−1
m
y1 1 (t), . . . , ynmn (t)
Global Stability of Dynamic Systems of High Order
13
we get the solution
αi t
ψ3,i (t, x) = xi e
1+
i
− mi xm
i
Z
βi mi
α m t
x (1 − e i i )
αi i
t
! −1
αi mi s
[ψ3,i (s, x)]−1−mi Z3,i (ψs3 (x))e
mi
ds
0
and the implicit form of the Y3 -flow reads as
ψt3 (x)
3.7
αt
= xe
β
1 + xm (1 − eαmt ) − mxm
α
Z
t
−1−m
ψs3 (x)
Z3 (ψs3 (x))eαms ds
− m1
.
(29)
0
Estimation of the Y3 -f low
By the same arguments as in the previous, we get the following estimates of the Y3 -flow.
Lemma 10. If the following assumptions are true
i) all the coefficients αi are negative, −a ≤ αi ≤ −b < 0;
ii) all the coefficients β i are non positive, −a0 ≤ βi ≤ −b0 ;
iii) the exponents mi are even natural numbers with 0 < m0 ≤mi ≤ m00 ;
iv)
kZ3i (x)k ≤ c00 |xi |2+mi if x ∈ B (0, 1) ,
kZ3i (x)k ≤ c000 |xi |1+mi if x ∈ Rn \ B (0, 1)
with c0 = max {c00 , c000 }, b0 = b0 − c0 > 0, a0 = a0 + c0 .
Then
1) there exist constants m > 0 and m0 > 0 such that the Y3 −flow has the estimates, ∀ t ≥ 0
− 1
a0
m
kxk e−at 1 +
kxkm (1 − e−amt )
a
− 1 0
m
b0
m0
3
−bt
−bm0 t
≤ kψt (x)k ≤ kxk e
1 + kxk (1 − e
)
;
b
2) for any t > 0 there are positive constants c1 and c2 such that
c1 kxk e−at ≤ kψt3 (x)k ≤ c2 kxk e−bt ;
(30)
3) the vector field Y3 is semi-complete.
By similar calculations as in previous lemmas, we get the following estimates to the first
derivative of the Y3 -flow.
Lemma 11. Suppose that
i) all the coefficients αi are negative, −a ≤ αi ≤ −b < 0;
ii) all the coefficients β i are non positive, −a0 ≤ βi ≤ −b0 ;
iii) the exponents mi are even natural numbers such that 0 < m0 ≤ mi ≤ m00 ;
iv)
kDl Z3i (x)k ≤ c0l |xi |2−l+mi if x ∈ B (0, 1) ,
kDl Z3i (x)k ≤ c00l |xi |1−l+mi if x ∈ Rn \ B (0, 1)
with l = 0, 1;
14
M. Benalili and A. Lansari
v)
a0 = a0 + c0 ,
b0 = b0 − c0 > 0
and
a1 = a0 (1 + m0 ) + c1 ,
b1 = b0 (1 + m0 ) − c1 > 0
with cl = max {c0l , c00l }.
Then there exist constants m > 0 and m0 > 0 such that for any t ≥ 0
−at
− a1
b0 m
b0
m
−bmt
1 + kxk (1 − e
)
b
− b1 0
0
a0
0
a0 m
≤ kDψt3 (x)k ≤ e−bt 1 +
kxkm (1 − e−am t )
a
e
and for any t ≥ 0, there is a constant M1 > 0 such that
Dψt3 (x) ≤ M1 e−bt .
4
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Global stability of prolongations of f lows
With notations of the previous sections, we will give global stability of some flows.
4.1
Global stability of the Y1 -f low
Lemma 12. Let the vector fields
Y1 =
n
X
(αi xi + Z1i (x))
i=1
∂
∂xi
with the following assumptions
i) all the coefficients are negative, −a ≤ αi ≤ −b < 0;
ii)
kZ1 (x)k ≤ c00 kxk1+m
kZ1 (x)k ≤
c000 kxk
∀ x ∈ B (0, 1) and ∀m ≥ 1,
∀ x ∈ Rn \ B (0, 1) ;
iii) b0 = b − c0 > 0, where c0 = max {c00 , c000 }.
Then the origin 0 is a globally asymptotically stable equilibrium to the Y1 -flow ψt1 on Rn .
Proof . Let ψt1 = exp(tY1 ) be the Y1 -flow, then by the assumptions and the estimates given by
Lemma 2 we get that
1 ψt (x) ≤ kxk e−b0 t ∀ t ≥ 0 and ∀x ∈ Rn
and by Proposition 1, the origin 0 is G.A.S. for ψt1 on Rn .
Example 3. We consider the vector field
X3 =
n X
i=1
i
αi xi + βi x1+m
i
∂
∂xi
Global Stability of Dynamic Systems of High Order
15
of Example 1 with a ≤ αi ≤ b < 0, a0 ≤ βi ≤ b0 ≤ 0. The X3 -flow φ3t = exp (tX3 ) is then given
by
φ3t (x)
αt
= xe
β
1 + xm 1 − eαmt
α
−1
m
.
Let ρ > 0 be arbitrary and fixed real number. By the estimates (13), we have for any x ∈ B(0, ρ)
and any t ≥ t0 ≥ 0
3 φt (x) ≤ kxk e−bt .
By Proposition1 the origin 0 is a G.A.S. for the flow φ3t on Rn .
4.2
Global stability of the f irst prolongation of the Y1 -f low
Lemma 13. With the same assumptions as in Lemma 12 and the following conditions
kDZ1 (x)k ≤ c01 kxkm
kDZ1 (x)k ≤
c001
∀ x ∈ B (0, 1) and ∀ m ≥ 1,
∀ x ∈ Rn \ B (0, 1)
with b1 = b − c1 > 0 and c1 = max {c01 , c001 }.
Then the origin 0 is a globally asymptotically stable for the first prolongation of the Y1 -flow
ψt1 on Rn .
Proof . By the estimates (18) and the hypothesis we deduce that
kDψt1 (x)νk ≤ kνk e−b1 t
∀ t > 0, ∀ ν ∈ Rn
and by Proposition 1,we obtain that the origin 0 is a G.A.S. equilibrium on Rn for η11 (t, x, v) =
Dψt1 (x)ν.
4.3
Global stability of the kth prolongation of the Y1 -f low
Suppose that
i) all the coefficients are negative, −a ≤ αi ≤ −b < 0;
ii) for any l = 1, . . . , k − 1
kDl Z1 (x)k ≤ c0l kxk1−l+m
l
kD Z1 (x)k ≤
c00l
for any x ∈ B (0, 1) and for any integer m ≥ l − 1,
n
∀ x ∈ R \ B (0, 1) ,
a0 = a + c0 ,
b0 = b − c0 > 0,
a1 = a + c1 ,
b1 = b − c1 > 0
with cl = max {c0l , c00l }, bl = cl ∀ l ≥ 2.
Put η1l (t, x, ν, . . . , ν) = Dk ψt1 (x)ν k , where ν ∈ Rn . Since by Lemmas 12 and 13 the origin 0 is
an G.A.S. equilibrium for η1l , with l = 0, 1, on Rn , we suppose that this property remains true
for l = 0, 1, . . . , k − 1 with k ≥ 2 i.e. for any ρ > 0 and any x ∈ B(0, ρ) there exist constants
Ml > 0 such that for any t ≥ t0 > 0
kDl ψt1 (x)k ≤ Ml e−b1 t .
We will show that the origin 0 is a G.A.S. equilibrium for η1k on Rn . η1k (t, x, ν, . . . , ν) =
Dk ψt1 (x)ν k is solution of the dynamic system
d k
η = Dy Y1 · η1k + Gk1 (t, x, ν),
dt 1
η1k (0, x, ν, . . . , ν) = ν
16
M. Benalili and A. Lansari
with y = ψt1 (x) and
Gk1 (t, x, ν) =
k
X

X
Dyl Y1 (y)


Dij ψt1 (x)ν ij 
j=1
i1 +···+il =k
l=2
l
Y
ij >0
=
k−1
X

X
Dyl Z1 (y)
l
Y

l=2

Dij ψt1 (x)ν ij  + Dyk Z1 (y) Dψt1 (x)ν
k
.
j=1
i1 +···+il =k
ij >0
Consequently we get
η1k (t, x, ν, . . . , ν)
=
Dψt1 (x)ν
Z
+
t
DΨ1t−s (ψs1 (x))Gk1 (s, x, ν)ds.
0
The integral is well defined at s = 0, since
1
lim Dψt−s
(ψs1 (x)) = Dψt1 (x)
s→0+
and there exist constants Al > 0 such that
lim Gk1 (s, x, ν) =
s→0+
k
X
Al Dyl Z1 (y)ν k .
l=2
We will show that it converges uniformly with respect to x as t + ∞. Put
Z t
1
Ik =
kDψt−s
(ψs1 (x))kkGk1 (s, x, ν)kds.
0
kDl Z1 (x)k
Since
≤ cl ∀ l ≥ 1, ∀ x ∈ Rn , there are constants bl > 0 such that ∀ y ∈ Rn ,
kDyl Y1 (y)k ≤ bl and by the assumption of recurrence there exist constants Ml > 0 such that
kDl ψt1 (x)k ≤ Ml e−b1 t
∀ t ≥ 0.
We deduce that there is a constant Ck > 0 such that
Z t
k
X
b l Ml
e−b1 (t−s+sl) ds ≤ Ck e−b1 t .
Ik ≤
0
l=2
So for any x ∈
Rn
lim Ik ≤
t→+∞
one has
k
X
l
Z
Ml bl kνk
−b1 s
e
0
l=2
+∞
k
1 X
Ml bl kνkl
ds =
b1
l=2
and the integral Ik is uniformly convergent with respect to x ∈ Rn as t → +∞. Consequently
Z +∞
1
lim kη1k k = lim kDψt1 (x)νk +
lim kDψt−s
(ψs1 (x))kkGk1 (s, x, ν)kds = 0
t→+∞
t→+∞
0
t→+∞
and there is a constant Mk0 > 0 such that
Z +∞
1
kη1k k ≤ kDψt1 (x)νk +
kDψt−s
(ψs1 (x))kkGk1 (s, x, ν)kds ≤ Mk0 kνkk e−b1 t .
0
This show by Proposition 1 that the origin 0 is a G.A.S. equilibrium to η1k on Rn . We formulate
our proving as follows
Proposition 3. Let k ≥ 0 be any integer. The origin 0 is a G.A.S. equilibrium of order k for
the Y1 -flow and there is a constant Mk > 0 such that ∀ t > 0
kDk ψt1 (x)k ≤ Mk e−b1 t ,
1
kDk ψ−t
(x)k ≤ Mk ea1 t .
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Global Stability of Dynamic Systems of High Order
5
17
Global stability of a f low generated
by nonlinear perturbed vector f ields
First we will start with monomial vector fields.
5.1
Global stability of the X2 -f low
Let
X2 =
n
X
i=1
i
βi x1+m
i
∂
∂xi
with
(i) all the coefficients β i ≤ 0 such that −a0 ≤ βi ≤ −b0 ;
(ii) all the exponents mi are even natural integers with 0 < m0 ≤mi ≤ m00 .
Let φ2t = exp (tX2 ) be the X2 -flow. By the estimations (19) we obtain
2 −1
φt (x) ≤ kxk 1 + a0 m00 t kxkm00 m00 .
Let ρ > 0 be arbitrary fixed, for any x ∈ B(0, ρ) and any t ≥ t0 > 0 there is a constant
M0 > 0 such that
kφ2t (x)k ≤ M0 kxk t
− m10
0
.
By Proposition 1, the origin is a globally asymptotically stable equilibrium to the flow φ2t on Rn .
Let l = 1, 2, . . . any positive integer. By Proposition 2, we have: for any fixed ρ > 0, and all
x ∈ B(0, ρ) and t ≥ t0 > 0, there exist constants Ml > 0 and Ml0 > 0 such that
kDl φ2t (x)k ≤ Ml t
−1− m10
0
and
kDl φ20 (x)k ≤ Ml0 .
So the origin 0 is a G.A.S. equilibrium for Dl φ2t (x) on Rn .
Resuming our proving, we get
Proposition 4. Let k ≥ 0 be any integer. Under the above conditions (i) and (ii), the origin 0
is a G.A.S. of order k for the X2 -flow on Rn .
5.2
Global stability of high order of the Y2 -f low
Let
Y2 =
n
X
i=1
∂
i
βi x1+m
+ Z2,i (x)
i
∂xi
be a smooth vector field on Rn such that
i) all the coefficients β i ≤ 0 are non negative with −a0 ≤ βi ≤ −b0 ;
ii) mi are even natural numbers with 0 < m0 ≤mi ≤ m00 ;
iii) for k = 0, . . . , 1 + mi
kDk Z2i (x)k ≤ c0k |xi |2−k+mi
if x ∈ B (0, 1) ;
kDk Z2i (x)k ≤ c00k |xi |1−k+mi
if x ∈ Rn \ B (0, 1) ;
18
M. Benalili and A. Lansari
iv) for any k ≥ 2 + mi
kDk Z2i (x)k ≤ ck ;
v)
a0 = a0 + c0 ,
a1 = a0 (1 + m0 ) + c1 ,
b0 = b0 − c0 > 0,
b1 = b0 (1 + m0 ) − c1 > a0 m00
with ck = max {c0k , c00k }.
Remark 1. If x ∈ B (0, 1) then kDk Z2i (x)k ≤ c0k |xi |2−k+mi ≤ c0k |xi |1−k+mi . Putting cl =
max {c0l , c00l }, we deduce that for any x ∈ Rn have kDk Z2i (x)k ≤ ck |xi |1−k+mi .
Global stability of the Y2 -f low on Rn
5.2.1
Let ψt2 = exp (tY2 ) be the Y2 -flow and let ρ > 0 be arbitrary and fixed, so by the estimates (25)
for all x ∈ B(0, ρ) and all t ≥ t0 > 0 there is a constant M0 > 0 such that
kψt2 (x)k ≤ M0 kxk t
− m10
0
.
So by Proposition 1, the origin 0 is a G.A.S. equilibrium for the Y2 -flow ψt2 on Rn .
Global stability of prolongation of the Y2 -f low on Rn
5.2.2
We proceed by recurrence. Since it is already true for k = 0, we suppose that for any l =
1, . . . , k − 1, with k ≥ 2, the origin 0 is a G.A.S. to Dl ψt2 (x) on Rn that is to say for any fixed
ρ > 0, all x ∈ B(0, ρ) and all t ≥ t0 > 0 there are constants Ml > 0 such that
kDl ψt2 (x)k ≤ Ml t
b1
0
0 m0
−a
kDl ψ02 (x)k ≤ Ml0 .
and
We will show that 0 is a G.A.S. for Dk ψt2 (x) on Rn .
Put η2k (t, x, ν, . . . , ν) = Dk ψt2 (x)ν k ∀ ν ∈ Rn which is solution of the dynamic system
d k
η = Dy Y2 · η2k + Gk2 (t, x, ν),
dt 2
η2k (0, x, ν, . . . , ν) = ν
with y = ψt2 (x) and
Gk2 (t, x, ν) =
k
X
l=2

Dyl Y2 (y)
X
l
Y

i1 +···+il =k

Dij ψt2 (x)ν ij  .
j=1
ij >0
By the method of the resolvent, we deduce
Z t
2
η2k (t, x, ν, . . . , ν) = Dψt2 (x)ν +
Dψt−s
(ψs2 (x))Gk2 (s, x, ν)ds.
0
Clearly the integral
Ik1
Z
=
0
1
2
kDψt−s
(ψs2 (x))kkGk2 (s, x, ν)kds
Global Stability of Dynamic Systems of High Order
19
is well defined at s = 0 and s = t, since
2
lim Dψt−s
(ψs2 (x)) = Dψt2 (x).
s→0+
By the recurrent assumption Dl ψ02 (x) are bounded and there exist constants Al > 0 such that
lim kGk2 (s, x, ν)k ≤
s→0+
k
X
Al kDxl Y2 (x)ν l k.
l=2
In the same way
2
lim Dψt−s
(ψs2 (x)) = identity.
s→t−
Now, we have to show that
Z t
2
2
kDψt−s
(ψs2 (x))kkGk2 (s, x, ν)kds
Ik =
1
converges uniformly on any compact set K ⊂ Rn as t → 0.
Let x ∈ K, by the relations (26) and (28) we get for all t ≥ 0
−1
−1
0 0
kxk (1 + a0 m0 t kxkm0 ) m0 ≤ kψt2 (x)k ≤ kxk 1 + b0 m00 t kxkm0 m0 ,
b1
a
0 −
0
− 1
(1 + b0 m0 t kxkm0 ) b0 m0 ≤ kDψt2 (x)k ≤ 1 + a0 m00 t kxkm0 a0 m0 .
2 (ψ 2 (x)) is bounded. Since for any x ∈ Rn and any
So kyk = kψt2 (x)k ≤ kxk and Dψt−s
s
l = 1, . . . , 1 +mi , kDl Z2i (x)k ≤ cl |xi |1−l+mi then Dyl Y2 (y) are bounded. Now by the assumption
of recurrence there exist constants Ml > 0 such that for any t > 0
kDl ψt2 (x)k ≤ Ml t
with a0 m00 < b1 i.e.
lim I 2
t→+∞ k
≤
b1
0
0 m0
−a
b1
a0 m00
k
X
> 1, and we deduce the existence of constants Cl > 0 such that
Z
Cl
+∞ − lb1
a m0
s
0
0
k
X
ds ≤
1
l=2
l=2
Cl
−1
lb1
−1
.
a0 m00
The integral Ik2 converges uniformly on any compact K ⊂ Rn as t → +∞.
Now since the integral is well defined at s = 0, then
lim kη2k (t, x, ν, . . . , ν)k ≤ lim kDψt2 (x)νk = kνk
t→0
t→0
hence there is a constant Mk0 > 0 such that
kDk ψ02 (x)k ≤ Mk0 .
In the same way as above the integral
putting τ = st we obtain
η2k (t, x, ν, . . . , ν)
=
Dψt2 (x)ν
Z
+t
0
Rt
0
2 (ψ 2 (x))kkGk (s, x, ν)kds is well defined and
kDψt−s
s
2
1
2
2
k
Dψt(1−τ
) (ψtτ (x))G2 (tτ, x, ν)dτ.
20
M. Benalili and A. Lansari
Since b1 = b0 (1 + m0 ) − c1 > a0 m00 , by the estimates (26) and (28), we deduce the existence of
a constant Mk > 0 such that
kη2k (t, x, ν, . . . , ν)k
kDψt2 (x)νk
≤
Z
1
+t
kGk2 (tτ, x, ν)kdτ
0
kDψt2 (x)νk
≤
+t
k Z
X
l=2
0
1
lb1
0
0 m0
−a
0
b1
−a m
kxk1+m0 −l
0
0 0.
dτ
≤
M
t
k
1+m00 −l
0
m00
1 + b0 m00 tτ kxkm0
(tτ )
Which shows that the origin 0 is a G.A.S. equilibrium for η2k on Rn . We formulate this fact as
Proposition 5. Let k ≥ 0 be any integer. Under the above conditions (i), (ii), (iii), (iv)
and (v), the origin 0 is a G.A.S. of order k on Rn for the Y2 -flow and there is a constant
Mk > 0 such that for any t ≥ t0 > 0
kDk ψt2 (x)k ≤ Mk t
5.3
b1
0
0 m0
−a
.
(33)
Global stability of prolongations of the Y3 -f low
Let
Y3 =
n
X
i=1
∂
i
αi xi + βi x1+m
+ Z3i (x)
i
∂xi
with
i) all the coefficient αi are negative with −a ≤ αi ≤ −b;
ii) all the coefficients βi ≤ 0 and −a0 ≤ βi ≤ −b0 ;
iii) the exponents mi are even natural numbers with 0 < m0 ≤ mi ≤ m00 ;
iv) For any k = 0, . . . , 1 + mi
kDk Z3i (x)k ≤ c0k |xi |2−k+mi
if x ∈ B (0, 1) ,
kDk Z3i (x)k ≤ c00k |xi |1−k+mi
if x ∈ Rn \ B (0, 1) ;
v) for any k ≥ 2 + mi
kDk Z3i (x)k ≤ ck ;
vi)
a0 = a0 + c0 ,
a1 = a0 (1 + m0 ) + c1 ,
b0 = b0 − c0 > 0,
b1 = b0 (1 + m0 ) − c1 > 0
with ck = max {c0k , c00k } .
Remark 2. If x ∈ B (0, 1) then kDk Z3i (x)k ≤ c0k |xi |2−k+mi ≤ c0k |xi |1−k+mi .
Let cl = max {c0l , c00l }, for any x ∈ Rn one has kDk Z3i (x)k ≤ ck |xi |1−k+mi .
Global Stability of Dynamic Systems of High Order
5.3.1
21
Global stability of the Y3 -f low Rn
Denote by ψt3 = exp(tY3 ), by the estimates (30), we have
kψt3 (x)k ≤ Ckxke−bt
∀ t > 0 and ∀ x ∈ Rn ,
where C > 0 is a constant. So by Proposition 1, 0 is a G.A.S. on Rn . We proceed by recurrence;
since the property is true in case k = 0, we assume that the property remains true for any
l = 1, . . . , k − 1, with k fixed i.e. 0 is a global G.A.S. of η3l (t, x, ν, . . . ν) = kDl ψt3 (x)ν k k on Rn
and there exist constants Ml > 0 such that for any t > 0
kDl ψt3 (x)k ≤ Ml e−bt .
We will show that 0 is a G.A.S. equilibrium to η3k on Rn .
η3k (t, x, ν, . . . , ν) is a solution to the dynamic system
d k
η = Dy η3k + Gk3 (t, x, ν)
dt 3
with y = ψt3 (x) and
Gk3 (t, x, ν) =
k
X

X
Dyl Y3 (y)
l=2
l
Y

i1 +···+il =k

Dij ψt3 (x)ν ij  .
j=1
ij >0
By the method of the resolvent, we get
Z t
k
3
3
η3 (t, x, ν, . . . , ν) = Dψt (x)ν +
Dψt−s
(ψs3 (x))Gk3 (s, x, ν)ds
0
and by the same argument as for the Y 1 -flow, we deduce that for any integer k ≥ 0 there exist
a constant Mk such that ∀ t ≥ 0
kDk ψt3 (x)k ≤ Mk kxk e−bt .
By Proposition 1, we have
Proposition 6. Under the above conditions (i), (ii), (iii), (iv), (v) and (vi), the origin 0 is a
G.A.S. equilibrium of order k on Rn to the Y3 -flow .
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