Global Stability of Dynamic Systems of High Order
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.
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2007
|
| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/147373 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Global Stability of Dynamic Systems of High Order / M. Benalili, A. Lansari // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 10 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-147373 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1473732019-02-15T01:25:02Z Global Stability of Dynamic Systems of High Order Benalili, M. Lansari, A. 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. 2007 Article Global Stability of Dynamic Systems of High Order / M. Benalili, A. Lansari // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 10 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37C10; 34D23 http://dspace.nbuv.gov.ua/handle/123456789/147373 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
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. |
| format |
Article |
| author |
Benalili, M. Lansari, A. |
| spellingShingle |
Benalili, M. Lansari, A. Global Stability of Dynamic Systems of High Order Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Benalili, M. Lansari, A. |
| author_sort |
Benalili, M. |
| title |
Global Stability of Dynamic Systems of High Order |
| title_short |
Global Stability of Dynamic Systems of High Order |
| title_full |
Global Stability of Dynamic Systems of High Order |
| title_fullStr |
Global Stability of Dynamic Systems of High Order |
| title_full_unstemmed |
Global Stability of Dynamic Systems of High Order |
| title_sort |
global stability of dynamic systems of high order |
| publisher |
Інститут математики НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147373 |
| citation_txt |
Global Stability of Dynamic Systems of High Order / M. Benalili, A. Lansari // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 10 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT benalilim globalstabilityofdynamicsystemsofhighorder AT lansaria globalstabilityofdynamicsystemsofhighorder |
| first_indexed |
2025-07-11T01:56:39Z |
| last_indexed |
2025-07-11T01:56:39Z |
| _version_ |
1837313837128220672 |
| fulltext |
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 BelKäıd, 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 ∂
∂x1
and the solution is given by
Y (x1, . . . , xn) =
∫ x1
−r
Z(t, x2, . . . , xn)dt,
where ‖x‖ = max
1≤i≤n
|xi| < r. In the case of singular vector fields, i.e. X(a) = 0 little is known.
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
dt
(φt)∗Y = [(φt)∗X, (φt)∗Y ] .
mailto:m_benalili@mail.univ-tlemcen.dz
mailto:a_lansari@mail.univ-tlemcen.dz
http://www.emis.de/journals/SIGMA/2007/077/
2 M. Benalili and A. Lansari
Put Yt = −
∫ t
0 (φs)∗Zds, then
[X,Yt] = − d
dt
∣∣
t=0
(φt)∗
∫ t
0
(φs)∗Zds = −
∫ t
0
d
ds
(φs)∗Zds = Z − (φt)∗Z.
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 ‖·‖ be
the Euclidean norm on Rn, K ⊂ Rn is a compact set and f any smooth function on Rn, we put
‖f‖K
r = sup
x∈K
max
|α|≤r
‖Dαf(x)‖ . (1)
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 ‖φt (x)− a‖ ≤ ε 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
∀ ε > 0, ∃ TK > 0 such that ∀ t ≥ TK ⇒ ‖φt − aI‖K
r ≤ ε,
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
‖φt(x)‖ ≤ a(‖x‖)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
t→+∞
b(t) = 0.
3 Estimates of prolongations of flows
We start with some perturbations of linear vector fields.
Global Stability of Dynamic Systems of High Order 3
3.1 Perturbation of linear vector fields
Consider the following linear vector field
X1 =
n∑
i=1
αixi
∂
∂xi
,
where the coefficients αi ∈ [a, b] ⊂ R and are not all 0.
The X1-flow, ψ1
t = exp(tX1) is then
ψ1
t (x) = xeαt =
(
x1e
α1t, . . . , xne
αnt
)
∀ t ∈ R (3)
and its estimates are given by
‖x‖ eat ≤ ‖ψ1
t (x)‖ ≤ ‖x‖ 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
ψ1
t (x) = xeAt +
∫ t
0
Z1
(
ψ1
s(x)
)
ds, (5)
where A =
α1 · · · 0
...
. . .
...
0 · · · αn
.
Lemma 1. If the perturbation Z1 fulfills
‖Z1(x)‖ ≤ c0 ∀ x ∈ Rn (6)
then the vector field Y1 is complete and the Y1-flow satisfies the estimates(
‖x‖ − c0
a
)
ebt +
c0
a
≤ ‖ψ1
t (x)‖ ≤
(
‖x‖+
c0
b
)
ebt − c0
b
.
Proof. Clearly the Y1-flow ψ1
t is bounded for any t ∈ [0, T ] with T < +∞ and any x ∈ Rn.
The same is true if we replace t by −t. Then ψ1
t is complete.
Consider now the equation
1
2
d
dt
∥∥ψ1
t (x)
∥∥2 =
〈
ψ1
t (x), αψ
1
t (x) + Z1
(
ψ1
t (x)
)〉
. (7)
Letting y = ‖ψ1
t (x)‖, we deduce
ay2 − c0y ≤
1
2
d
dt
y2 ≤ by2 + c0y, y(0) = ‖x‖
and by integrating we obtain(
‖x‖ − c0
a
)
ebt +
c0
a
≤ y ≤
(
‖x‖+
c0
b
)
ebt − c0
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
‖Z1(x)‖ ≤ c′0 ‖x‖
1+m ∀ x ∈ B (0, 1) and any integer m ≥ 1,
‖Z1(x)‖ ≤ c′′0 ‖x‖ for every x ∈ Rn \B (0, 1) , (8)
then Y1 is complete and the Y1-flow fulfills the following estimates for ant t ≥ 0
‖x‖ ea0t ≤
∥∥ψ1
t (x)
∥∥ ≤ ‖x‖ eb0t,
‖x‖ e−b0t ≤
∥∥ψ1
−t(x)
∥∥ ≤ ‖x‖ e−a0t (9)
with c0 = max {c′0, c′′0}, a0 = a− c0 and b0 = b+ c0.
Proof. Taking account of the explicit form of the flow (5) and the estimates (8), we deduce
that Y1 is complete. If x ∈ B (0, 1) then ‖Z1(x)‖ ≤ c′0 ‖x‖
1+m ≤ c′0 ‖x‖, letting c0 = max {c′0, c′′0}
then ‖Z1(x)‖ ≤ c0 ‖x‖ for any x ∈ Rn. If we put y =
∥∥ψ1
t (x)
∥∥ the equation (7) leads to
(a− c0)y ≤
d
dt
y ≤ (b+ c0)y, y(0) = ‖x‖
and putting b0 = b+ c0, a0 = a− c0, we deduce the following estimates
‖x‖ ea0t ≤ y ≤ ‖x‖ eb0t 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
‖Z1(x)‖ ≤ c0 ‖x‖1+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
‖x‖ eat
(
1− c0
a
‖x‖m (1− eamt)
)− 1
m (11)
≤
∥∥ψ1
t (x)
∥∥ ≤ ‖x‖ebt
(
1− c0
b
‖x‖m (1− ebmt)
)− 1
m
.
Proof. By the relation (5) and the estimates (10), we deduce that the vector field Y1 is semi-
complete. Letting y =
∥∥ψ1
t (x)
∥∥ and taking into account the equation (7) and the estimates (10)
we deduce that
ay − c0y
1+m ≤ d
dt
y ≤ by + c0y
1+m, y(0) = ‖x‖
and by integration we have
‖x‖ eat
(
1− c0
a
‖x‖m (1− eamt)
)− 1
m ≤ y ≤ ‖x‖ ebt
(
1− c0
b
‖x‖m (1− ebmt)
)− 1
m
. �
Example 1. Let the vector field
X3 =
n∑
i=1
(
αixi + βix
1+mi
i
) ∂
∂xi
such that all the coefficients fulfilling
a ≤ αi ≤ b < 0, a′ ≤ βi ≤ b′ ≤ 0
Global Stability of Dynamic Systems of High Order 5
and all the exponents mi are even positive integers with 0 < m′
0 ≤ mi ≤ m0. The associated
flow φ3
t = exp(tX3) is the solution of the dynamic system
d
dt
φt(x) = X3 ◦ φt(x), φ0(x) = x
or in coordinates
d
dt
(φt(x))i = αi (φt(x))i + βi (φt(x))
1+mi
i , φ0(x) = x.
This latter is a Bernoulli type equation and its solution is given by
(
φ3
t (x)
)
i
= xie
αit
(
1 +
βi
αi
xmi
i
(
1− eαimit
))−1
mi
. (12)
The X3-flow φ3
t = exp (tX3) then has the explicit form
φ3
t (x) = xeαt
(
1 +
β
α
xm
(
1− eαmt
))−1
m
and the following estimates are true, ∀ t ≥ 0
‖x‖ eat ≤
∥∥φ3
t (x)
∥∥ ≤ ‖x‖ ebt. (13)
3.2 Estimation of the kth prolongation of the Y1-f low
Denote by η1
1(t, x, ν) = Dψ1
t (x)ν, where ν ∈ Rn, the first derivative with respect to x of the
Y1-flow, solution of the dynamic system
d
dt
η1
1(t, x, ν) = (DyX1 +DyZ1) η1
1(t, x, ν), η1
1(0, x, ν) = ν
with y = ψ1
t (x).
Lemma 4. If the perturbation Z1 fulfills the estimate
‖DZ1(x)‖ ≤ c1 for any x ∈ Rn, (14)
then the derivative of the Y1-flow is complete and has the following estimates, for any t ≥ 0
ea1t ≤
∥∥Dψ1
t (x)
∥∥ ≤ eb1t, e−b1t ≤
∥∥Dψ1
−t(x)
∥∥ ≤ e−a1t (15)
with a1 = a− c1 and b1 = b+ c1 .
Proof. Consider as in previous lemmas the following equation
1
2
d
dt
∥∥η1
1(t, x, ν)
∥∥2 =
〈
η1
1(t, x, ν), (α+DZ1) η1
1(t, x, ν)
〉
(16)
and put z =
∥∥η1
1(t, x, ν)
∥∥, so
(a− c1)z2 ≤ 1
2
d
dt
z2 ≤ (b+ c1)z2, z(0) = ‖ν‖ (17)
and then
‖ν‖ ea1t ≤ z ≤ ‖ν‖ eb1t for any t ≥ 0 and ν ∈ Rn. �
6 M. Benalili and A. Lansari
Lemma 5. If the perturbation Z1 fulfils the estimates∥∥DlZ1(x)
∥∥ ≤ c′l ‖x‖
1−l+m for any x ∈ B (0, 1) and all integers m ≥ 1,∥∥DlZ1(x)
∥∥ ≤ c′′l ‖x‖
1−l ∀ x ∈ Rn \B (0, 1)
with l = 0, 1, then the first derivative of the Y1-flow is complete and is estimated by, for any
t ≥ 0
ea1t ≤ ‖Dψ1
t (x)‖ ≤ eb1t, e−b1t ≤ ‖Dψ1
−t(x)‖ ≤ e−a1t (18)
with cl = max {c′l, c′′l }, al = a− cl and bl = b+ cl, l = 0, 1.
Proof. For any x ∈ B (0, 1) we have ‖DlZ1(x)‖ ≤ c′l ‖x‖
1−l+m ≤ c′l ‖x‖
1−l and letting cl =
max {c′l, c′′l }, we get for any x ∈ Rn ‖DlZ1(x)‖ ≤ cl ‖x‖1−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
‖Z1(x)‖ ≤ c0 ‖x‖1+m , ‖DZ1(x)‖ ≤ c1 ‖x‖m 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
eat
(
1− c0
a
‖x‖m (1− eamt)
)− c1
mc0 ≤
∥∥Dψ1
t (x)
∥∥ ≤ ebt
(
1− c0
b
‖x‖m (1− ebmt)
)− c1
mc0 .
Proof. Letting y =
∥∥ψ1
t (x)
∥∥ and z =
∥∥η1
1(t, x, ν)
∥∥ in equation (16), we get
(a− c1y
m)z2 ≤ 1
2
d
dt
z2 ≤ (b+ c1y
m)z2, z(0) = ‖ν‖
and taking into account the estimates given by the relation (11), we obtain
‖x‖m emat
(
1− c0
a
‖x‖m (1− eamt)
)−1
≤ ym ≤ ‖x‖m embt
(
1− c0
b
‖x‖m (1− ebmt)
)−1
consequently
‖ν‖ exp
(
at− c1
∫ t
0
‖x‖m emasds
1− c0
a ‖x‖
m (1− eams)
)
≤ z ≤ ‖ν‖ exp
(
bt+ c1
∫ t
0
‖x‖m embsds
1− c0
b ‖x‖
m (1− ebms)
)
which has the solution
‖ν‖ eat
(
1− c0
a
‖x‖m (1− eamt)
)− c1
mc0
≤ z ≤ ‖ν‖ ebt
(
1− c0
b
‖x‖m (1− ebmt)
)− c1
mc0 for ν ∈ Rn. �
Example 2. We consider the same vector field as in Example 1. Denote by ξ13(t, x, ν) =
Dφ3
t (x)ν, ∀ ν ∈ Rn, the first derivation of the X3-flow. In coordinates, we have for any i, j =
1, . . . , n,
(
φ3
t (x)
)
i
= xie
αit
(
1 +
βi
αi
xmi
i
(
1− eαimit
))−1
mi
Global Stability of Dynamic Systems of High Order 7
so we deduce that
∂
∂xj
(
φ3
t (x)
)
i
= eαit
(
1 +
βi
αi
xmi
i
(
1− eαimit
))−1− 1
mi
δi
j
and by the estimates (13) we get
eat ≤
∥∥Dφ3
t (x)
∥∥ ≤ ebt.
The second derivative is
∂2
∂x2
i
(
φ3
t (x)
)
i
= −(1 +mi)
βi
αi
x−1+mi
i eαit
(
1− eαimit
)(
1 +
βi
αi
xmi
i
(
1− eαimit
))−2− 1
mi
.
Consequently, for l = 1, 2 and any x ∈ B (0, ρ) with ρ > 0 arbitrary fixed, there are constants
Ml > 0 such that∥∥Dlφ3
t (x)
∥∥ ≤Mle
bt.
3.3 Perturbation of a nonlinear vector field
Consider the nonlinear vector field
X2 =
n∑
i=1
βix
1+mi
i
∂
∂xi
with all mi > 0 and all βi ≤ 0.
The explicit form of the X2-flow is then given by
φ2
t (x) = x(1−mβtxm)
−1
m (19)
for any t ≥ 0 in the sense(
φ2
t (x)
)
i
= xi(1−miβitx
mi
i )
−1
mi , 1 ≤ i ≤ n.
Lemma 7. If the following assumptions are true
i) all the coefficients βi are non positive, −a′ ≤ βi ≤ −b′ ≤ 0
ii) all the exponents mi are even positive integers; 0 < m0 ≤mi ≤ m′
0.
Then the vector field X2 is semi-complete and the X2-flow satisfies the estimates
‖x‖
(
1 + b′m0t ‖x‖m0
) −1
m0 ≤ ‖φ2
t (x)‖ ≤ ‖x‖
(
1 + a′m′
0t ‖x‖
m′
0
) −1
m′
0 for any t ≥ 0. (20)
Proof. Clearly the flow φ2
t = exp(tX2) given by (19) is semi-complete i.e. defined for all t ≥ 0.
Consider the equation
1
2
d
dt
‖φ2
t (x)‖2 =
〈
φ2
t (x), β
(
φ2
t (x)
)1+m
〉
and put y = φ2
t (x), then
b′y2+m0 ≤ 1
2
d
dt
y2 ≤ a′y2+m′
0 , y(0) = ‖x‖
and we get the estimates given in (20). �
8 M. Benalili and A. Lansari
3.4 Estimation of the kth order derivation of the X2-f low
Let ξ12(t, x, ν) = Dφ2
t (x)ν, ∀ ν ∈ Rn be the first derivation of the X2-flow.
By formula (19), we get in coordinates
∂
∂xj
(
φ2
t (x)
)
i
= (1−miβitx
mi
i )−1− 1
mi δj
i with δj
i =
{
1 if i = j,
0 if i 6= j,
where i, j = 1, . . . , n.
Consequently(
1 + b′mt ‖x‖m0
)−1− 1
m0 ≤ ‖Dφ2
t (x)‖ ≤
(
1 + a′m′
0t ‖x‖
m′
0
)−1− 1
m′
0 . (21)
To get the estimates of the second derivative, we put
wi = 1−miβitx
mi
i ,
so
d
dxi
wi = mi(wi − 1)x−1
i and
∂
∂xi
(
φ2
t (x)
)
i
= wi
−1− 1
mi .
Consequently
∂2
∂x2
i
(
φ2
t (x)
)
i
= (1 +mi)x−1
i wi
− 1
mi
(
w−2
i − w−1
i
)
= x−1
i wi
− 1
mi
(
a2
1
wi
+
a2
2
w2
i
)
,
where a2
1 and a2
2 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∥∥Dlφ2
t (x)
∥∥ ≤Mlt
−1− 1
m′
0 .
Suppose that for l = 1, . . . , k − 1, with fixed k, there exist constants al
j and Ml > 0 such that
∂l
∂xl
i
(
φ2
t (x)
)
i
= x1−l
i wi
− 1
mi
l∑
j=1
al
j
wj
i
,
where al
j are real constants and
∥∥Dlφ2
t (x)
∥∥ ≤Mlt
−1− 1
m′
0 ∀ t > 0.
For the estimates of the kth derivative, we compute
∂k
∂xk
i
(
φ2
t (x)
)
i
= x1−k
i wi
− 1
mi
k∑
j=1
ak
j
wj
i
,
∂k
∂xk
i
(
φ2
t (x)
)
i
=
d
dxi
x2−k
i wi
− 1
mi
k−1∑
j=1
ak−1
j
wj
i
= x1−k
i wi
− 1
mi
k−1∑
j=1
(
ak−1
j
wj
i
(1− k − jmi) +
ak−1
j
wj+1
i
(1 + jmi)
)
= x1−k
i wi
− 1
mi
k∑
j=1
ak
j
wj
i
,
where ak
j are real constants.
So we resume
Global Stability of Dynamic Systems of High Order 9
Proposition 2. Suppose that
i) all the coefficients satisfy βi ≤ 0, −a′ ≤ βi ≤ −b′,
ii) the exponents mi are even natural numbers such that 0 < m0 ≤ mi ≤ m′
0.
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∥∥Dkφ2
t (x)
∥∥ ≤Mkt
−1− 1
m′
0 . (22)
3.5 Estimates of the Y2-f low
Let
Y2 =
n∑
i=1
(
βix
1+mi
i + Z2i(x)
) ∂
∂xi
the perturbation of the nonlinear vector field X2 and denote by ψ2
t = exp(tY2) the solution of
the dynamic system
d
dt
ψ2
t (x) = Y2 ◦ ψ2
t (x), ψ2
0(x) = x.
In coordinates we have, i = 1, . . . , n,
∂
∂t
ψ2,i(t, x) = βiψ
1+mi
2,i (t, x) + Z2i
(
ψ2
t (x)
)
, ψ2,i(0, x) = xi.
Putting
yi(t) = ψ−mi
2,i (t, x)
and
ψ2
t (x) = y
−1
m (t) =
(
y
−1
m1
1 (t), . . . , y
−1
mn
n (t)
)
we get
y′i(t) = −miψ
−1−mi
2,i (t, x)
∂
∂t
ψ2,i(t, x).
The Cauchy problem reads as
y′i(t) = −miβi −mi (yi(t))
1+ 1
mi Z2i(y
−1
m (t)), yi(0) = x−mi
i
and has the following solution
yi(t) = x−mi
i −miβit−mi
∫ t
0
yi(s)
1+ 1
mi Z2i(y
−1
m (s)ds,
i.e.
ψ2,i(t, x) = xi
(
1−miβitx
mi
i −mix
mi
i
∫ t
0
ψi(s, x)−1−miZ2i(ψ2
s(x))ds
)− 1
mi
,
so we have the explicit form of the Y2-flow
ψ2
t (x) = x
(
1−mβtxm −mxm
∫ t
0
ψ2
s(x)
−1−mZ2(ψ2
s(x))ds
)− 1
m
. (23)
Now we will estimate the Y2-flow.
10 M. Benalili and A. Lansari
Lemma 8. Suppose that
i) all the coefficients satisfy βi ≤ 0, −a′ ≤ βi ≤ −b′;
ii) the exponents mi are even natural numbers with 0 < m0 ≤ mi ≤ m′
0;
iii)
‖Z2i(x)‖ ≤ c′0 |xi|2+mi if x ∈ B (0, 1) ,
‖Z2i(x)‖ ≤ c′′0 |xi|1+mi if x ∈ Rn \B (0, 1)
with c0 = max {c′0, c′′0}, b0 = b′ − c0 > 0, a0 = a′ + c0.
Then
1) the vector field Y2 is semi-complete;
2) the Y2-flow has the estimates
‖x‖ (1 + a0m0t ‖x‖m0)
−1
m0 ≤ ‖ψ2
t (x)‖ ≤ ‖x‖
(
1 + b0m
′
0t ‖x‖
m′
0
) −1
m′
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
‖ψ2
t (x)‖ ≤M0 ‖x‖ t
− 1
m′
0 . (25)
Proof. Let x ∈ B (0, 1), by assumption we have ‖Z2i(x)‖ ≤ c′0 |xi|2+mi ≤ c′0 |xi|1+mi , put
c0 = max {c′0, c′′0} then for any x ∈ Rn we deduce ‖Z2i(x)‖ ≤ c0 |xi|1+mi . Now taking account of
the relation (23) we deduce that for any t ∈ [0, T ]∥∥ψ2
t (x)
∥∥ ≤ ‖x‖
(
1 +mt ‖x‖m (b′ − c0)
)− 1
m ≤ ‖x‖
hence the vector Y2 is semi-complete, i.e. defined for all t ≥ 0.
Consider the equation
1
2
d
dt
‖
(
ψ2
t (x)
)
i
‖2 =
〈(
ψ2
t (x)
)
i
, βi
(
ψ2
t (x)
)1+mi
i
+ Z2i
(
ψ2
t (x)
)〉
we get yi = ‖
(
ψ2
t (x)
)
i
‖ and yi(0) = |xi|, so we deduce
1
2
d
dt
y2
i ≤ (βi + c0)y2+mi
i ≤ −(b′ − c0)y2+mi
i
and
1
2
d
dt
y2
i ≥ (βi − c0)y2+mi
i ≥ −(a′ + c0)y2+mi
i .
We put b0 = b′ − c0 and a0 = a′ + c0, the solutions are estimated as
(|xi|−mi + a0mit)
− 1
mi ≤ ‖
(
ψ2
t (x)
)
i
‖ ≤ (|xi|−mi + b0mit)
− 1
mi . (26)
Hence, we have the estimate (25). �
Now, we estimate the first derivation of the Y2-flow. Let η1
2(t, x, ν) = Dψ2
t (x)ν, ∀ ν ∈ Rn
the solution of the dynamic system
d
dt
η1
2(t, x, ν) = (DyX2 +DyZ2) η1
2(t, x, ν), η1
2(0, x, ν) = ν
with y = ψ2
t (x).
Global Stability of Dynamic Systems of High Order 11
Lemma 9. Suppose that
i) the coefficients are such that βi ≤ 0, −a′ ≤ βi ≤ −b′;
ii) the coefficients mi are even natural numbers, 0 < m0 ≤ mi ≤ m′
0;
iii)
‖DlZ2i(x)‖ ≤ c′l |xi|2−l+mi if x ∈ B (0, 1) ,
‖DlZ2i(x)‖ ≤ c′′l |xi|1−l+mi if x ∈ Rn \B (0, 1)
with l = 0, 1;
iv)
a0 = a′ + c0, b0 = b′ − c0 > 0
and
a1 = a′(1 +m0) + c1, b1 = b′(1 +m0)− c1 > 0
with cl = max {c′l, c′′l }.
Then the first derivation of the Y2-flow has the following estimates, for any t > 0(
1 + b0m0t ‖x‖m0
)− a1
b0m0 ≤ ‖Dψ2
t (x)‖ ≤
(
1 + a0m
′
0t ‖x‖
m′
0
)− b1
a0m′
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
‖Dψ2
t (x)‖ ≤M1t
− b1
a0m′
0 . (28)
Proof. Let x ∈ B (0, 1), for l = 0, 1 we have
‖DlZ2i(x)‖ ≤ c′l |xi|2−l+mi ≤ c′l |xi|1−l+mi .
Let cl = max {c′l, c′′l } then for x ∈ Rn one has
‖DlZ2i(x)‖ ≤ cl |xi|1−l+mi .
Consider the equation
1
2
d
dt
‖η1
2(t, x, ν)‖2 =
〈
η1
2(t, x, ν), (DyX2 +DyZ2) η1
2(t, x, ν)
〉
and put z(t) = ‖η1
2(t, x, ν)‖ with z(0) = ‖ν‖, then
1
2
d
dt
z2 ≤ sup
i=1,...,n
(
((1 +mi)βi + c1)‖
(
ψ2
t (x)
)
i
‖mi
)
z2 ≤ z2 sup
i=1,...,n
(
−b1‖
(
ψ2
t (x)
)
i
‖mi
)
and
1
2
d
dt
z2 ≥ inf
i=1,...,n
(
((1 +mi)βi − c1)‖
(
ψ2
t (x)
)
i
‖mi
)
z2 ≥ z2 inf
i=1,...,n
(
−a1‖
(
ψ2
t (x)
)
i
‖mi
)
.
The solutions fulfill the following estimates
‖ν‖ exp inf
i=1,...,n
(
−a1
∫ t
0
‖
(
ψ2
s(x)
)
i
‖mids
)
≤ z(t) ≤ ‖ν‖ exp sup
i=1,...,n
(
−b1
∫ t
0
‖
(
ψ2
s(x)
)
i
‖mids
)
12 M. Benalili and A. Lansari
with, by (26)
|xi|mi
1 + a0mit |xi|mi
≤ ‖
(
ψ2
t (x)
)
i
‖mi ≤ |xi|mi
1 + b0mit |xi|mi
.
So we deduce
‖ν‖ exp inf
i=1,...,n
(
−a1
∫ t
0
|xi|mi
1 + b0mis |xi|mi
ds
)
≤ z(t) ≤ ‖ν‖ exp sup
i=1,...,n
(
−b1
∫ t
0
|xi|mi
1 + a0mis |xi|mi
ds
)
.
Consequently the solutions satisfy
‖ν‖ inf
i=1,...,n
(1 + b0mit |xi|mi)−
a1
b0mi ≤ z(t) ≤ ‖ν‖ sup
i=1,...,n
(1 + a0mit |xi|mi)−
b1
a0mi .
Then there are constants m0 > 0 and m′
0 > 0 such that
‖ν‖ (1 + b0m0t ‖x‖m0)−
a1
b0m0 ≤ ‖Dψ2
t (x)ν‖
≤ ‖ν‖
(
1 + a0m
′
0t ‖x‖
m′
0
)− b1
a0m′
0 ∀ ν ∈ Rn and for any t > 0.
Hence, we have the estimate (28). �
3.6 Perturbation of binomial vector fields
Let
Y3 =
n∑
i=1
(
αixi + βix
1+mi
i + Z3i(x)
) ∂
∂xi
with a ≤ αi ≤ b < 0, a′ ≤ βi ≤ b′ ≤ 0 and 0 < m0 ≤ mi ≤ m′
0, be the perturbation of
the binomial vector field X3 and let ψ3
t = exp(tY3) be the Y3-flow which is the solution of the
dynamic system
d
dt
ψt(x) = Y3 ◦ ψt(x), ψ0(x) = x
and in coordinates, we get
∂
∂t
ψ3,i(t, x) = αiψ3,i(t, x) + βiψ
1+mi
3,i (t, x) + Z3,i
(
ψ3
t (x)
)
, 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
yi(t) = ψ−mi
3,i (t, x)
and
ψ3
t (x) = y
−1
m (t) =
(
y
−1
m1
1 (t), . . . , y
−1
mn
n (t)
)
Global Stability of Dynamic Systems of High Order 13
we get the solution
ψ3,i(t, x) = xie
αit
(
1 +
βi
αi
xmi
i (1− e
αimit
)
−mix
mi
i
∫ t
0
[ψ3,i(s, x)]
−1−mi Z3,i(ψ3
s(x))e
αimis
ds
)−1
mi
and the implicit form of the Y3-flow reads as
ψ3
t (x) = xeαt
(
1 +
β
α
xm(1− eαmt)−mxm
∫ t
0
[
ψ3
s(x)
]−1−m
Z3(ψ3
s(x))e
αmsds
)− 1
m
. (29)
3.7 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 βi are non positive, −a′ ≤ βi ≤ −b′;
iii) the exponents mi are even natural numbers with 0 < m0 ≤mi ≤ m′
0;
iv)
‖Z3i(x)‖ ≤ c′0 |xi|2+mi if x ∈ B (0, 1) ,
‖Z3i(x)‖ ≤ c′′0 |xi|1+mi if x ∈ Rn \B (0, 1)
with c0 = max {c′0, c′′0}, b0 = b′ − c0 > 0, a0 = a′ + c0.
Then
1) there exist constants m > 0 and m′ > 0 such that the Y3−flow has the estimates, ∀ t ≥ 0
‖x‖ e−at
(
1 +
a0
a
‖x‖m (1− e−amt)
)− 1
m
≤ ‖ψ3
t (x)‖ ≤ ‖x‖ e−bt
(
1 +
b0
b
‖x‖m′
(1− e−bm′t)
)− 1
m′
;
2) for any t > 0 there are positive constants c1 and c2 such that
c1 ‖x‖ e−at ≤ ‖ψ3
t (x)‖ ≤ c2 ‖x‖ 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 βiare non positive, −a′ ≤ βi ≤ −b′;
iii) the exponents mi are even natural numbers such that 0 < m0 ≤ mi ≤ m′
0;
iv)
‖DlZ3i(x)‖ ≤ c′l |xi|2−l+mi if x ∈ B (0, 1) ,
‖DlZ3i(x)‖ ≤ c′′l |xi|1−l+mi if x ∈ Rn \B (0, 1)
with l = 0, 1;
14 M. Benalili and A. Lansari
v)
a0 = a′ + c0, b0 = b′ − c0 > 0
and
a1 = a′(1 +m0) + c1, b1 = b′(1 +m0)− c1 > 0
with cl = max {c′l, c′′l }.
Then there exist constants m > 0 and m′ > 0 such that for any t ≥ 0
e−at
(
1 +
b0
b
‖x‖m (1− e−bmt)
)− a1
b0m
≤ ‖Dψ3
t (x)‖ ≤ e−bt
(
1 +
a0
a
‖x‖m′
(1− e−am′t)
)− b1
a0m′
and for any t ≥ 0, there is a constant M1 > 0 such that∥∥Dψ3
t (x)
∥∥ ≤M1e
−bt. (31)
4 Global stability of prolongations of flows
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∑
i=1
(αixi + Z1i(x))
∂
∂xi
with the following assumptions
i) all the coefficients are negative, −a ≤ αi ≤ −b < 0;
ii)
‖Z1(x)‖ ≤ c′0 ‖x‖
1+m ∀ x ∈ B (0, 1) and ∀m ≥ 1,
‖Z1(x)‖ ≤ c′′0 ‖x‖ ∀ x ∈ Rn \B (0, 1) ;
iii) b0 = b− c0 > 0, where c0 = max {c′0, c′′0}.
Then the origin 0 is a globally asymptotically stable equilibrium to the Y1-flow ψ1
t on Rn.
Proof. Let ψ1
t = exp(tY1) be the Y1-flow, then by the assumptions and the estimates given by
Lemma 2 we get that∥∥ψ1
t (x)
∥∥ ≤ ‖x‖ e−b0t ∀ t ≥ 0 and ∀x ∈ Rn
and by Proposition 1, the origin 0 is G.A.S. for ψ1
t on Rn. �
Example 3. We consider the vector field
X3 =
n∑
i=1
(
αixi + βix
1+mi
i
) ∂
∂xi
Global Stability of Dynamic Systems of High Order 15
of Example 1 with a ≤ αi ≤ b < 0, a′ ≤ βi ≤ b′ ≤ 0. The X3-flow φ3
t = exp (tX3) is then given
by
φ3
t (x) = xeαt
(
1 +
β
α
xm
(
1− eαmt
))−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∥∥φ3
t (x)
∥∥ ≤ ‖x‖ e−bt.
By Proposition1 the origin 0 is a G.A.S. for the flow φ3
t on Rn.
4.2 Global stability of the first prolongation of the Y1-f low
Lemma 13. With the same assumptions as in Lemma 12 and the following conditions
‖DZ1(x)‖ ≤ c′1 ‖x‖
m ∀ x ∈ B (0, 1) and ∀ m ≥ 1,
‖DZ1(x)‖ ≤ c′′1 ∀ x ∈ Rn \B (0, 1)
with b1 = b− c1 > 0 and c1 = max {c′1, c′′1}.
Then the origin 0 is a globally asymptotically stable for the first prolongation of the Y1-flow
ψ1
t on Rn.
Proof. By the estimates (18) and the hypothesis we deduce that
‖Dψ1
t (x)ν‖ ≤ ‖ν‖ e−b1t ∀ t > 0, ∀ ν ∈ Rn
and by Proposition 1,we obtain that the origin 0 is a G.A.S. equilibrium on Rn for η1
1(t, x, v) =
Dψ1
t (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
‖DlZ1(x)‖ ≤ c′l ‖x‖
1−l+m for any x ∈ B (0, 1) and for any integer m ≥ l − 1,
‖DlZ1(x)‖ ≤ c′′l ∀ x ∈ Rn \B (0, 1) ,
a0 = a+ c0, b0 = b− c0 > 0,
a1 = a+ c1, b1 = b− c1 > 0
with cl = max {c′l, c′′l }, bl = cl ∀ l ≥ 2.
Put ηl
1(t, x, ν, . . . , ν) = Dkψ1
t (x)ν
k, where ν ∈ Rn. Since by Lemmas 12 and 13 the origin 0 is
an G.A.S. equilibrium for ηl
1, 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
‖Dlψ1
t (x)‖ ≤Mle
−b1t.
We will show that the origin 0 is a G.A.S. equilibrium for ηk
1 on Rn. ηk
1 (t, x, ν, . . . , ν) =
Dkψ1
t (x)ν
k is solution of the dynamic system
d
dt
ηk
1 = DyY1 · ηk
1 +Gk
1(t, x, ν), ηk
1 (0, x, ν, . . . , ν) = ν
16 M. Benalili and A. Lansari
with y = ψ1
t (x) and
Gk
1(t, x, ν) =
k∑
l=2
Dl
yY1(y)
∑
i1+···+il=k
ij>0
l∏
j=1
Dijψ1
t (x)ν
ij
=
k−1∑
l=2
Dl
yZ1(y)
∑
i1+···+il=k
ij>0
l∏
j=1
Dijψ1
t (x)ν
ij
+Dk
yZ1(y)
(
Dψ1
t (x)ν
)k
.
Consequently we get
ηk
1 (t, x, ν, . . . , ν) = Dψ1
t (x)ν +
∫ t
0
DΨ1
t−s(ψ
1
s(x))G
k
1(s, x, ν)ds.
The integral is well defined at s = 0, since
lim
s→0+
Dψ1
t−s(ψ
1
s(x)) = Dψ1
t (x)
and there exist constants Al > 0 such that
lim
s→0+
Gk
1(s, x, ν) =
k∑
l=2
AlD
l
yZ1(y)νk.
We will show that it converges uniformly with respect to x as t+∞. Put
Ik =
∫ t
0
‖Dψ1
t−s(ψ
1
s(x))‖‖Gk
1(s, x, ν)‖ds.
Since ‖DlZ1(x)‖ ≤ cl ∀ l ≥ 1, ∀ x ∈ Rn, there are constants bl > 0 such that ∀ y ∈ Rn,
‖Dl
yY1(y)‖ ≤ bl and by the assumption of recurrence there exist constants Ml > 0 such that
‖Dlψ1
t (x)‖ ≤Mle
−b1t ∀ t ≥ 0.
We deduce that there is a constant Ck > 0 such that
Ik ≤
k∑
l=2
blMl
∫ t
0
e−b1(t−s+sl)ds ≤ Cke
−b1t.
So for any x ∈ Rn one has
lim
t→+∞
Ik ≤
k∑
l=2
Mlbl ‖ν‖l
∫ +∞
0
e−b1sds =
1
b1
k∑
l=2
Mlbl ‖ν‖l
and the integral Ik is uniformly convergent with respect to x ∈ Rn as t→ +∞. Consequently
lim
t→+∞
‖ηk
1‖ = lim
t→+∞
‖Dψ1
t (x)ν‖+
∫ +∞
0
lim
t→+∞
‖Dψ1
t−s(ψ
1
s(x))‖‖Gk
1(s, x, ν)‖ds = 0
and there is a constant M ′
k > 0 such that
‖ηk
1‖ ≤ ‖Dψ1
t (x)ν‖+
∫ +∞
0
‖Dψ1
t−s(ψ
1
s(x))‖‖Gk
1(s, x, ν)‖ds ≤M ′
k‖ν‖ke−b1t.
This show by Proposition 1 that the origin 0 is a G.A.S. equilibrium to ηk
1 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
‖Dkψ1
t (x)‖ ≤Mke
−b1t, ‖Dkψ1
−t(x)‖ ≤Mke
a1t. (32)
Global Stability of Dynamic Systems of High Order 17
5 Global stability of a flow generated
by nonlinear perturbed vector fields
First we will start with monomial vector fields.
5.1 Global stability of the X2-f low
Let
X2 =
n∑
i=1
βix
1+mi
i
∂
∂xi
with
(i) all the coefficients βi ≤ 0 such that −a′ ≤ βi ≤ −b′;
(ii) all the exponents mi are even natural integers with 0 < m0 ≤mi ≤ m′
0.
Let φ2
t = exp (tX2) be the X2-flow. By the estimations (19) we obtain
∥∥φ2
t (x)
∥∥ ≤ ‖x‖
(
1 + a′m′
0t ‖x‖
m′
0
) −1
m′
0 .
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
‖φ2
t (x)‖ ≤M0 ‖x‖ t
− 1
m′
0 .
By Proposition 1, the origin is a globally asymptotically stable equilibrium to the flow φ2
t 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 M ′
l > 0 such that
‖Dlφ2
t (x)‖ ≤Ml t
−1− 1
m′
0 and ‖Dlφ2
0(x)‖ ≤M ′
l .
So the origin 0 is a G.A.S. equilibrium for Dlφ2
t (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∑
i=1
(
βix
1+mi
i + Z2,i(x)
) ∂
∂xi
be a smooth vector field on Rn such that
i) all the coefficients βi ≤ 0 are non negative with −a′ ≤ βi ≤ −b′;
ii) mi are even natural numbers with 0 < m0 ≤mi ≤ m′
0;
iii) for k = 0, . . . , 1 +mi
‖DkZ2i(x)‖ ≤ c′k |xi|2−k+mi if x ∈ B (0, 1) ;
‖DkZ2i(x)‖ ≤ c′′k |xi|1−k+mi if x ∈ Rn \B (0, 1) ;
18 M. Benalili and A. Lansari
iv) for any k ≥ 2 +mi
‖DkZ2i(x)‖ ≤ ck;
v)
a0 = a′ + c0, a1 = a′(1 +m0) + c1,
b0 = b′ − c0 > 0, b1 = b′(1 +m0)− c1 > a0m
′
0
with ck = max {c′k, c′′k}.
Remark 1. If x ∈ B (0, 1) then ‖DkZ2i(x)‖ ≤ c′k |xi|2−k+mi ≤ c′k |xi|1−k+mi . Putting cl =
max {c′l, c′′l }, we deduce that for any x ∈ Rn have ‖DkZ2i(x)‖ ≤ ck |xi|1−k+mi .
5.2.1 Global stability of the Y2-f low on Rn
Let ψ2
t = 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
‖ψ2
t (x)‖ ≤M0 ‖x‖ t
− 1
m′
0 .
So by Proposition 1, the origin 0 is a G.A.S. equilibrium for the Y2-flow ψ2
t on Rn.
5.2.2 Global stability of prolongation of the Y2-f low on Rn
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 Dlψ2
t (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
‖Dlψ2
t (x)‖ ≤Mlt
− b1
a0m′
0 and ‖Dlψ2
0(x)‖ ≤M ′
l .
We will show that 0 is a G.A.S. for Dkψ2
t (x) on Rn.
Put ηk
2 (t, x, ν, . . . , ν) = Dkψ2
t (x)ν
k ∀ ν ∈ Rn which is solution of the dynamic system
d
dt
ηk
2 = DyY2 · ηk
2 +Gk
2(t, x, ν), ηk
2 (0, x, ν, . . . , ν) = ν
with y = ψ2
t (x) and
Gk
2(t, x, ν) =
k∑
l=2
Dl
yY2(y)
∑
i1+···+il=k
ij>0
l∏
j=1
Dijψ2
t (x)ν
ij
.
By the method of the resolvent, we deduce
ηk
2 (t, x, ν, . . . , ν) = Dψ2
t (x)ν +
∫ t
0
Dψ2
t−s(ψ
2
s(x))G
k
2(s, x, ν)ds.
Clearly the integral
I1
k =
∫ 1
0
‖Dψ2
t−s(ψ
2
s(x))‖‖Gk
2(s, x, ν)‖ds
Global Stability of Dynamic Systems of High Order 19
is well defined at s = 0 and s = t, since
lim
s→0+
Dψ2
t−s(ψ
2
s(x)) = Dψ2
t (x).
By the recurrent assumption Dlψ2
0(x) are bounded and there exist constants Al > 0 such that
lim
s→0+
‖Gk
2(s, x, ν)‖ ≤
k∑
l=2
Al‖Dl
xY2(x)νl‖.
In the same way
lim
s→t−
Dψ2
t−s(ψ
2
s(x)) = identity.
Now, we have to show that
I2
k =
∫ t
1
‖Dψ2
t−s(ψ
2
s(x))‖‖Gk
2(s, x, ν)‖ds
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
‖x‖ (1 + a0m0t ‖x‖m0)
−1
m0 ≤ ‖ψ2
t (x)‖ ≤ ‖x‖
(
1 + b0m
′
0t ‖x‖
m′
0
) −1
m′
0 ,
(1 + b0m0t ‖x‖m0)−
a1
b0m0 ≤ ‖Dψ2
t (x)‖ ≤
(
1 + a0m
′
0t ‖x‖
m′
0
)− b1
a0m′
0 .
So ‖y‖ = ‖ψ2
t (x)‖ ≤ ‖x‖ and
∥∥Dψ2
t−s(ψ
2
s(x))
∥∥ is bounded. Since for any x ∈ Rn and any
l = 1, . . . , 1+mi, ‖DlZ2i(x)‖ ≤ cl |xi|1−l+mi then Dl
yY2(y) are bounded. Now by the assumption
of recurrence there exist constants Ml > 0 such that for any t > 0
‖Dlψ2
t (x)‖ ≤Mlt
− b1
a0m′
0
with a0m
′
0 < b1 i.e. b1
a0m′
0
> 1, and we deduce the existence of constants Cl > 0 such that
lim
t→+∞
I2
k ≤
k∑
l=2
Cl
∫ +∞
1
s
− lb1
a0m′
0 ds ≤
k∑
l=2
Cl
(
lb1
a0m′
0
− 1
)−1
.
The integral I2
k converges uniformly on any compact K ⊂ Rn as t→ +∞.
Now since the integral is well defined at s = 0, then
lim
t→0
‖ηk
2 (t, x, ν, . . . , ν)‖ ≤ lim
t→0
‖Dψ2
t (x)ν‖ = ‖ν‖
hence there is a constant M ′
k > 0 such that
‖Dkψ2
0(x)‖ ≤M ′
k.
In the same way as above the integral
∫ t
0 ‖Dψ
2
t−s(ψ
2
s(x))‖‖Gk
2(s, x, ν)‖ds is well defined and
putting τ = s
t we obtain
ηk
2 (t, x, ν, . . . , ν) = Dψ2
t (x)ν + t
∫ 1
0
Dψ2
t(1−τ)(ψ
2
tτ (x))G
k
2(tτ, x, ν)dτ.
20 M. Benalili and A. Lansari
Since b1 = b′(1 +m0)− c1 > a0m
′
0 , by the estimates (26) and (28), we deduce the existence of
a constant Mk > 0 such that
‖ηk
2 (t, x, ν, . . . , ν)‖ ≤ ‖Dψ2
t (x)ν‖+ t
∫ 1
0
‖Gk
2(tτ, x, ν)‖dτ
≤ ‖Dψ2
t (x)ν‖+ t
k∑
l=2
∫ 1
0
(tτ)
− lb1
a0m′
0 ‖x‖1+m′
0−l(
1 + b0m′
0tτ ‖x‖
m′
0
) 1+m′
0−l
m′
0
dτ ≤Mkt
− b1
a0m′
0 .
Which shows that the origin 0 is a G.A.S. equilibrium for ηk
2 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
‖Dkψ2
t (x)‖ ≤Mkt
− b1
a0m′
0 . (33)
5.3 Global stability of prolongations of the Y3-f low
Let
Y3 =
n∑
i=1
(
αixi + βix
1+mi
i + Z3i(x)
) ∂
∂xi
with
i) all the coefficient αi are negative with −a ≤ αi ≤ −b;
ii) all the coefficients βi ≤ 0 and −a′ ≤ βi ≤ −b′;
iii) the exponents mi are even natural numbers with 0 < m0 ≤ mi ≤ m′
0;
iv) For any k = 0, . . . , 1 +mi
‖DkZ3i(x)‖ ≤ c′k |xi|2−k+mi if x ∈ B (0, 1) ,
‖DkZ3i(x)‖ ≤ c′′k |xi|1−k+mi if x ∈ Rn \B (0, 1) ;
v) for any k ≥ 2 +mi
‖DkZ3i(x)‖ ≤ ck;
vi)
a0 = a′ + c0, a1 = a′(1 +m0) + c1,
b0 = b′ − c0 > 0, b1 = b′(1 +m0)− c1 > 0
with ck = max {c′k, c′′k} .
Remark 2. If x ∈ B (0, 1) then ‖DkZ3i(x)‖ ≤ c′k |xi|2−k+mi ≤ c′k |xi|1−k+mi .
Let cl = max {c′l, c′′l }, for any x ∈ Rn one has ‖DkZ3i(x)‖ ≤ ck |xi|1−k+mi .
Global Stability of Dynamic Systems of High Order 21
5.3.1 Global stability of the Y3-f low Rn
Denote by ψ3
t = exp(tY3), by the estimates (30), we have
‖ψ3
t (x)‖ ≤ C‖x‖e−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 ηl
3(t, x, ν, . . . ν) = ‖Dlψ3
t (x)ν
k‖ on Rn
and there exist constants Ml > 0 such that for any t > 0
‖Dlψ3
t (x)‖ ≤Mle
−bt.
We will show that 0 is a G.A.S. equilibrium to ηk
3 on Rn.
ηk
3 (t, x, ν, . . . , ν) is a solution to the dynamic system
d
dt
ηk
3 = Dyη
k
3 +Gk
3(t, x, ν)
with y = ψ3
t (x) and
Gk
3(t, x, ν) =
k∑
l=2
Dl
yY3(y)
∑
i1+···+il=k
ij>0
l∏
j=1
Dijψ3
t (x)ν
ij
.
By the method of the resolvent, we get
ηk
3 (t, x, ν, . . . , ν) = Dψ3
t (x)ν +
∫ t
0
Dψ3
t−s(ψ
3
s(x))G
k
3(s, x, ν)ds
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
‖Dkψ3
t (x)‖ ≤Mk ‖x‖ 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 .
References
[1] Benalili M., Lansari A., Ideal of finite codimension in contact Lie algebra, J. Lie Theory 11 (2001), 129–134.
[2] Benalili M., Lansari A., Une propriété des idéaux de codimension finie en algèbre de Lie des champs de
vecteurs, J. Lie Theory 15 (2005), 13–26.
[3] Fenichel N., Asymptotic stability with rate conditions, Indiana Univ. Math. J. 23 (1973/1974), 1109–1137.
[4] Gutierrez C., A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. H. Poincaré
Anal. Non Linéaire 12 (1995), 627–671.
[5] Hahn W., Stability of motions, Springer-Verlag, New York, 1967.
[6] Libre J., Teixeria M., Global asymptotic stability for a class of discontinuous vector fields in R2, Dyn. Syst.
122 (2007), 133–146.
[7] Shafer D.S., Structure and stability of gradient polynomial vector fields, J. London Math. Soc. (2) 41 (1990),
109–121.
[8] Meisters G.H., Global stability of dynamical systems, SIAM Rev. 31 (1989), 147–151.
[9] Zajtz A., Global version of a Sternberg linearization theorem, Roczik Nauk. Dydact. Prace Mat. (2000),
no. 17, 265–269.
[10] Zajtz A., Some division theorems for vector fields, Ann. Polon. Math. 58 (1993), 19–28.
1 Introduction
2 Generalities
3 Estimates of prolongations of flows
3.1 Perturbation of linear vector field
3.2 Estimation of the kth prolongation of the Y1-flow
3.3 Perturbation of a nonlinear vector field
3.4 Estimation of the kth order derivation of the X2-flow
3.5 Estimates of the Y2-flow
3.6 Perturbation of binomial vector fields
3.7 Estimation of the Y3-flow
4 Global stability of prolongations of flows
4.1 Global stability of the Y1-flow
4.2 Global stability of the first prolongation of the Y1-flow
4.3 Global stability of the kth prolongation of the Y1-flow
5 Global stability of a flow generated by nonlinear perturbed vector fields
5.1 Global stability of the X2-flow
5.2 Global stability of high order of the Y2-flow
5.2.1 Global stability of the Y2-flow on Rn
5.2.2 Global stability of prolongation of the Y2-flow on Rn
5.3 Global stability of prolongations of the Y3-flow
5.3.1 Global stability of the Y3-flow Rn
References
|