Construction Of Lyapunov Functions For Some Fourth Order

Transcription

INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 10, OCTOBER 2014
ISSN 2277-8616
Construction Of Lyapunov Functions For Some Fourth Order
Nonlinear Ordinary Differential Equations By Method Of Integration
Orie Bassey O.
Abstract: In giving adequate attention to some qualitative properties of solutions in ordinary differential equations, Lyapunov functions is quite
indis-
pensable. The process of tackling some problems in the application is the construction of appropriate Lyapunov functions for some nonlinear fourth order
differential equation. Our focus is finding V(x), a quadratic form and positive definite also finding U(x) which is positive definite such that the
derivative
of V with respect to time would be equal to the negative value of U(x) In this work, we adopted the pre-multiplication of the given differential equation by

x and thereafter we integrated with respect to t
form
t  0 to t  T . We obtained a Lyapunov function candidate for a fourth order differential equa-
tion or its scalar equation.
Index Terms: Lyapunov Functions, Linear systems, non linear systems Negative definite, Positive definite, Linear and non linear systems
————————————————————
1. Introduction
Stability is a very important problem in the theory and ap-
Lyapunov functions are useful tools in determining stability,
plication of differential equations, and an effective method
asymptotic stability, uniform stability, global stability or out-
for studying the stability of differential equation is the second
right instability of differential system and boundedness of
method of Lyapunov [Omeike, 2008] Lyapunov functions
solution of a real scalar fourth-order differential equation
originated from a Russian mathematician Aleksandra M.
(Ogundare 2012, Ezeilo et al 2010, Omeike, 2008, Afuwape,
Lyapunov. The basis for all what he was proving is entirely
2010) Asymptotic stability is intimately linked to the exis-
on the well known fact that towards the equilibrium position,
tence of a ‗Lyapunov‘ function, that is, a proper,
the total energy in a system is either constant or decreasing.
non-negative function varnishing only on an invariant set
Lyapunov functions have been constructed for linear equa-
and decreasing along those trajectories of the system not
tions on the platform that given any quadratic positive defi-
evolving in the invariant set. Lyapunov proved that the ex-
nite V , there exists another positive definite function U
istence of a Lyapunov function guarantees asymptotic sta-
such that V  U and for the nonlinear case, a correlation
bility and for linear time-invariant systems also showed the
is taken between the constant
converse statement that asymptotic stability implies the
linear and nonlinear equations which leads to the appropri-
existence of a Lyapunov function (Lars et al, 2006).
ate Lyapunov functions for the nonlinear case [Ezeilo et al,
coefficient equations of
2010] Problems and complex computations are encountered in trying to construct appropriate Lyapunov function
candidates for nonlinear fourth order differential systems.
_____________________________
However, our method for construction of Lyapunov functions

Orie Bassey
is currently pursuing Doctorate
de-
gree in Applied Mathematics. He is currently a lecturer at the department of Mathematics and
for nonlinear fourth order differential lies on the fact that
given any real system
Sta-
x  Ax
tistics, Federal Polytechnic, Nekede, Owerri, Imo
.
.
.
(1.0.0)
State, Nigeria.
E-mail: [email protected]
x   n , where A is a constant matrix and that A has all its
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eigen values with negative real parts. Then it is known from
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V  x T Px  xT Px
the general theory that corresponding to any positive defi-


 xT AT P  PA x
nite quadratic form U  x  there exists another positive defi-
Since X  AX and X T   AX T  xT AT
nite quadratic form V  x  such that
V  U
.
.
.
Let AT P  PA  Q , where Q is positive definite. If Q is posi(1.0.1)
tive definite then the linear system is globally asymptotically
stable. AT P  PA  Q is called the Lyapunov equation. So, in
along the solution paths of (1.0.0). This result in (1.0.1) has
been extended to hold for positive semi definite quadratic
U  x
as well. As a matter of fact, our basis for construction
solving for P, we need to first of all accept that A is stable, so
that any Q  0 will yield a P  0 , the usual approach though
is to set Q  1 , then solve for P (Hedrick and Girard, 2005).
In Lyapunov‘s second method, the theorem provides a suf-
of Lyapunov function in this work would ultimately satisfy
ficient condition, not a necessary condition. Consider the
equation (1.0.1) We consider V to be positive definite func-
system
tion, then if V always decreases, then it must reach zero
x  f  x, t  , f  0, t   0  t
eventually. That is, for a stable system, all trajectories must
move so that the values of V are decreasing. This is simi-
If a scalar function is defined such that
lar to the energy argument for of mechanical systems. To
relate V to the system dynamics, we compute V
V 
V
t

(i)
(ii)
V
xi
V
t
V  x, t  is positive, that is, there exists a conti-
xi
i

V  0, t   0
nuous non decreasing scalar function   0   0
 V T f  x 
and  x  0,0    x   V  x, t 
where
(iii)
 f1   x1 
 f   x 
f  x   2    2 
 f3   x3 
   
 f 4   x4 
V  x, t 
is
negative
V  x, t     x   0 where
(iv)
V    x 
where

decrescent, the Lyapunov
to be negative definite for our expectation to be
(v)
is
a
continuous
function is upper
V is radically unbounded, that is,
(1.0.0), and we want to investigate the stability of the system
  x    as
 x  
using Lyapunov theory, for the sake of argument, let
V  x Px
is a continuous
bounded.
true. Also, if we consider the autonomous linear system
T
is
non-decreasing function and   0   0 that is V is
V T   XV1 , XV2 , XV3 , XV4 
V

that
non-decreasing scalar function such that   0   0
and
We need
definite,
We are only trying to bring to mind that Asymptotic stability
requires V to be negative definite, stability requires V to
be negative semi definite, condition (iv) yields uniformity for
where P is a positive definite matrix. Then,
the time-varying system and Global stability is given by
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condition (v). Ordinarily, no matter where one starts, equilibrium is returned we refer to it as uniform. For asymptotic
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nuously differentiable function V  x  satisfying the condi-
stability, we say that no matter how large the perturbation,
tions for stability is called a Lyapunov function. The surface
the system returns back to origin; and in the large (globally),
V  x   c for some c  0 , is called a Lyapunov surface or a
it is true everywhere (Ademola et al,2011) A more general
statement for globally asymptotically stable autonomous
system (1.0.2)with f continuous and let V be a scalar
level surface. A geometric integrator for a system of ordinary
with continuous first partial derivatives. Assume that
preserve V as a Lyapunov function for the discrete system
differential equations with a Lyapunov function V should
(Grimm and Quispel, 2005) (Bainov, 1997) by means of
(i)
for some K  0 , the region  k
V  x  K
(ii)
V  0
defined by
gues of the classical Lyapunov‘s functions, obtained suffi-
is bounded
for all
piecewise continuous auxiliary functions which are analocient conditions for the existence of periodic solutions of a
x
in
k
linear system of impulsive differential difference equations.
The impulse takes place at fixed moments. Their investiga-
let R be the set of all points within i where V  0 . Let M
tions were carried out by using minimal subsets of a suitable
x t 
space of piecewise continuous functions, by the elements of
originating in k tends to M as t tends to infinity. Since Lya-
functions are estimated. The Lyapunov stability of periodic
punov proposed his famous theory on stability of motion,
solutions were established from the generalized high-
numerous methods have been proposed for deriving suita-
er-order neutral functional differential equation.
be the largest invariant set in R. then every solution
which the derivatives of the piecewise continuous auxiliary
ble Lyapunov functions to study the stability and boundedness of solutions of certain second-third-forth-fifth and sixth
order non-linear differential equations; see for example
  x t   Cx t    
l n 1
p

 F t , x  t  , x  t  , . . ., xl 1  t 

[Anderson 1968, Barbasin 1968] The other Lyapunov‘s
into a system of first-order differential equations and then
theorem considered V  x  such that V  0   0 and
V  x   0 x  D / 0
applying Mawhin‘s continuation theory and some new in-
V  x   0, x  D
equalities, with this sufficient conditions for the existence of
periodic solution for the equation was established (Jingli etal,
then the origin is stable, moreover if
2011). Mohammed etal (2013) in their lecture on dynamic
systems and control opined that a continuous-time system
V  x  , x  D / 0
x  t   Ax  t 
then the origin is asymptotically stable. Furthermore, if
is asymptotically stable if and only if all the eigenvalues of A
V  x   0, x  0
are in open left half plane, and also showed that this result
x    V  x  
can be inferred from Lyapunov theory. A consequence of
and V  x   0, x  0 , then the origin is globally asymptoti-
this is that stability can be assessed by methods that may
cally stable. So we can observe that the origin is stable if
dratic Lyapunov functions and the associated mathematics
there is a continuously differentiable positive definite func-
turn up are worth mastering in the context of stability evalu-
tion V  x  so that V  x  is negative definite, it is globally
ation. In this paper, we address the issue of construction of
asymptotically stable if the conditions for asymptotic stability
present and verify Lyapunov indirect methods for proving
holds globally and V  x  is radially unbounded. A conti-
be computationally simpler than Eigen analysis. So, qua-
Lyapunov function for the linear systems and then we
stability of a non-linear system with interest in fourth-order.
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2. Statement of Problems. Preliminaries, Definitions.
Consider the fourth-order differential equation

  bx  cx  dx  0
x  ax
.
. .

.
. .
(2.0.5)
.
. .
(2.0.6)
(ii) A continuous function
V  x, t   V  x1, x2 ,..., xn , t 
(2.0.1)
where a, b, c and d are constants with
a  0, b  0, c  0 and d  0 . . .
V  x, t     x
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is positive semi definite if
lim V  x, t   0
(2.0.2)
x 0
and there exist
The equation (2.0.1) is equivalent to the following four sys-
 x
tems.

such that
x  y
y  z
.
. .
V  x, t     x
(2.0.3)
z  w
w  aw  bz  cy  dx

(iii) A continuous function
V  x, t   V  x1, x2 ,..., xn , t 
is negative definite if
The
system (2.0.3) have negative real parts if and only if
a  0, b  0, c  0 and d  0 . There is therefore need to have
a positive definite continuous quadratic function V and
lim V  x, t   0
x 0
and there exists
 x
another positive quadratic form U such that

such that
V  U
.
. .
(2.0.4)
V  x, t     x
along the solution paths of (2.0.1) or (2.0.3). Before now, the

.
. .
(2.0.7)
(iv) A continuous function
result in equation (2.0.4) has been extended and is estabV  x, t   V  x1, x2 ,..., xn , t 
lished to hold for positive semi definite quadratic U  x  as
well. It is our interest therefore to construct a Lyapunov
is negative semi definite if
function that would ultimately satisfy equation (2.0.4)
lim V  x, t   0
x 0
2.1 Definitions
(i)
A continuous function
and there exists
 x
V  x, t   V  x1, x2 ,..., xn , t 

such that
is positive definite if
V  x, t     x
lim V  x, t   0

.
. .
(2.0.8)
x 0
(v) A continuous function
and there exists
V  x, t   V  x1, x2 ,..., xn , t 
 x

is indefinite if it assumes both positive and negative values
such that
in any arbitrary neighbourhood of the origin in a Domain D.
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at a glance we have
origin
V  0  0, V  x   0 for x  0 (Positive semi definite)
V  x0   0
V  0  0, V  x   0 for x  0 (Positive definite)
x0
this context. Given a differential equation
(2.0.9)
V : Rn  R
V is positive definite
(b)
The time derivative of V,
Then the trivial solution
Suppose there exists a C  function
along the solution
Let
(b)
The time derivative of V, V along the solution path
be a sphere in

equation (2.0.9)
R2
x 
f : D R2 ; D x:
stable in the sense of
Lyapunov.
(vii) Suppose there exists a
S  
S    x R2 :
of (2.0.9) is negative semi definite. Then the tri-
x  0 is
x  0 of
Proof:
(a)
(a)
V
is asymptotically stable.
V : R  R with the following properties
V : Rn  R
with the following properties
(a)
(c)
n
vial solution
of equation (2.0.9) is unstable in
path of equation (2.0.9) is negative definite.
f is continuous in  t , x 
(vi)
, then the trivial solution
C  functional
In another Lyapunov theorem, we reaffirm the definitions in
. .
that
Given the system (2.0.9) and suppose there exists a
(Radially undounded)
.
such
2.2 Fundamental Theorem
V  0  0, V  x   0 for x  0 (Negative definite)
x  f  t , x  , f t , o   0
xo
exists
the sense of Lyapunov.
V  0   0, V  x   0 for x  0 (Negative semi definite)
x  ,V  x   
there
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
x H
C  function
S  
(b)
V
along the so-
S  
lution path of equation (2.0.9) is negative
definite. Then the trivial solution
x0
is asymptotically stable in the sense of
Lyapunov.
(viii)
Suppose there exists a
V :R R
n
(a)
(b)
There exists
C  function
x  t , x0 
with
x0  
Since V is positive definite in D and it is continuous,
V  x  t , x0  
such that V is a non increasing func-
tion along all the solution of
V
 0
approaches a limit
along the solu-
tion path of equation (2.0.9) is positive definite and in every neighbourhood of the
assert that , then since
V  0
as
t .
We
V  0   V  x   0 , x  0 ,
then we have proved that
X  t , x0   0
as
t 
.
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Suppose not, that is
V  0  0 ,
V  x   V  0
such that
x 
if
S  
where

will never enter
d
V  x, t    M
dt
V
is positive definite,
...
2V  K1 x2  K2 y 2  K3 z 2  K4 w2  2K5 xy  2K6 xz  2K7 xw  2K8 yz  2K9 yw  2K10 zw
(3.0.2)
There is a need to obtain the derivative with respect to
2V  2K1xx  2K2 yy  2K3 zz  2K4 ww  2K5  yx  xy   2K6  zx  xz 
 2K7  wx  xw   2K8  zy  yz   2K9  wy  yw   2K10  wz  zw 
is the radius of the sphere
From the hypothesis
therefore
 0
(from continuity). We
x  t , x0 
assert that the trajectory
S   : x  ,
then there exists
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t
...(3.0.3)
V  K1 xx  K 2 yy  K3 zz  K 4 ww  K5  yx  xy   K6  zx  xz   K7  wx  xw  ...(3.0.4)
 K8  zy  yz   K9  wy  yw   K10  wz  zw 
and on integrating we have,
But from equation (2.0.3)
V  x, t   V  x,0    Mdt
t
V  K1 xy  K 2 yz  K3 zw  K 4 w  aw  bz  cy  dx   K5  yy  xz   K6  zy  xw   K 7  wy  x  aw  bz  cy  dx  
0
 K8  zz  yw  K9  wz  y  aw  bz  cy  dx    K10  ww  z  aw  bz  cy  dx  
Therefore,
V  x  t    V  x0   mt
bringing the terms and their respective coefficients
t   , then
Terms
V  x  t    V  x0   mt  
x2
dk7
y2
k5  ck9
z2
k8  bk10
w2
k10  ak4
xy
k1  ck7  dk9
If we let
Hence a contradiction to the assumption that
V  0  0
is
impossible
Coefficients
3. Methodology and Discussions
The system (2.1) under investigation is

  bx  cx  dx  0
x  ax
resulting to the four scalar first order shown as follows:
x  y
y  z
z  w
w  aw  bz  cy  dx
leading to the matrix
 0

0

X  AX  
 0

 d
0  x 
 
0 1
0  y 
...
0 0
1  z 
 
c b a  w 
1
0
xz
k5  bk7  dk10
xw
k6  dk4  ak7
yz
k2  k6  bk9  ck10
yw
k7  ck4  k8  ak9
zw
k3  bk4  k9  ak10
. . .3.0.6
(3.0.1)
Next is to determine
where a, b, c, d > 0 for the system to have a negative real
V such that one of the following con-
ditions hold:
path. Consider,
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k1  dbk4  dca k4
(i)
(ii)
V   x2
k2  ack4  bca k4  dk4  b2 k4
V   y 2
k3  a 2 k4  ac k4
(iii)
V   z 2
(iv)
V   w
(v)
V   x 2  y 2  z 2  w2
k4 . 1
. . . 3.0.7

. .
3.0.7
k5  adk4
2
k6  dk4

k7  0
For the realization of any of these cases, one is required
k8  abk4
to impose conditions by realization of any of these
k9  bk4  ac k4
terms as follows:
k10  ak4
k7  0
.
. .
(3.0.8)a
ak4  k10  0
.
. .
(3.0.8)b
k8  bk
.
. .
(3.0.8)c
10  0
. . .
k5  ck9  0
k10  ak4 from equation (3.0.8b)
Then ploughing them back into equation (3.0.2) gives


 2K 4 dxz  2K 4 abyz  2K 4  b 
k8  bk10  k8  abk4
Since k7  0
and from equation
(3.0.6)
k1  dk9
.
. .
(3.0.9a)
k5  dk10
.
. .
(3.0.9b)
k6  dk4
.


2V  dK 4  b  ac  x 2  K 4 ac  bca  d  b2 y 2  K 4 a 2  ac z 2  w2  2K 4 adxy
(3.0.8d)
By setting
k4  1
c
a
. . . …..(3.1.1)
 yw  2K4awz
and dividing both sides by 2, we have




V  d2  b  ac  x 2  12 ac  bca  d  b2 y 2  12 a 2  ac z 2
 w  adxy  dxz  abyz   b   yw  awz
1
2
2
c
a
. .
(3.0.9c)
k2  c k1 0  k6  b k9 . . .
(3.0.9d)
k8  c k4  a k9
(3.0.9e)
This is positive definite provided that ab  c and if we so
(3.0.9f)
choose that a  b  c  d  1 This is an indication of posi-
.
. .
k3  ak10  bk4  k9
.
. .
tive definiteness and the corresponding time derivative
Recall that one interest is on equation (3.0.8d)
k5  ck9  0


V   a 2 d  c 2  abc y 2
In (3.0.9b), k5  dk10 , thus
. . .
dk10  ck9  0
.
. .
(3.1.3)
since equation (4.1.3) satisfies V defined by equation (3.1.2)
(3.1.0)
satisfied
equation
(1.0.1)
if
U
is
In (3.0.9f), k9  ak10  bk4  k3 or In (3.0.9e) k8  ck4  ak9
a d  c
But
function for the fourth order system (2.0.1)
2
2
replaced
by

 abc y 2 , then V defined by (3.1.2) is a Lyapunov
k8  abk4 , thus

abk4  ck4  ak9
3.1 Lyapunov Function Candidate for the Most Ge-
k9  bk4  k4
neralized Form of the Fourth Order Nonlinear Equa-
c
a
tion
From (3.1.0) and (3.0.8b)
Consider the fourth order nonlinear differential equation of
the form
adk4  c bk4  ac k4   0

x  f 
x   g  
x   h  x   j  x   0
a 2 dk4  acbk4  c 2 k4  0
a d  c
2
2

By setting k4  1 and substituting into equation (3.0.2) yields
V
as one substitute the following
. .
(3.1.4)
or the equivalent four system
 abc k4  0
positive definite function
.
x  y, y  z, z  w, w   f 
x   g  
x   h  x   j  x  . . . (3.1.5)
Our objective is to construct an integrated equation denoted
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by V for the four system (2.0.3). So, we consider the con-
2  dx 
V  12 
x 2  12 bx
x
ISSN 2277-8616
.
.
.
(3.2.9)
stant coefficient fourth order differential equation
for the system (3.1.4) or (3.1.5) Next, we generalized the V

  bx  cx  dx  0
x  ax
.
. .
in equation (3.2.9) to the corresponding nonlinear system
(3.1.6)
(3.1.4). However, the comparison between equations (3.1.5)
and (3.1.7) shows that equation (3.1.5) is equivalent to equation (3.1.7) if
  ax
  bx  cx  dx . . .
x  y, y  z, z  w, w
(3.1.7)

f 
x  is replaced by ax
where a, b, c, d are positive constants. We therefore proceed as follows: Multiply equation (4.1.6) by
grate with respect to t from
T
t 0
T

x
and inte-
t  T which yields,
to
T
 12 
2 ds   cx
  0
x 2  12 dx2  dxx   ax
xds   dxds
0
0
0
.
. .
g  x is replaced by
bx
is replaced by
cx
h  x 
(3.1.8)
j  x  is replaced by
dx
Observe that the term ‗a‘ does not appear in equation (3.2.9),
V  
x  bx  dxx , so that
Let
1
2
2
1
2
T
T
T
0
0
0
2
2 ds   cx
  0
V   ax
xds   dxds
T
T
T
0
0
0
2 ds   cx
  0
V   ax
xds   dxds
  cx
V  ax
x  dx  0
2


2  cx
V   ax
x  dx  0
which suggest that our V for equation (3.1.5) is defined by
.
. .
(3.1.9)
.
. .
(3.2.0)
V  V  x, y, z, w  12 
x 2  G  
x   xJ  
x
. .
(3.2.1)
.
. .
(3.2.2)
. .
(3.3.0)
where
G  
x    g  s  ds
x
.
.
0
J  
x    j  s  ds
x
0





. .
.
(3.3.1)
From equation (1.0.1), that is
3.2 Lyapunov Function Candidate for Some Special
V  U
Form of Fourth Order Nonlinear Differential Equation
  cx
U  ax
x  dx
2
.
. .
(3.2.3)
Consider
Similarly the derivative of V  12 
x 2  12 bx2  dx 
x , along the
solution paths of equation (3.1.7) is given by
V  
x 
x  bx 
x  dx 
x  dx

x  f 
x   g  x   h t , x   a4 x  0
a4  0
.
. .
(3.3.2)
is a constant of the equivalent system
.
. .
(3.2.4)
V   aw  bz  cy  dx  w  bzw  dxw  dz .
. .
(3.2.5)
V  aw  bzw  cyw  dxw  bzw  dxw  dz .
. .
(3.2.6
the four system. So consider the fourth order differential
 aw  cwy  dz
. .
(3.2.7)
system (2.0.1) or the equivalent four system (2.0.3) in which
2
2
.
x  y, y  z, z  w, w   f x   g  x   h t , x   a4 x . . . (3.3.3)
Our concern here is to construct an integrated equation for
a, b, c, d are positive constants. Multiply equation (2.0.1) by
From equations (3.1.8), (3.1.9), (3.2.0) and (3.2.1), we infer

x
and integrate with respect to t, t from [0 to T]. That is
that
T
2  cx 
V  ax
x  dx
.
. .
(3.2.8)
Therefore, we have gotten a positive definite function
T
 12 ax2  12 bx x  12 cx x  dx2    
x 2ds   cx2ds  0
0
0
. . . (3.3.4)
V  V  x, y, z, w  12 ax2  12 bx 
x  12 cx x  dx2 . . . (3.3.5)
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ht , x 
then the equation (6.2) becomes
T
T
0
0
V   
x 2 ds   cx2 ds  0
.
. .
T
T
0
0
.
. .
2
V  
x 2  cx


.
. .
DER DIFFERENTIAL EQUATION
(3.3.8)
Consider the non linear differential equation
x  ax  bx  cx  j x   0

u  
x  cx
2
Again differentiating
.
. .
(3.3.9)
v along the solution paths of equation
x  y, y  z, z  w, w  ax  bx  cx  jx
j x 
. .
(3.3.9)
is replaced by
dx
It is clearly known that the expression
v  aw  aw  bz  cy  dx   12 bw  12 bz (aw  bz  cy  dx)  12 czw 
1
1
V  x2  bx2  dxx
2
2
2
cy  aw  bz  cy  dx   2dw
is an established integration equation for equation (2.0.3).
v  a w  abwz  acwy  adwx  12 bw  12 abwz  12 b z 
2
(3.4.3)
uation (3.4.3). If
v along these solution paths
1
2
..
and (3.4.3) shows that equation (2.0.3) is equivalent to eq-
this differentiation is done to verify the positive definiteness
.
(3.4.2)
We remark that a close approximation of equation (2.0.3)
2  12 bx
  dx2
v  12 ax
x  12 cxx
  12 bxx
  12 bxx
  12 cxx
  12 cxx
  2dx
v  axx
...
(2.0.3)
of
. . . (3.4.1)
3.3 OTHER FORMS OF NON-LINEAR FOURTH OR-
v  u
2
x
1
1
V   x f s ds  g xx  ht , x x  a4 x2
2
2

(3.3.7)
Differentiating equation (3.3.7) with respect to t gives
2
V   
x 2  cx
cx
a 4 is replaced by d
(3.3.6)
Therefore
2 ds
V   
x 2 ds   cx
is replaced by
ISSN 2277-8616
2
2
1
2
bczy  12 bdzx  12 czw  12 acwy  12 bczy  12 c 2 y 2 
1
2
dcxy  2dw
2 2
Therefore,
V  12 
x 2  12 bx2  j  x  
x
...
(3.4.4)
This corresponds to the four system (4.4.3) as an integrated
equation. Furthermore, considering the non linear fourth
2a 2  b, 2ab  c, a  2
order equation
x  f x  bx  cx  dx  0
b
c


v   w2  a 2    12 z 2b 2  12 y 2c 2  23 wz  ab   
2
2


3
1
1
2 acwy  2 bdzx  2 dcxy  bczy  wd  ax  2 
Our next work is to generalize
v in
...
(3.4.5)
x
equation 4.3.5) to the
corresponding nonlinear four system (3.3.3). However, the
 y, y  z, z  w, w   f x  bx  cx  dx
.
.
.
(3.4.6)
comparison between equation (2.0.1) and (3.3.2)
The comparison between equation (2.0.3) and (3.4.6) yields
Shows that equation (2.0.1) is equivalent to equation (3.3.2)
that equation (2.0.3) is equivalent to equation (3.4.6) if
if
f x is replaced by ax
g x is replaced by bx
f x is replaced by ax thus
v   x f s ds  bx2  dxx
.
.
.
(3.4.8)

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ISSN 2277-8616
This corresponds to the four system (3.4.3) as an integrated
rential Equations. Computational & Applied Mathemat-
equation.
ics, Vol. 29, N. 3 pp 329-342, 2010
[3] Bainov, D.D ―Second Method of Lyapunov and Exis-
4.0 SIGNIFICANCE OF STUDY
The integrated equations or the Lyapunov functions are
tence of Periodic Solutions of Linear Impulsive Diffe-
useful in the study of qualitative properties of ordinary dif-
rential – Difference Equations‖. Divulgaciones Mathe-
ferential equation. Essentially, the study of the hypothesis of
maticas. V.S, No. ½ (1997), 29-36.
an ordinary differential equation that clearly determines
[4] Grimm, V. Quispel, G.R.W. ―Geometric Integration Me-
whether the solutions are stable, bounded or periodic with-
thods that Preserve Lyapunov Functions, BIT Numerical
out actually going through the solutions. Stability, boun-
Mathematics, 2005 Vol. 45, No. 1, pp 709-723.
dedness and periodicity of solutions is indispensable in social sciences, sciences and engineering. Hence, the study of
[5] Hedrick, J.K, Girarad A. ―Stability of Non Linear Sys-
these properties is a useful tool in every technological in-
tems‖ Control of Non linear Dynamic Systems. Theory
novation and is results oriented. We are all concerned with
and Applications, 2005.
stable economy, stable government, stability of continued
energy generation, stability of peace and sound stable eco-
[6] Jingl, Ren, Wing-Sum Cheung, Zhibo Cheng ―Existence
logical system to mention just a few. In engineering the
and Lyapunov stability of periodic solutions for Genera-
strength of materials is such that can withstand stress and
lized higher-order Neutral Differential equations‖ Hin-
strain as the materials remain stable. Little wonder, stability
dawi Publishing Corporation. Boundary value problems,
theory has become a major area of interest to a great
volume 2011 Article ID635767, 21 pages, doi: 10: 1155
number of mathematicians.
2011/635767
[7] Ezeolu J.O.C, Ogbu, H.M ―Construction of Lyapunov –
5.0 CONCLUSION
The main key in construction of Lyapunov function of non
Type of Functions for some third order Non linear ordi-
linear differential equation by the method of integration is
nary differential equations by the method of integration‖,
the fact that corresponding to any positive definite quadratic
Journal of Science Teachers Association of Nigeria 45,
1 & 2, April & September 2010, 49-58.
form u (x) , there exists another positive definite quadratic
form
v(x) such that the time derivative of v(x) along the
[8] Lars Grune,
Fabian R. Wirth. ―Lyapunov‘s Second method for Non
autonomous Differential Equations‖ Communicated by
solution paths of the four scales system is equal to the negative
u (x), that is
Peter, E. Kloedem, Stefan Siegmund,
Aim Sciences Support by the Science Foundation‘s
V  U
Ireland grants 04-IN3-1460 and 00/PI.1/6067, 2006.
is satisfied; leading us to obtain the Lyapunov type of functions or ―integrated equations‖.
[9] Mohammed, Dahleh, Munthier A. Dahlel, George
Verghese ―Lectures on Dynamic Systems and control‖
6.0 REFERENCES
Department of Electrical Engineering and Computer
[1] Ademola, A.T., Aramowo, P.O. ―Asymptotic Behavior of
Science. Massachusetts Institute of Technology.
Solutions of Third Order Non-Linear Differential Equation‖ Acta Univ. Sapientiae, Mathematica, 3,2, (2011),
[10] Mohammed, Harmouche, Salah Laghrouche, Yacine
Chitour ―Stabilization of Perturbed Integrator Chains
197-211.
using Lyapunov Based Homogeneous Controllers, ar Xi
[2] Afuwape, A.U, Omeike, M.O, ―Stability and Bounded-
v: 1303.5330VZ (Math O.C) 30th May 2013.
ness of Solutions of a Kind of Third Order Delay Diffe284
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[11] Ogundare, B.S ―On Boundaries and Stability of Solutions of Certain Third Order Delay Differential Equation
―Journal of the Nigerian Mathematical Society, Vol. 31
pp 55-68, 2012.
[12] Omeike, M.O ―Further Results on Global Stability of
Solutions of Certain Third Order Non Linear Differential
Equations‖. Acta Univ. Palacki Olomuc, Fac, rer, nat.
Mathematica 47 (2008) 121-127.
285
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Construction Of Lyapunov Functions For Some Fourth Order

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