STABILITY OF SOME POLINOMIAL EQUATIONS
WITH DELAY
V. R. Nosov, J. A. Ortega Herrera
Instituto Politécnico Nacional, ESIME-Zacatenco, México D. F., MÉXICO
Abstract- For different types of scalar arbitrary odd degree equations with delay and
without linear terms delay-independent and delay–dependent conditions of asymptotic
stability are established. For general nonlinear equations with principal part giving by
arbitrary odd degree expressions with delays asymptotic stability conditions and
attraction domain estimation are obtained. All stability conditions are expressed
directly in terms of equations coefficients and nonlinearities bounds.
1. INTRODUCTION
The stability of time-delay systems has been widely investigated by different
authors (1-5). This class of systems is described by retarded functional differential
equations. A number of important scientific and technical problems demands for their
adequate description to take into account existing different delays (1-5). The use of
the Liapunov´s direct method for equations with delays encountered some principal
difficulties. Krasovskii (6) suggested the use of functionals defined on retarded
equations´ trajectories instead of Liapunov function and proved general stability
theorems based on the use of functionals. Now such functionals are commonly called
Liapunov-Krasovskii functionals. Another important class of functionals used in
stability study of different functional differential equations is the class of degenerated
functionals introduced firstly in (7). The degenerated functionals are non negative but
not positive definite functionals. In the case of degenenerated functionals the stability
investigation consists of two stages. At the first stage a positive functional which
derivative is negative definite is constructed. The second stage is connected with the
stability study of some functional inequalities generated by zero set of non negative
functional constructed earlier (1, 2, 4).
There exist a sufficiently little number of equations with delay whose stability
conditions are expressed directly in terms of their coefficients. The most studied are
autonomous and non autonomous linear equations for which different sufficient
conditions of stability depending or not on delay are established (1-5). If the nonlinear
equation contains linear terms its stability can be studied by use of the theorem on
stability in the first approximation (1). The stability problems for nonlinear equations
without linear terms are more complicated. In our previous paper (8) stability of scalar
fifth degree equation with delay has been studied in details. Stability conditions as
well as delay-independent and also delay-dependent have been established.
In present paper scalar equations with delay of arbitrary odd degree without linear
terms are studied. Using Liapunov-Krasovskii functionals some sufficient delayindependent conditions of global asymptotic stability for different equations of such
types are derived. Delay-dependent conditions of asymptotic stability and estimations
of attraction domains are obtained by use of degenerated functionals. Nonlinear
equations with principal arbitrary odd degree part are also investigated. All stability
conditions are expressed directly in terms of equations coefficients. In the following
common stability definitions and some theorems from (1, 2) are used.
7-11
2. GENERAL DEFINITIONS AND THEOREMS
Consider delay functional differential equation
x (t )  f (t , xt ), x(t )  R n ,
xt  x(t  s ),  h  s  0, xt  C[h,0],
(1)
x( s )   ( s ),  h  s  0.
Suppose the solution of the initial value problem (1) locally exists. In general the
solution of the initial value problem (1) is not continuable on the half–axis
0,   (1-4). But if the asymptotic stability of the trivial solution is established,
then all solutions with initial functions from attraction domain are continuable.
Let us remember three definitions from (1, 2).
Definition 1. The trivial solution x(t)=0 of Eq.(1) is called stable if for any ε>0 there
exist δ>0 such that the solution x(t,φ) of Eq.(1) corresponding to initial function φ
satisfies
: 
the
inequality
C  h ,0
x(t ,  )   , t  0 for any initial function
  .
Definition 2. The trivial solution x(t) =0 of Eq.(1) is called asymptotically stable if
(1) it is stable,
(2) there exist Δ>0 such that
lim x(t ,  )  0,

t 
C  h , 0 
 .
(2)
The set of all initial functions φ such that condition (2) holds is called the
attraction domain of the trivial solution. If the attraction domain coincide with all
space C[`-h,0], then the trivial solution is called globally asymptotically stable.
Definition 3.The continuous functional
V (t ,  ) : R1  C h,0  R1
is called positive definite (negative definite) if there exist a scalar, continuous, non
decreasing function ω(u) such that ω(u)>0, u>0 and ω(0)=0 which satisfies the
inequality
(3)
V (t ,  )  (  (0) ), V (t ,  )  (  (0) )
 (s)  C h,0 .
The following proposition is established in (2) (Th. 2.4 p. 103 and Th. 3.3 p.118)
Proposition 1. Let for retarded equation (1) exist the continuous positive definite
functional (3) such that
1) 1 (  0  )  V (t ,  )  2 (  ( s )
2) 1 (u)  , u  ,
7-12
C[  h,0]
),
3) the derivative of functional V along the solution of considered retarded equation
is negative definite:
V   3 ( x(t ) ).
Then the trivial solution of considered retarded equation is globally asymptotically
stable.
The proof of some theorems below is based on the use of the degenerated (non
negative but not positive definite) functionals of the form
(4)
V (t , xt , X (t , xt )), X (t , xt )  x(t )  G(t , xt ).
Let us cite the following proposition (Th. 2.4 p.175 from (2)).
Proposition 2. Let for retarded equation (1) exist the continuous degenerated
functional of form (4) such that in the ball S H  xt  C h, o : xt  H :
1) 1 ( X (t , xt ) )  V (t , xt , X (t , xt ))  2 ( xt
C[  h,0]
),
2) functional G(t , xt ) satisfies Lipschitz condition
G(t , xt )  G(t , yt )   xt  yt , 0    1, xt , yt  SH ,
3) the derivative of functional (4) along the solution of retarded equation (1) is
negative definite:
V   3 ( x(t ) ).
Then the trivial solution of retarded equation (1) is asymptotically stable and the
ball S H  xt  C h, o : xt  H  lies in its attraction domain.
3. DELAY-INDEPENDENT CONDITIONS OF GLOBAL ASYMPTOTIC
STABILITY OF ARBITRARY ODD DEGREE EQUATIONS
Consider a scalar arbitrary odd order equation with one delay of the form
x( t )  a2k 1,0 ( t )x 2k 1( t )  a2k ,1( t )x 2k ( t )x( t  h ) 
a2k 1,2 ( t )x 2k 1( t )x 2 ( t  h )  ...  a2 ,2k 1( t )x 2 ( t )x 2k 1( t  h ) 
a1,2k ( t )x( t )x 2k ( t  h )  a0 ,2k 1( t )x 2k 1( t  h ),
1
k  3,4,5,...,h  const, h  0, x( t )  R , t  0
Theorem 1.If conditions
7-13
(3)
a0 ,2k 1( t )   ,
2a2k 1,0 ( t )  a2k ,1( t )      0 ,
2a2k 1,2 ( t )  a2k ,1( t )  a2k  2 ,3( t )  0 ,
....,
(4)
2a1,2k ( t )  a2 ,2k 1( t )  a0 ,2k 1( t )  0 , t  0
are satisfied then the trivial solution of Eq.(23) is globally asymptotically stable.
The proof of this theorem is analogous to Th. 1 proof and is based on the
1
1
Liapunov- Krasovskii functional V ( t , x t ) : R  C  h ,0  R of the form


t
V ( t , xt )  x 2 ( t )  

x 2k  2 ( s )ds
(5)
t h
The derivative of positive defined functional (5) computed along the trajectories
of. (3) is:
V   2 x(t ) x (t )  x 2 k  2 (t )  x 2 k  2 (t  h)  2a 2 k 1,0 (t ) x 2 k  2 (t )  2a 2 k ,1 (t ) x 2 k 1 (t ) x(t  h) 
2a 2 k 1, 2 (t ) x 2 k 1 (t ) x 2 (t  h)  ...  2a 0, 2 k 1 (t ) x(t ) x 2 k 1 (t  h)  x 2 k  2 (t )  x 2 k  2 (t  h) 
2a2 k 1, 0 (t ) x 2 k 2 (t )  a2 k ,1 (t ) x 2 k 2 (t )  a2 k ,1 (t ) x 2 k (t ) x 2 (t  h)  ... 
2a1, 2 k (t ) x 2 (t ) x 2 k (t  h)  a0, 2 k 1 (t ) x 2 (t ) x 2 k (t  h)  a0, 2 k 1 (t ) x 2 k  2 (t  h) 
x 2 k  2 (t )  x 2 k  2 (t  h)  x 2 k  2 (t )
Under conditions (4) this derivative is negative defined, V   x
is sufficient to prove Th.4.
Investigate the special case of Eq.( 3) for which Th.1 is not valid
2k  2
( t ) and this
x(t )   a2 k 1, 2 (t ) x 2 k 1 (t ) x 2 (t  h)  ... 
a3, 2 k  2 (t ) x 3 (t ) x 2 k  2 (t  h)  a1, 2 k (t ) x(t ) x 2 k (t  h),
(6)
a2k 1,2 ( t )  0,...,a3,2k  2 ( t )  0, a1,2k ( t )  0,k  3,4,5,...,
h  const, h  0, x( t )  R1 , t  0
Using the positive definite functional V  x 2 (t ) we can prove as in (8) the
following theorem.
Theorem 2. The trivial solution of Eq.(7) is globally asymptotically stable if
7-14
a2k 1,2( t )  0,...,a1,2k ( t )  0 ,
( a2k 1,2 ( t )  ... a1,2k ( t )    0
(7)
4. DELAY-DEPENDENT CONDITIONS OF ASYMPTOTIC STABILITY
Consider special type of odd order equation with one delay
x(t )  b(t ) x 2 k 1 (t  h), b(t )  0, k  4,5,6....
(8)
For Eq. (8) stability conditions dependent on delay are established.
Theorem 11. The trivial solution of Eq.(54) is asymptotically stable if
b(t )    0,
t
b1 (t ) 
 b( s  h)ds  
1
 1, t  0.
(9)
t h
The attraction domain of the trivial solution contains the ball
x( s )
C  h ,0
 ( 2k  1 )
 1
( 2k  2 )
.
(10)
The proof of Th.10 is based on the degenerated (positive but not positive definite)
functional of the form
t
1
V ( t , xt )  X 2 ( t , xt ) 
k
t
 b( t  2h )dt  b( s  h )x
1
t h
1
2k ( 2k 1 )
( s )ds,
t1
X ( t , xt )  x( t )  G( t , xt ),
(11)
t
G( t , xt ) 
 b( s  h )x
2k 1
( s )ds.
t h
(12)
Functional (11) does not verify inequality (3) from definition 3 of positive definite
funcfional, but it verifies the following weakened condition
V ( t , xt )  X 2 ( t , xt ).
(13)
Functional G from (12) satisfies in the ball (10) Lipschitz condition with Lipschitz
constant less than 1:
7-15
t
G( t , xt )  G( t , yt ) 
 b( s  h )x
t
2k 1
( s )ds 
t h
 b( s  h )x( s )  y( s )x
 b( s  h ) y
2k 1
( s )ds 
t h

t

2k 2
( s )  x 2k 3 ( s ) y( s )  ...  x( s ) y 2k 3 ( s )  y 2k 2 ( s ) ds
t h
Applying Hölder inequality
 
 p q 1 1

,  1
p
q p q
2k 3
( s ) y( s ), y 2k 3( s )x( s ) ,
with p=(2k-2)/(2k-3) and q=(2k-2) to products x
2k  4
( s ) y 2( s ), y 2k 4( s )x2( s )
with p=(2k-2)/(2k-4) and q=(2k-2)/2 to products x
and so on, we have following estimations
x
2k 2

( s )  x 2k 3 ( s ) y( s )  ...  x( s ) y 2k 3 ( s )  y 2k 2 ( s ) 
2k  3 2 k  2
1
2k  4 2 k  2
x
(s)
y 2k 2 ( s ) 
x
(s)
2k  2
2k  2
2k  2
2
1
2k  3 4
y 2k 2 ( s )  ... 
x 2k 2 ( s ) 
y ( s )  y4( s ) 
2k  2
2k  2
2k  2
1  2 k  2  2k  3 2k  4
1 
 2k  3 2k  4
x 2k 2 ( s )1 

 ... 
( s )1 

 ... 
 y

2k  2 
2k  2 
 2k  2 2k  2
 2k  2 2k  2
x 2k 2 ( s ) 




2k  1 2 k  2
 2k  3  2 k  2
( s )  y 2k 2 ( s ) 
x
( s )  y 2k 2 ( s ) .
1 
x
2 
2

Finally we obtain in ball (10)
t
G( t , xt )  G( t , yt )  x( s )  y( s )
C  h ,0

b( s  h )
t h


2k  1 2 k  2
x
( s )  y 2k  2 ( s ) ds 
2
t
x( s )  y( s )
C  h ,0

b( s  h )ds  1 x( s )  y( s )
C  h ,0
, 1  1 .
(14)
t h
Functional (11)-(12) derivative computed along the trajectories of Eq. (8) is equal
to
7-16
t




2
k

1
V   2 x( t )  b( s  h )x
( s )ds  x( t )  b( t  h )x 2k 1( t )  b( t )x 2k 1( t  h ) 


t h





t

1
1
b( t  h ) b( s  h )x 2k ( 2k 1 )( s )ds  b( t  h )x 2k ( 2k 1 )( t )b1( t  h ) 
k
k
t h
t
2k
 2b( t  h )x ( t )  2b( t  h )x
2k 1

( t ) b( s  h )x 2k 1( s )ds 
t h
(15)
t

1
1
b( t  h ) b( s  h )x 2k ( 2k 1 )( s )ds  b( t  h )x 2k ( 2k 1 )( t )b1( t  h )
k
k
t h
Appling Hölder inequality with p=2k/(2k-1) and q=2k to the product
x 2k 1( t )x 2k 1( s ) we have
x 2k 1( t )x 2k 1( s ) 
2k  1 2k
1 2k( 2k 1 )
x (t )
x
(s)
2k
2k
As b(t+h)>0 the following estimation of second term of functional derivative (15)
holds:
t
2b( t  h )x 2k 1( t )
 b( s  h )x
2k 1
( s )ds 
t h
t
2
 b( t  h )x
2k 1
( t )b( s  h )x 2k 1( s )ds 
t h
2k  1
1
b( t  h )x 2k ( t )b1( t )  b( t  h )
k
k
(16)
t

b( s  h )x 2k ( 2k 1 )( s )ds.
t h
Replacing second term in (15) by its estimation (16) and taking into account that in
2
ball (10) x ( t )  1 we have in ball (10) following estimation of functional (11)
derivative
7-17
V ( t , x t )  2b( t  h )x 2k ( t ) 
2k  1
b( t  h )x 2k ( t )b1 ( t ) 
k
1
b( t  h )x 2k ( 2k 1 ) ( t )b1 ( t  h ) 
k
5
1


 b( t  h )x 2k ( t )2  1  1 x 2k ( 2k  2 ) ( t ) 
3
3


(17)
 2( 1  1 )x 2k ( t ) .
Conditions (13),(14) and (17) show that all conditions of Proposition 2 are
satisfied. Hence, the conclusion of Th.11 follows now from this proposition.
Remark 1. Th.3 is valid without any changes in its formulation also for more
general equation
x( t )  a2k 3,2( t )x 2k 3( t )x 2( t  h )  a2k 5,4( t )x 2k 5( t )x 4( t  h )  ...
 a1.2k 2( t )x( t )x 2k 2( t  h )  b( t )x3( t  h ),
a2k 3,2( t )  0, a2k 5,4( t )  0,...,a1.2k 2( t )  0, t  0
(18)
Only change in the proof is the addition in (17) of terms
 2a2k 3,2( t )x2k 2( t )x2( t  h )  2a2k 5,4( t )x2k 4( t )x4( t  h )  ...
 2a1.2k 2( t )x2( t )x2k 2( t  h )  0
(19)
5. NONLINEAR EQUATIONS WITH ARBITRARY ODD DEGREE
PRINCIPAL PART
Consider the general nonlinear equation with (2k+1)-degree principal part
x( t )  a2k 1,0 ( t )x 2k 1( t )  a2k ,1( t )x 2k ( t )x( t  h ) 
a2k 1,2 ( t )x 2k 1( t )x 2 ( t  h )  ...  a2 ,2k 1( t )x 2 ( t )x 2k 1( t  h ) 
a1,2k ( t )x( t )x 2k ( t  h )  a0 ,2k 1( t )x 2k 1( t  h )  f ( t , x( t ), x( t  h )),
k  3,4,5,...,h  const, h  0, x( t )  R1 , t  0
By use of functional (5) the following theorem can be derived.
Theorem 4. Let conditions (4) where first condition is replaced by condition
a0 ,2k 1( t )     ,   0
hold and let the continuous function f(t,x(t),x(t-h)) be such that
7-18
(20)
f ( t , x( t ),x( t  h ))  x2k 2( t )  x2k 2( t  h ),   0,   0
(21)
Then the trivial solution of Eq.(66) is asymptotically stable and the ball
 
 
x( s )
 m in
,
 4 2 

C  h ,0


lies in its attraction domain.
ACKNOWLEDGEMENT
This work was partially sponsored by the S.N.I. and the EDI scholarships granted
by the CONACyT, and the Instituto Politécnico Nacional de México, IPN,
respectively.
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