Research

Stability of planar nonautonomous dynamic systems

Gro Hovhannisyan

Author Affiliations

Kent State University at Stark, 6000 Frank Ave. NW, Canton, OH, 44720-7599, USA

Advances in Difference Equations 2013, 2013:144  doi:10.1186/1687-1847-2013-144

 Received: 28 February 2013 Accepted: 6 May 2013 Published: 23 May 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We are describing the stable nonautonomous planar dynamic systems with complex coefficients by using the asymptotic solutions (phase functions) of the characteristic (Riccati) equation. In the case of nonautonomous dynamic systems, this approach is more accurate than the eigenvalue method. We are giving a new construction of the energy (Lyapunov) function via phase functions. Using this energy, we are proving new stability and instability theorems in terms of the characteristic function that depends on unknown phase functions. By different choices of the phase functions, we deduce stability theorems in terms of the auxiliary function of coefficients R A ( t ) , which is invariant with respect to the lower triangular transformations. We discuss some examples and compare our theorems with the previous results.

MSC: 34D20.

Keywords:
nonautonomous dynamic system; stability; attractivity to the origin; asymptotic stability; asymptotic solutions; characteristic function; Lyapunov function; energy function

1 Introduction

We are interested in the behavior of a given solution u ( t ) of the nonlinear planar dynamic system

u ( t ) = A ( t , u ) u ( t ) , A ( t , u ) = ( a 11 ( t , u ( t ) ) a 12 ( t , u ( t ) ) a 21 ( t , u ( t ) ) a 22 ( t , u ( t ) ) ) , t T , (1.1)

where a k j ( t , u ( t ) ) are complex-valued functions from C 2 ( T , ) , and u ( t ) = ( u 1 ( t ) u 2 ( t ) ) . Since we are assuming that the solution u ( t ) of (1.1) is given (fixed), system (1.1) may be considered as a linear nonautonomous system with coefficients A ( t ) = A ( t , u ) depending only on a time variable.

Here and further, C k ( T , ) is the set of k times differentiable functions on ( T , ) , L 1 ( T , ) is the set of Lebesgue absolutely integrable functions on ( T , ) , and BV ( T , ) is the set of functions of bounded variation on ( T , ) .

Dynamic system (1.1) is said to be stable if for any ε > 0 and for any solution u ( t ) of (1.1) there exists δ ( T , ε ) > 0 such that u ( t ) < ε for all t T , whenever u ( T ) = | u 1 ( T ) | 2 + | u 2 ( T ) | 2 < δ ( T , ε ) . Dynamic system (1.1) is said to be attractive (to the origin) if for every solution u ( t ) of (1.1)

lim t u ( t ) = 0 . (1.2)

Dynamic system (1.1) is asymptotically stable if it is stable and attractive.

A solution u ( t ) of (1.1) is stable if for any ε > 0 there exists δ ( T , ε ) > 0 such that u ( t ) < ε for all t T , whenever u ( T ) = | u 1 ( T ) | 2 + | u 2 ( T ) | 2 < δ ( T , ε ) .

A solution of (1.1) u ( t ) is asymptotically stable (attractive to the origin) if (1.2) is true.

It is well-known that for a nonautonomous system with the complex eigenvalues λ j ( t ) , j = 1 , , n , the classical Routh-Hurvitz condition of stability Re [ λ j ] 0 , j = 1 , , n , fails. Indeed, nonautonomous system (1.1) with

A ( t ) = ( λ 1 e t μ 0 λ 2 ) , Re [ λ 1 ] 0 , Re [ λ 2 ] 0 (1.3)

is unstable if Re [ μ ] > Re [ λ 2 ] , although the Routh-Hurvitz condition is satisfied. Necessary and sufficient conditions of asymptotic stability of this system,

Re [ λ 1 ] < 0 , Re [ λ 2 ] < 0 , Re [ λ 2 ] < Re [ μ ] , (1.4)

could be found from the explicit solutions

u 1 ( t ) = C 2 e t ( μ + λ 2 ) μ + λ 2 λ 1 + C 1 e t λ 1 , u 2 ( t ) = C 2 e t λ 2 . (1.5)

This example shows that the description of stability of nonautonomous dynamic systems in terms of the eigenvalues is not accurate.

The usual method of investigation of asymptotic stability of differential equations is the Lyapunov direct method that uses energy functions and Lyapunov stability theorems [1-3].

The asymptotic representation of solutions and error estimates in terms of the characteristic function was used in [4-6] to prove asymptotic stability. In this paper we describe the stable dynamic systems by using energy approach with the use of the characteristic function (see (1.7) below), which is a more accurate tool than the eigenvalues.

The main idea of this paper is to construct the energy function in such a way that the time derivative of this energy is the linear combination of the characteristic functions (see (2.15) below). Using this energy, we prove main stability theorems for two-dimensional systems in terms of unknown phase functions (see Theorems 3.1-3.3).

Theorems 3.1-3.3 are similar to Lyapunov stability theorems with additional construction of an energy function in terms of the phase functions. Theorems 3.1-3.3 are applicable to a wide range of nonlinear systems with complex-valued coefficients (see Example 5.2 below or the linear Dirac equation with complex coefficients in [7]) since they have the flexibility in the choice of an energy function.

To show that our theorems are useful, we deduce different versions of stability theorems (old well-known and some new ones) by using different phase functions as asymptotic solutions of the characteristic equation (see (2.8) below). Moreover, we formulate some of the conditions of stability in terms of the auxiliary function R A ( t ) (see (2.10) below), which is invariant with respect to the lower triangular transformations (see Theorem A.1). Note that there is no universal stability theorem in terms of coefficients for nonautonomous system (1.1) since there is no universal formula for an asymptotic solution of the characteristic equation.

As an application (see Example 5.5), we prove the asymptotic stability of the nonlinear Matukuma equation from astrophysics [8,9].

Consider the second-order linear equation

L [ v ] = v ( t ) + 2 P ( t ) v ( t ) + Q ( t ) v ( t ) = 0 . (1.6)

Define the characteristic (Riccati) equation of (1.6)

C L j ( t ) = e T t χ j ( s ) d s L ( e T t χ j ( s ) d s ) = χ j ( t ) + χ j 2 ( t ) + 2 P ( t ) χ j ( t ) + Q ( t ) = 0 , (1.7)

where C L j ( t ) is said to be the characteristic function, and χ 1 , 2 ( t ) are the phase functions. In Section 6 (see Lemma 6.1) the following lemma is proved.

Lemma 1.1Assume that every solution v ( t ) C 2 ( T , ) of (1.6) approaches zero as t , then

lim t T t [ 2 P ( s ) + χ 1 ( s ) χ 2 ( s ) χ 1 ( s ) χ 2 ( s ) ] d s = , (1.8)

where χ 1 , 2 ( t ) C 1 ( T , ) are solutions of characteristic equation (1.7).

In the proof of Lemma 1.1, it is shown that (1.8) is also a sufficient condition of attractivity of solutions of (1.6) to the origin under additional condition

ln C T ( [ χ 1 ( s ) χ 2 ( s ) ] ) d s ln C , C = const > 0 . (1.9)

If the asymptotic behavior of χ 1 ( t ) χ 2 ( t ) as t is known, then the condition of attractivity (1.8) could be clarified. Unfortunately, there is no a simple formula for asymptotic behavior of χ 1 ( t ) χ 2 ( t ) depending on the behavior of P ( t ) , Q ( t ) as t . Anyway, under some restrictions, one can obtain stability theorems for (1.6) by considering different asymptotic expansions of χ 1 ( t ) χ 2 ( t ) .

Assume that for some positive constants P 1 , Q 0 , Q m ,

| Q ( t ) | + | P ( t ) | P 1 , Q 0 Q ( t ) Q m . (1.10)

Theorem 1.2 (Ignatyev [10])

Suppose that the functions P ( t ) C ( T , ) , Q ( t ) C 1 ( T , ) are real, and they satisfy conditions (1.10) and

2 P ( t ) + Q ( t ) 2 Q ( t ) m > 0 for some   m = const . (1.11)

Then linear equation (1.6) is asymptotically stable.

Condition that | Q ( t ) | is bounded above in (1.10) was removed in [11].

Note that if

χ 1 ( t ) χ 2 ( t ) χ 1 ( t ) χ 2 ( t ) Q ( t ) 2 Q ( t ) L 1 ( T , ) , (1.12)

then condition (1.8) turns to

T [ 2 P ( s ) + Q ( t ) 2 Q ( t ) ] d s = , (1.13)

and is an integral version of (1.11).

In [12] Ballieu and Peiffer introduced a more general condition than Ignatyev’s one (1.11) for the attractivity (see (1.15), (1.16) below) of a nonlinear second-order equation.

Theorem 1.3 (Pucci-Serrin [9], Theorem B)

Suppose that functions P ( t ) C ( T , ) , Q ( t ) C 1 ( T , ) are real, and there exists a non-negative continuous function k ( t ) of bounded variation on ( T , ) such that

v f ( v ) > 0 for   v 0 , (1.14)

2 P ( t ) + Q ( t ) 2 Q ( t ) k ( t ) Q ( t ) , t T , (1.15)

T k ( t ) Q ( t ) d t = , (1.16)

lim t inf ( T t k 2 ( s ) [ 2 P ( s ) + Q ( s ) 2 Q ( s ) ] d s T t k ( s ) Q ( s ) d s ) < , (1.17)

then every bounded solution of the nonlinear equation

v ( t ) + 2 P ( t ) v ( t ) + Q ( t ) f ( v ) = 0 , t T , (1.18)

tends to zero as t .

In this paper we prove general stability Theorems 3.1-3.3 in terms of unknown phase functions. Using these theorems we derive the versions of stability theorem of Pucci-Serrin [9], Smith [13], and some new ones.

2 Energy and some other auxiliary functions

Assuming a 12 ( t ) 0 , consider the following second-order nonlinear equation associated with system (1.1):

L [ u 1 ] = u 1 ( t ) + 2 P ( t , u ) u 1 ( t ) + Q ( t , u ) u 1 ( t ) = 0 , (2.1)

where

2 P ( t , u ) = Tr ( A ( t ) ) a 12 ( t , u ( t ) ) a 12 ( t , u ( t ) ) , Q ( t , u ) = det ( A ( t ) ) + W [ a 11 , a 12 ] a 12 ( t , u ( t ) ) , (2.2)

Tr ( A ( t ) ) a 11 ( t , u ( t ) ) + a 22 ( t , u ( t ) ) , det ( A ( t ) ) a 11 ( t , u ( t ) ) a 22 ( t , u ( t ) ) a 12 ( t , u ( t ) ) a 21 ( t , u ( t ) ) , (2.3)

W [ a 11 , a 12 ] a 11 ( t , u ( t ) ) a 12 ( t , u ( t ) ) a 11 ( t , u ( t ) ) a 12 ( t , u ( t ) ) . (2.4)

Remark 2.1 Note that using equation (1.1), one can eliminate dependence a 12 ( t , u ( t ) ) on u ( t ) . Indeed a 12 ( t , u ( t ) ) = a 12 t + j = 1 2 a 12 u j u j ( t ) t = a 12 t + j = 1 2 a 12 u j ( a j 1 u 1 + a j 2 u 2 ) . Similar calculations show that a 12 ( t , u ( t ) ) depends only on t, u ( t ) , coefficients a k j ( t , u ( t ) ) , and their derivatives.

Here and further, often we suppress the dependence on t and u ( t ) for simplicity.

Introduce the characteristic function of (2.1) that depends on an unknown phase function θ j ( t ) :

C L j ( t ) = C L ( θ j ) = L [ e θ j ( t ) ] e θ j ( t ) = θ j ( t ) + θ j 2 ( t ) + 2 P ( t , u ) θ j ( t ) + Q ( t , u ) , (2.5)

and the auxiliary function:

H L ( t ) = C L 1 ( t ) C L 2 ( t ) θ 1 ( t ) θ 2 ( t ) = θ 1 ( t ) + θ 2 ( t ) + θ ( t ) θ ( t ) + 2 P ( t , u ) , (2.6)

where

e θ j ( t ) = e T t θ j ( s ) d s , j = 1 , 2 , θ ( t ) θ 1 ( t ) θ 2 ( t ) 2 . (2.7)

Define the characteristic (Riccati) function of system (1.1)

C A j ( t ) = C A ( θ j ) = θ j ( t ) + θ j 2 ( t ) θ j ( t ) [ Tr ( A ( t ) ) + a 12 ( t ) a 12 ( t ) ] + det ( A ( t ) ) + W [ a 11 , a 12 ] a 12 ( t ) . (2.8)

Equation C A j ( t ) = 0 is the characteristic equation of system (1.1). For diagonal system (1.1), formulas (2.8) fail (for this case, see (A.23)).

Introduce the auxiliary functions

H A ( t ) = C A 1 ( t ) C A 2 ( t ) ( θ 1 ( t ) θ 2 ( t ) ) = θ 1 ( t ) θ 2 ( t ) θ 1 ( t ) θ 2 ( t ) + θ 1 ( t ) + θ 2 ( t ) Tr ( A ( t ) ) a 12 ( t ) a 12 ( t ) , (2.9)

R A ( t ) = det ( A ( t ) ) ( Tr A ( t ) ) 2 4 + W [ t , a 11 a 22 , a 12 ] 2 a 12 ( t ) + a 12 ( t ) 2 a 12 ( t ) 3 a 12 2 ( t ) 4 a 12 2 ( t ) . (2.10)

To explain the motivation for the choice of an energy function for system (1.1) (assuming a 12 ( t , u ) 0 ), consider a representation of solutions of (1.1) in Euler form (see [6]):

u 1 = C 1 e χ 1 ( t ) + C 2 e χ 2 ( t ) , u 2 = C 1 U 1 ( t ) e χ 1 ( t ) + C 2 U 2 ( t ) e χ 2 ( t ) , (2.11)

where χ j ( t ) , j = 1 , 2 , are exact solutions of the characteristic equation C A j ( χ j ) = 0 , e χ j ( t ) are defined as in (2.7), and

U j ( t ) = χ j ( t ) a 11 ( t , u ( t ) ) a 12 ( t , u ( t ) ) , j = 1 , 2 . (2.12)

For the case of linear system (1.1), representation (2.11) gives the general solution of (1.1), where C 1 , C 2 are constants. For a nonlinear system, C 1 , C 2 depend on a solution u ( t ) . Solving equations (2.11) for C 1 , 2 , we get

C 1 = a 12 u 2 ( χ 2 a 11 ) u 1 ( χ 1 χ 2 ) e χ 1 ( t ) , C 2 = a 12 u 2 ( χ 1 a 11 ) u 1 ( χ 1 χ 2 ) e χ 2 ( t ) . (2.13)

Replacing χ j ( t ) by arbitrary differentiable functions θ j ( t ) , we define auxiliary energy functions

E j ( t ) = E j ( θ j ( t ) ) = | C j | 2 = | a 12 u 2 ( θ j a 11 ) u 1 | 2 | ( θ 1 θ 2 ) e θ 3 j | 2 , j = 1 , 2 . (2.14)

Remark 2.2 Although (2.14) are not constants for a nonlinear or nonautonomous system, they are useful for the study of stability. One can expect that for an appropriate choice of θ j ( t ) these energy functions are approximately conservative expressions for some nonlinear systems that are close to linear.

The derivative of the energy functions (2.14) may be written (see (6.23) below) as a linear combination of the characteristic functions:

E j ( t ) = 2 [ ( θ j a 11 ) ¯ | u 1 | 2 C A j u 2 a 12 ¯ u 1 C A j H A | ( θ j a 11 ) u 1 a 12 u 2 | 2 ] | θ e θ 3 j | 2 . (2.15)

From (2.15) it follows that if for any given solution u ( t ) of (1.1) the phase functions θ j ( t ) satisfy characteristic equation, that is, θ j ( t ) = χ j ( t ) , j = 1 , 2 , then energy conservation laws E j ( t ) = const , j = 1 , 2 are satisfied.

Otherwise, (2.15) means that the error of asymptotic solutions is measured by the characteristic function.

Define (total) energy function as a non-negative quadratic form

E ( t ) = E 1 ( t ) + E 2 ( t ) . (2.16)

Remark 2.3 If the phase functions are chosen as

θ 1 , 2 ( t ) = ± θ ( t ) θ ( t ) 2 θ ( t ) + Tr ( A ( t ) ) 2 + a 12 ( t ) 2 a 12 ( t ) , (2.17)

where θ ( t ) is an arbitrary differentiable function, then

H A ( t ) 0 . (2.18)

3 Stability theorems in terms of unknown phase functions

In this section we formulate the main Theorems 3.1-3.3 of the paper.

Theorem 3.1Suppose that for a solution u ( t ) of (1.1), we have A ( t , u ) C 1 ( T , ) , and there exist the complex-valued functions p 1 ( t ) , p 2 ( t ) ; θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) and the real numbers c > 0 , αsuch that for all t T we have a 12 ( t , u ) 0 and

| θ 1 ( t ) a 11 ( t , u ) a 12 ( t , u ) | 2 + | θ 2 ( t ) a 11 ( t , u ) a 12 ( t , u ) e s ( t ) | 2 + 1 + | e s ( t ) | 2 c | θ ( t ) | 2 α , (3.1)

( 1 + | e s ( t ) | 2 ) [ J 0 ( t ) + H A ( t ) ] > ( 1 | e s ( t ) | 2 ) [ p 1 ( t ) p 2 ( t ) ] , (3.2)

T J ( t , u ( t ) ) d t c < , (3.3)

where s ( t ) = θ 1 ( t ) θ 2 ( t ) p 1 ( t ) + p 2 ( t ) , J ( t , u ( t ) ) = J 1 ( t ) , 2 θ ( t ) θ 1 ( t ) θ 2 ( t )

J 1 ( t ) = [ 2 θ ( t ) + ( 2 α 1 ) θ ( t ) θ ( t ) + Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) + p 2 ( t ) p 1 ( t ) ] + J 0 ( t ) , (3.4)

J 0 ( t ) = | C A 1 ( t ) | e s ( t ) | θ 1 ( t ) θ 2 ( t ) C A 2 ( t ) ¯ | e s ( t ) | θ 1 ( t ) θ 2 ( t ) ¯ | 2 + [ ( p 2 ( t ) p 1 ( t ) + C A 1 ( t ) + C A 2 ( t ) θ 1 ( t ) θ 2 ( t ) ) ] 2 . (3.5)

Then the solution u ( t ) of system (1.1) is stable.

Remark 3.1 Since stability conditions (3.1)-(3.3) of Theorem 3.1 are given in terms of estimates with constants that depend on solutions of (1.1), system (1.1) is stable if these estimates are satisfied uniformly for all solutions (with constants that do not depend on solutions).

Remark 3.2 Note that for a linear nonautonomus system (1.1) with the choice θ j ( t ) = χ j ( t ) , j = 1 , 2 , [ p 2 ( t ) p 1 ( t ) ] 0 , the error function J 0 ( t ) 0 and conditions (3.1), (3.3) are close to the necessary and sufficient condition of the stability.

Theorem 3.2Suppose that for a solution u ( t ) of (1.1) A ( t , u ) C 1 ( T , ) , there exist the complex-valued functions p 1 ( t ) , p 2 ( t ) ; θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) , and the real numbers c > 0 , αsuch that for all t T , a 12 ( t , u ) 0 and conditions (3.1), (3.2),

T J ( t , u ( t ) ) d t = (3.6)

are satisfied with J ( t , u ( t ) ) = J 1 ( t ) as in (3.4), (3.5).

Then the solution u ( t ) of system (1.1) is asymptotically stable.

Theorem 3.3Suppose that for a solution u ( t ) of (1.1), we have A ( t , u ) C 1 ( T , ) , and there exist the complex-valued functions p 1 ( t ) , p 2 ( t ) ; θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) such that for all t T we have a 12 ( t , u ) 0 ,

[ H A ( t ) ] < J 0 ( t ) | [ p 1 ( t ) p 2 ( t ) ] | , (3.7)

lim t exp T t J 2 ( s , u ) d s | ( θ 1 a 11 ) e s | 2 + | θ 2 a 11 | 2 + | a 12 | 2 ( 1 + | e s | 2 ) ( t ) = , (3.8)

where J 0 ( t ) is defined in (3.5), and

J 2 ( t , u ) = [ θ 1 ( t ) θ 2 ( t ) + p 2 ( t ) p 1 ( t ) + θ ( t ) θ ( t ) + Tr ( A ( t ) ) + a 12 ( t ) a 12 ( t ) ] J 0 ( t ) . (3.9)

Then the solution u ( t ) of system (1.1) is unstable.

Example 3.1 From Theorem 3.3 it follows that the linear canonical equation

v ( t ) + 2 b t γ 1 v ( t ) + c t 2 β 2 v ( t ) = 0 , c > 0 , b < 0 , β > γ > 0 (3.10)

is unstable.

Remark 3.3 If

Re [ θ 1 ( t ) θ 2 ( t ) ] Re [ p 1 ( t ) p 2 ( t ) ] 0 , (3.11)

then Re [ s ] 0 , | e s ( t ) | 1 and condition (3.2) is satisfied if Re [ H A ( t ) + J 0 ( t ) ] > 0 .

Otherwise (3.2) is satisfied if J 0 ( t ) > 0 , Re [ H A ] | Re [ p 1 ( t ) p 2 ( t ) ] | .

Under condition (3.11), condition (3.1) turns to

| θ 1 a 11 | 2 + | ( θ 2 a 11 ) e s | 2 + | a 12 e s | 2 | a 12 | 2 ( c | θ 1 θ 2 | 2 α 1 ) ,

which is satisfied if

| θ 1 a 11 | 2 + | ( θ 2 a 11 ) | 2 | a 12 | 2 ( c | θ 1 θ 2 | 2 α 2 )

or

3 | θ 1 ( t ) a 11 ( t ) | 2 + 2 | θ 1 ( t ) θ 2 ( t ) | 2 | a 12 | 2 ( c | θ 1 ( t ) θ 2 ( t ) | 2 α 2 ) . (3.12)

Sometimes it is convenient to use other than (3.4) formula for J 1 ( t ) :

J 1 ( t ) = [ 2 θ 1 ( t ) + 2 α θ ( t ) θ ( t ) H A ( t ) + p 2 ( t ) p 1 ( t ) ] + J 0 ( t ) . (3.13)

Remark 3.4 If p 1 ( t ) p 2 ( t ) 0 , and there exists a function θ ( t ) C 1 ( T , ) such that

T | e ± 2 θ ( t ) C A j ( t ) θ ( t ) | d t < , j = 1 , 2 , (3.14)

then H A ( t ) L 1 ( T , ) , J 0 ( t ) L 1 ( T , ) . In this case formula (3.5) is simplified

J 0 ( t ) = ( [ H A ( t ) ] ) 2 + | C A 1 ( t ) | e s ( t ) | + C A 2 ( t ) | e s ( t ) | θ 1 ( t , u ) θ 2 ( t , u ) | 2 , (3.15)

and we get Re [ J 0 + H A ] 0 . From Theorem 3.1 it follows that in this case the solution u ( t ) of system (1.1) is asymptotically stable if for some real numbers α, l

[ θ 1 ( t ) + α θ ( t ) θ ( t ) ] l < 0 , t T (3.16)

are satisfied (see (3.13), (3.6)).

Note that (3.16) is a nonautonomous analogue of the classical asymptotic stability criterion of Routh-Hurvitz.

If the phase functions θ 1 , 2 are chosen by formula (2.17), then H A ( t ) 0 , and

J 1 ( t ) = [ 2 θ 1 ( t ) + 2 α θ ( t ) θ ( t ) + p 2 p 1 ] + | C A 1 ( t ) | e 2 θ ( t ) | + C A 1 ( t ) | e 2 θ ( t ) | 2 θ ( t ) | . (3.17)

From Theorems 3.1-3.3 one can deduce stability theorems for second-order equation (2.1). The attractivity to the origin for the solution of equation (2.1) is valid even by removing condition (3.1) (compare Theorem 3.2 with the following theorem).

Theorem 3.4Suppose that for a given solution u 1 ( t ) of (2.1), there exist the complex-valued functions p 1 , p 2 ; θ 1 , 2 C 1 ( T , ) such that conditions (3.2), (3.6) are satisfied with J ( t , u ) = J 3 ( t ) defined as

J 3 ( t ) = [ θ 1 ( t ) θ 2 ( t ) θ ( t ) θ ( t ) 2 P ( t , u ) + p 2 ( t ) p 1 ( t ) ] + J 0 ( t ) , (3.18)

J 0 ( t ) = | C L 1 | e s | θ 1 θ 2 C L 2 ¯ | e s | θ 1 θ 2 ¯ | 2 + [ ( p 2 p 1 + C L 1 + C L 2 θ 1 θ 2 ) ] 2 . (3.19)

Then the solution u 1 ( t ) of (2.1) approaches zero as t .

Choosing

p 1 ( t ) 2 θ ( t ) , p 2 ( t ) 0 , α = 1 (3.20)

from Theorem 3.1 (in view of s = 0 ), we obtain the following theorem.

Theorem 3.5Suppose that for a given solution u ( t ) of (1.1), A ( t , u ) C 1 ( T , ) , and there exist complex-valued functions θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) such that for all t T we have a 12 ( t , u ) 0 ,

| θ 1 ( t ) a 11 ( t , u ) | 2 + | θ 2 ( t ) a 11 ( t , u ) | 2 + 2 | a 12 ( t , u ) | 2 c | a 12 ( t , u ) | 2 | θ ( t ) | 2 , (3.21)

J 0 ( t ) + [ H A ( t ) ] = J 0 ( t ) + [ θ 1 ( t ) + θ 2 ( t ) + θ ( t ) θ ( t ) Tr ( A ) a 12 ( t , u ) a 12 ( t , u ) ] 0 , (3.22)

and (3.6) are satisfied, where J ( t , u ) = J 4 ( t ) ,

J 4 ( t ) = ( θ ( t ) θ ( t ) + Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) + J 0 ( t ) ) , (3.23)

J 0 ( t ) = | C A 1 ( t ) 2 θ ( t ) C A 2 ( t ) ¯ 2 θ ( t ) ¯ | 2 + [ ( C A 1 ( t ) + C A 2 ( t ) 2 θ ( t ) 2 θ ( t ) ) ] 2 . (3.24)

Then the solution u ( t ) of system (1.1) is asymptotically stable.

By choosing

α = 0 , p 1 ( t ) = p 2 ( t ) = 0 , (3.25)

we have s ( t ) = 2 θ ( t ) , and assuming (3.11) we get | e s ( t ) | 1 . From Theorem 3.2 we deduce the following theorem.

Theorem 3.6Suppose that for a given solution u ( t ) of (1.1), A ( t , u ) C 1 ( T , ) , and there exist complex-valued functions θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) such that for all t T we have a 12 ( t , u ) 0 ,

[ θ 1 ( t ) θ 2 ( t ) ] 0 , | θ 1 ( t ) a 11 ( t , u ) | 2 + | θ 2 ( t ) a 11 ( t , u ) | 2 C | a 12 ( t , u ) | 2 , (3.26)

J 0 ( t ) + [ H A ( t ) ] = J 0 ( t ) + [ θ ( t ) θ ( t ) + θ 1 ( t ) + θ 2 ( t ) Tr ( A ( t ) ) a 12 ( t , u ) a 12 ( t , u ) ] 0 , (3.27)

and (3.6) are satisfied with J 0 ( t ) is as in (3.5), and J ( t , u ) = J 5 ( t ) :

J 5 ( t ) = J 0 ( t ) + [ 2 θ 1 ( t ) H A ( t ) ] . (3.28)

Then the solution u ( t ) of system (1.1) is asymptotically stable.

Theorem 3.7Suppose that for a given solution u ( t ) of (1.1), A ( t , u ) C 1 ( T , ) , there exist complex-valued function θ 2 ( t ) L 1 ( T , t ) and the real numbers c > 0 , αsuch that for all t T we have a 12 ( t , u ) 0 and the conditions

[ θ ( t ) ] 0 , 2 θ ( t ) d d t ln ( 1 + 2 θ ( T ) T t e T s ( Tr A ( y ) + a 12 ( y ) a 12 2 θ 2 ( y ) ) d y d s ) , (3.29)

| 2 θ ( t ) | 2 + | 2 θ 2 ( t ) 2 a 11 ( t , u ) | 2 | a 12 ( t , u ) | 2 ( c | θ | 2 α 1 ) , (3.30)

equation (3.3) (or (3.6)) are satisfied, where J ( t , u ( t ) ) = J 6 ( t ) ,

J 6 ( t ) = J 0 ( t ) + [ 2 Tr ( A ( t ) ) + 2 a 12 ( t , u ) a 12 ( t , u ) 2 θ 2 ( t ) + 2 ( α 1 ) θ ( t ) θ ] (3.31)

or

J 6 ( t ) = J 0 ( t ) + [ 2 α ( Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) 2 θ 2 ( t ) 2 θ ( t ) ) + 4 θ ( t ) + 2 θ 2 ( t ) ] , J 0 ( t ) = | C A 2 ( t ) | [ 1 + | 1 + 2 θ ( T ) T t e T s ( Tr A ( y ) + a 12 ( y ) a 12 2 θ 2 ( y ) ) d y d s | 2 2 | θ ( T ) | e T t [ Tr A ( y ) + a 12 ( y ) a 12 2 θ 2 ( y ) ] d y ] . (3.32)

Then the solution u ( t ) of system (1.1) is stable (or asymptotically stable).

Theorem 3.8Suppose that for a solution u 1 ( t ) of (2.1), P ( t , u ) C 1 ( T , ) , Q ( t , u ) C ( T , ) , there exist the real numbers c > 0 , αand the complex-valued function θ 2 ( t ) L 1 ( T , t ) such that for all t T , conditions (3.29) and

T [ 2 P ( t , u ) + θ ( t ) θ ( t ) 2 θ ( t ) J 0 ( t ) ] d t = T [ 2 θ 2 ( t ) 4 θ ( t ) J 0 ( t ) ] d t = (3.33)

are satisfied, where θ ( t ) , J 0 ( t ) are given by (3.29), (3.32).

Then the solution u 1 ( t ) of equation (2.1) approaches zero as t .

4 Stability of the planar dynamic systems

From Theorems 3.1-3.3 one can deduce more useful asymptotic stability theorems in terms of coefficients of (1.1) by choosing the phase functions as asymptotic solutions of the characteristic equation.

Theorem 4.1Suppose that for a solution u ( t ) of (1.1), we have A ( t , u ) C 3 ( T , ) , and for all t T the conditions

[ s ( t ) ] 0 , [ p 1 ( t ) p 2 ( t ) ] 0 , s ( t ) = 2 R A ( t ) p 1 ( t ) + p 2 ( t ) , (4.1)

| R A ( t ) | + | Tr ( A ( t ) ) + a 12 ( t ) a 12 ( t ) 2 a 11 ( t ) | 2 | a 12 ( t ) | 2 ( c | R A ( t ) | α 1 ) , (4.2)

and (3.3) (or (3.6)) are satisfied, where J ( t , u ( t ) ) = J 7 ( t ) ,

J 7 ( t ) = J 0 ( t ) + [ 2 R A ( t ) + ( 2 α 1 ) R A ( t , u ) 2 R A ( t ) + Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) + p 2 ( t ) p 1 ( t ) ] , (4.3)

J 0 ( t ) = [ ( R A ( t , u ) 2 R A ( t ) ) ] 2 + ( Re [ p 2 p 1 ] ) 2 ( t ) + | R A ( t ) | 2 ( | e s ( t ) | | e s ( t ) | ) 2 16 | R A ( t ) | 2 . (4.4)

Then the solution u ( t ) of system (1.1) is stable (or asymptotically stable).

Theorem 4.2Suppose that for a solution u ( t ) of (1.1), we have A ( t , u ) C 2 ( T , ) , and for all t T we have a 12 ( t , u ) 0 and

1 t 2 + | Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) 2 a 11 ( t , u ) | 2 c | a 12 ( t , u ) | 2 , (4.5)

T [ 2 t + Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) + 2 t 2 T | R A ( t ) | ] d t = . (4.6)

Then the solution u ( t ) of system (1.1) is asymptotically stable.

Theorem 4.3Suppose that for a solution u ( t ) of (1.1), A ( t , u ) C 2 ( T , ) , for some numbers c > 0 , α, and for all t T , we have a 12 ( t , u ) 0 ,

[ ξ ( t ) ] 0 , ξ ( t ) 1 2 d d t ln ( 1 + 2 ξ ( T ) T t e T s T y 2 R A ( z , u ) d z d y d s ) , (4.7)

| ξ ( t ) | 2 + | Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) T t 2 R A ( s , u ) d s 2 a 11 ( t , u ) | 2 | a 12 ( t , u ) | 2 ( c | ξ ( t ) | 2 α 1 ) , (4.8)

and (3.3) (or (3.6)) are satisfied with J ( t , u ( t ) ) = J 8 ( t ) , where

J 8 ( t ) = J 0 ( t ) + [ Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) + ( 2 α 1 ) T t 2 R A ( s , u ) d s + 4 ( 1 α ) ξ ( t ) ] , (4.9)

J 0 ( t ) = | T t R A ( s , u ) d s | 2 [ 1 + | 1 + 2 ξ ( T ) T t e T s T y 2 R A ( z , u ) d z d y | 2 2 | ξ ( T ) | e T t T s [ 2 R A ( y , u ) ] d y d s ] . (4.10)

Then the solution u ( t ) of system (1.1) is stable (or asymptotically stable).

Example 4.1 From Theorem 4.3 it follows that system (1.1) with

a 11 = 0 , a 12 = 1 , a 21 = t 2 β β t β 1 ( γ + 1 ) ( γ + 2 ) t γ , a 22 = 2 t β , 1 < β 0 , γ β 2 2

(small damping) is asymptotically stable.

By using Jeffreys-Wentzel-Kramers-Brillouin (JWKB) approximation, we will prove the following theorem.

Theorem 4.4Suppose that for a solution u ( t ) of (1.1) A ( t , u ) C 4 ( T , ) , for all t T , the conditions a 12 ( t , u ) 0 , (4.1),

| R A ( t ) | + | Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) R A ( t ) 2 R A ( t ) 2 a 11 ( t , u ) | 2 | a 12 ( t ) | 2 ( c | R A ( t ) | α 1 ) (4.11)

and (3.3) (or (3.6)) are satisfied, where J ( t , u ( t ) ) = J 9 ( t ) ,

J 9 [ t ] = J 0 ( t ) + [ 2 i R A ( t ) + ( 2 α 1 ) R A ( t , u ) 2 R A ( t ) + Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) ] , (4.12)

J 0 ( t ) = 1 2 | R A 1 / 4 ( R A 1 / 4 ) ( t , u ) | ( | e 2 i R A ( t ) | + | e 2 i R A ( t ) | ) . (4.13)

Then the solution u ( t ) of system (1.1) is stable (or asymptotically stable).

The following theorem is proved by using the Hartman-Wintner approximation [14].

Theorem 4.5Suppose for a solution u ( t ) of system (1.1), A ( t , u ) C 3 ( T , ) , there exist the constants c > 0 , αsuch that and for t T , we have a 12 ( t , u ) 0 ,

[ s ] 0 , s = i R A ( t ) ( 1 r 2 ( t ) ) , r ( t ) R A ( t , u ) 4 R A 3 / 2 ( t , u ) , w ( t ) r ( t ) r ( t ) r 2 ( t ) 1 , (4.14)

| Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) R A ( t ) 2 R A ( t ) 2 a 11 ( t , u ) | 2 + | R A ( r 2 1 ) | | a 12 | 2 ( c | R A ( r 2 1 ) | α 1 ) (4.15)

and (3.3) (or (3.6)) are satisfied, where J ( t , u ( t ) ) = J 10 ( t ) ,

J 10 ( t ) = J 0 ( t ) + Re [ 2 R A ( r 2 1 ) ( t , u ) + ( 2 α 1 ) [ R A ( r 2 1 ) ] ( t , u ) 2 R A ( r 2 1 ) ( t , u ) + Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) ] , (4.16)

J 0 ( t ) = ( Re [ w ( t ) ] ) 2 + | w ( t ) | 2 4 | | e s ( t ) | + | e s ( t ) | + 1 r 2 ( t ) ( | e s ( t ) | | e s ( t ) | ) | 2 . (4.17)

Then the solution u ( t ) of system (1.1) is stable (or asymptotically stable).

Remark 4.1 Note that if R A ( t ) 0 and r 2 ( t ) < 1 , then | e s ( t ) | = 1 ,

J 0 ( t ) = ( Re [ w ( t ) ] ) 2 + | w ( t ) | 2 | w ( t ) | 2 , w ( t ) = H A ( t ) = r ( t ) r ( t ) r 2 ( t ) 1 .

In this case, asymptotic stability condition (3.6) is simplified:

T ( d d t ln | R A ( 1 r 2 ) | 1 / 2 α ( t , u ) | a 12 ( t , u ) | Tr ( A ( t ) ) | w ( t ) | 2 ) d t = . (4.18)

Remark 4.2 For the Euler equation u ( t ) + R A ( t ) u ( t ) = 0 with R A ( t ) = 1 4 t 2 , we have r ( t ) 1 , and the Hartman-Wintner approximation fails. To consider this case, one may consider the choice θ 2 = R A 4 R A = 1 2 t with the other phase function θ 1 = 1 2 t + 1 t ln ( t ) that could be found by solving the equation H A ( t ) = 0 (see (6.56)).

The following theorem is deduced from Theorem 4.1 by taking p 1 = θ 1 θ 2 , p 2 = 0 , α = 1 , s = 0 .

Theorem 4.6Suppose that for a solution u ( t ) of system (1.1), A ( t , u ) C 3 ( T , ) and for t T , we have a 12 ( t , u ) 0 and

[ R A ( t ) ] 0 , (4.19)

| Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) 2 a 11 ( t , u ) | 2 + | R A ( t ) | | a 12 | 2 ( c | R A ( t ) | 1 ) , (4.20)

T [ R A ( t ) 2 R A ( t ) + Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) + J 0 ( t ) ] d t = , (4.21)

where

J 0 ( t ) = 4 ( [ R A ( t ) ] ) 2 + ( [ R A ( t ) 2 R A ( t ) ] ) 2 . (4.22)

Then the solution u ( t ) of system (1.1) is asymptotically stable.

5 Stability theorems for the equations with real coefficients

Theorem 5.1Assume that for a solution u 1 ( t ) of (2.1), the coefficients P ( t , u 1 ) C 2 ( T , ) , Q ( t , u 1 ) C 1 ( T , ) are real-valued, for some positive constants c j , j = 1 , 2 , the conditions

R ( t , u 1 ) 0 , 1 + | P ( t , u 1 ) | 2 c 2 | R ( t , u 1 ) | , t T , T ( 2 P ( t , u 1 ) | R ( t , u 1 ) | + R ( t , u 1 ) 2 R ( t , u 1 ) ) d t = (5.1)

or

R ( t , u 1 ) 0 , R ( t , u 1 ) + | P ( t , u 1 ) | 2 c 3 , T ( 2 P ( t , u 1 ) | R ( t , u 1 ) | R ( t , u 1 ) 2 R ( t , u 1 ) ) d t = (5.2)

are satisfied.

Then the solution u 1 ( t ) of equation (2.1) is asymptotically stable.

Example 5.1 By Theorem 5.1 the canonical linear equation

v ( t ) + 2 b t γ 1 v ( t ) + c t 2 β 2 v ( t ) = 0 , b > 0 , c > 0 (5.3)

is asymptotically stable if one of the following conditions is satisfied:

(i) 0 < γ < β ,

(ii) β = γ > 0 , c b 2 > 0 ,

(iii) γ = 0 , b > β 1 > 0 ,

(iiii) γ = 0 , b = 1 , 0 < β < 2 .

A region of asymptotic stability of equation (5.3) described in Example 5.1 may be extended to

0 < γ 2 β (5.4)

by using another asymptotic solution of (5.3) (see Example 5.4 or [15,16]).

Theorem 5.2Assume that for a solution u 1 ( t ) of (2.1), the coefficients P ( t , u 1 ) C 2 ( T , ) , Q ( t , u 1 ) C 1 ( T , ) are real-valued, and for t T ,

R ( t , u 1 ) 0 , T ( 2 P ( t , u 1 ) + R ( t , u ) 2 R ( t , u 1 ) | R ( t , u 1 ) 2 R ( t , u 1 ) | ) d t = . (5.5)

Then the solution u 1 ( t ) of equation (2.1) approaches zero as t .

Theorem 5.3Assume that for a solution u 1 ( t ) of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) , Q ( t , u 1 ) C ( T , ) are real-valued, and for t T ,

| P ( t , u 1 ) | C , (5.6)

T ( 2 P ( t , u 1 ) | R ( t , u 1 ( t ) ) k k | ) d t = for some positive number   k . (5.7)

Then the solution u 1 ( t ) of equation (2.1) is asymptotically stable.

Theorem 5.4Suppose that for a solution u 1 ( t ) of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) , Q ( t , u 1 ) C ( T , ) are real functions, and condition (5.7) is satisfied. Then the solution u 1 ( t ) approaches zero as t .

Example 5.2 By Theorem 5.3 the equation

v ( t ) + 2 b t γ 1 v ( t ) + ( k 2 + ( σ + i μ ) t β ) v ( t ) = 0 , 1 β γ 1 , b > 1 2 k (5.8)

(where β, σ, μ are real numbers and b, k, γ are positive numbers) is asymptotically stable.

Theorem 5.5Assume that for a solution u 1 ( t ) of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) , Q ( t , u 1 ) C ( T , ) are real functions and

| P ( t , u 1 ) | C , t T > 0 , (5.9)

T ( 2 P ( t , u 1 ) 2 t 2 t 2 | R ( t , u 1 ) | T ) d t = , t T > 0 . (5.10)

Then the solution u 1 ( t ) is asymptotically stable.

Theorem 5.6Suppose that for a solution u 1 ( t ) of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) , Q ( t , u 1 ) C 2 ( T , ) are real and condition (5.10) is satisfied. Then the solution u 1 ( t ) approaches zero as t .

Example 5.3 By Theorem 5.5 the linear equation

v ( t ) + 2 a v ( t ) t + ( a 2 a t 2 + b t 3 ln 2 ( t ) ) v ( t ) = 0 , a > 1 (5.11)

is asymptotically stable.

Theorem 5.7Assume that for a solution u 1 ( t ) of (2.1), the coefficients P ( t , u 1 ) C 3 ( T , ) , Q ( t , u 1 ) C 2 ( T , ) are real functions, and for all t T ,

R ( t , u 1 ) = Q ( t , u 1 ) P 2 ( t , u 1 ) P ( t , u 1 ( t ) ) 0 , (5.12)

1 + | P ( t , u 1 ) + R ( t , u 1 ) 4 R ( t , u 1 ) | 2 c | R ( t , u 1 ) | , (5.13)

T ( 2 P R 2 R | ( ( R ) 1 / 4 ) ( R ) 1 / 4 2 R | ) d t = . (5.14)

Then the solution u 1 ( t ) of (2.1) is asymptotically stable.

Example 5.4 From Theorem 5.7 the asymptotic stability of the equation (see also [9,15,16]) follows:

v ( t ) + 2 b t γ 1 v ( t ) + c t 2 β 2 v ( t ) = 0 , b > 0 , c > 0 , 1 β < γ < 2 β . (5.15)

Example 5.5 By Theorem 5.7, the nonlinear Matukuma equation

u 1 + ( n 1 ) u 1 t + A u 1 | u 1 | 2 β 1 + t 2 = 0 , β > 0 , n > 3 (5.16)

is asymptotically stable.

Theorem 5.8Suppose that for a solution u 1 ( t ) of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) , Q ( t , u 1 ) C 2 ( T , ) are real functions, and the conditions

R ( t , u 1 ( t ) ) ( 1 r 2 ( t ) ) 0 , (5.17)

T ( 2 P ( t , u 1 ) + R ( t , u 1 ) 2 R ( t , u 1 ) 4 ξ ( t ) J 0 ( t , u 1 ) ) d t = (5.18)

are satisfied, where

J 0 ( t ) = | ( r 2 1 r ) | 1 + | 1 + 2 ξ 0 T t | R ( s , u 1 ) | e 2 i T s 2 R ( 1 r 2 ) ( y ) d y d s | 2 2 ξ 0 , (5.19)

ξ ( t ) = 1 2 d d t ln ( 1 + 2 ξ 0 T t | R ( s , u 1 ) | e 2 i T s R ( 1 r 2 ) ( y ) d y d s ) . (5.20)

Then the solution u 1 ( t ) of (2.1) approaches zero as t .

Remark 5.1 By taking r ( t ) = R ( t ) 4 R 3 / 2 ( t ) BV ( T , ) , r ( t ) β < 1 , we get J 0 ( t ) , ξ ( t ) L 1 ( T , ) , and Theorem 5.8 becomes a version of Pucci-Serrin Theorem 1.3. In this case, (5.18) is simplified to

T ( 2 P ( t ) + R ( t ) 2 R ( t ) ) d t = lim t ( T t 2 P ( s ) d s + 1 2 ln | R ( t ) R ( T ) | ) = . (5.21)

Example 5.6 Due to Theorem 5.8, every solution of (1.6) with

P ( t ) = 0 , Q ( t ) = R ( t ) = μ 2 t 2 γ ln 2 σ ( t ) , γ > 1  or  γ = 1 , σ > 0 ,

approaches zero as t , since

r ( t ) = R 4 R 3 / 2 = γ 2 μ t 1 + γ ln 1 + σ ( t ) + σ 2 μ t 1 + γ ln σ ( t ) 0 , t .

Theorem 5.9Suppose that for a solution u 1 ( t ) of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) , Q ( t , u 1 ) C 2 ( T , ) are real functions, and for some constant ξ 0 > 0 , we have

Re [ ξ ( t ) ] 0 , ξ ( t ) 1 2 d d t ln ( 1 + 2 ξ 0 T t | R ( s ) / R ( T ) | e 2 i T t R ( y ) d y d s ) , (5.22)

T ( 2 P ( t , u 1 ) + R ( t ) 2 R ( t ) + 2 i R ( t ) 4 ξ ( t ) J 0 ( t ) ) d t = , (5.23)

where

J 0 ( t ) = | 1 + 2 ξ 0 T t | R ( s ) / R ( T ) | e 2 i T s 2 R ( y ) d y d s | 2 + 1 2 | R 1 / 4 ( t ) ξ 0 R 1 / 2 ( T ) | | ( R 1 / 4 ) ( t ) | . (5.24)

Then the solution u 1 ( t ) approaches zero as t .

Theorem 5.10Suppose that for a solution u 1 ( t ) of (2.1), the functions P ( t , u 1 ) C ( T , ) , Q ( t , u 1 ) C ( T , ) are real and

T S ( t , u 1 ) ( 1 2 S ( t , u 1 ) T e s t ( 2 P ( y , u 1 ) 2 S ( y , u 1 ) ) d y d s ) d t = , (5.25)

S ( t , u 1 ) T t Q ( s , u 1 ) e t s 2 P ( y , u 1 ) d y d s . (5.26)

Then the solution u 1 ( t ) of (2.1) approaches zero as t .

If

S 2 ( t , u 1 ) T e s t ( 2 P ( y , u 1 ) 2 S ( y , u 1 ) ) d y d s L 1 ( T , ) , (5.27)

then the attractivity condition (5.25) is simplified

T S ( t , u 1 ) d t = T T t Q ( s , u 1 ) e s t 2 P ( y , u 1 ) d y d s d t = . (5.28)

Note that (5.28) is Smith’s [13] necessary and sufficient condition of asymptotic stability of (2.1) in the case of Q ( t ) = 1 , P ( t ) ε > 0 .

Theorems 5.1-5.10 are new versions of the stability theorem proved in [1-5,9-13,17-21] by a different technique of construction of the energy function.

6 Proofs

Lemma 6.1Assume that all the solutions of linear system (1.1) are attractive to the origin, and functions χ 1 , χ 2 C 1 ( T , ) are solutions of C A ( χ j ) = 0 , j = 1 , 2 . Then

lim t | a 12 ( t ) | exp { T t [ Tr ( A ( s ) ) ] d s } | χ 1 ( t ) χ 2 ( t ) | = 0 . (6.1)

Proof of Lemma 6.1 and Lemma 1.1 First, we derive formula (2.8) for the characteristic function. Solving for u 2 the first equation of (1.1), we get

u 2 ( t ) = u 1 ( t ) a 11 ( t , u ( t ) ) u 1 ( t ) a 12 ( t , u ( t ) ) . (6.2)

To eliminate u 2 , we substitute it in the second equation of (1.1) u 2 ( t ) = a 21 ( t , u ( t ) ) u 1 ( t ) + a 22 ( t , u ( t ) ) u 2 ( t ) , so we get (2.1): L [ u 1 ] = u 1 ( t ) + 2 P u 1 ( t ) + Q u 1 ( t ) = 0 , where P, Q are as in (2.2). From definition (2.5), we get (2.8). Formula (A.22) (see the Appendix) for C C A ( t ) is proved similarly by elimination of u 1 .

The first component of a solution of linear system (1.1) may be represented in the Euler form

u 1 ( t ) = C 1 e χ 1 ( t ) + C 2 e χ 2 ( t ) , (6.3)

where χ j , j = 1 , 2 are solutions of C A j = 0 . From H A ( t ) 0 we get

χ 1 ( t ) + χ 2 ( t ) = Tr ( A ( t ) ) + a 12 ( t ) a 12 ( t ) χ ( t ) χ ( t ) , χ ( t ) χ 1 ( t ) χ 2 ( t ) 2 . (6.4)

Since we are assuming that the solutions e χ j ( t ) , j = 1 , 2 of linear system (1.1) are attractive to the origin, we have

e T t ( χ 1 + χ 2 ) d s = e T t ( Tr ( A ( s , u ( s ) ) ) + a 12 ( s , u ( s ) ) / a 12 χ ( s ) / χ ) d s 0 (6.5)

as t , that is, (6.1) is satisfied. Note that if additional condition (1.9) is satisfied, then (6.1) is also a sufficient condition of attractivity of solutions of (1.6), since in view of (6.5) as t , we have

| e T t χ 1 d s | = e T t [ χ 1 χ 2 ] d s e T t [ χ 1 + χ 2 ] d s 0 , | e T t χ 2 d s | = e T t [ χ 2 χ 1 ] d s e T t [ χ 1 + χ 2 ] d s 0 .

To prove Lemma 1.1, rewrite equation (1.6) in the form of system (1.1)

d d t ( v ( t ) v ( t ) ) = ( 0 1 Q ( t ) 2 P ( t ) ) ( v ( t ) v ( t ) ) , (6.6)

which means that

a 11 ( t ) 0 , a 12 ( t ) = 1 , Tr ( A ) = a 22 ( t ) = 2 P ( t ) , det ( A ) = a 21 ( t ) = Q ( t ) . (6.7)

Then (1.8) follows from (6.1). □

Lemma 6.2If K ( t ) is a Hermitian 2 × 2 matrix with the entries k i j ( t ) such that

det ( K ( t ) ) k 11 ( t ) k 22 ( t ) | k 12 | 2 0 , k 22 ( t ) > 0 , t T , (6.8)

then the matrix K ( t ) is non-negative ( K ( t ) 0 ), and for any 2-vectoru

u K ( t ) u det ( K ( t ) ) Tr ( K ( t ) ) | u | 2 , Tr ( K ( t ) ) k 11 ( t ) + k 22 ( t ) . (6.9)

Remark 6.1 If

det ( K ( t ) ) k 11 ( t ) k 22 ( t ) | k 12 | 2 0 , k 22 ( t ) 0 , k 11 ( t ) 0 , t T , (6.10)

then k 12 ( t ) 0 , and

u K ( t ) u = k 11 | u 1 | 2 0 . (6.11)

Proof of Lemma 6.2 From the quadratic equation for the real eigenvalues of K ( t )

λ 2 λ Tr ( K ( t ) ) + det ( K ( t ) ) = 0 , (6.12)

we have

λ 1 = Tr ( K ( t ) ) + [ Tr ( K ( t ) ) ] 2 4 det ( K ( t ) ) 2 , λ 2 = det ( K ( t ) ) λ 1 . (6.13)

From det ( K ( t ) ) = k 11 ( t ) k 22 ( t ) | k 12 ( t ) | 2 0 , we have k 11 ( t ) 0 and

Tr ( K ( t ) ) = k 11 ( t ) + k 22 ( t ) > 0 . (6.14)

Further from

0 λ 1 Tr ( K ( t ) ) , λ 1 λ 2 det ( K ( t ) ) Tr ( K ( t ) ) (6.15)

we get

u K ( t ) u λ 2 | u | 2 det ( K ( t ) ) Tr ( K ( t ) )