Open Access

Stability of planar nonautonomous dynamic systems

Advances in Difference Equations20132013:144

DOI: 10.1186/1687-1847-2013-144

Received: 28 February 2013

Accepted: 6 May 2013

Published: 23 May 2013

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq1_HTML.gif, 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ1_HTML.gif
(1.1)

where a k j ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq3_HTML.gif are complex-valued functions from C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq4_HTML.gif, and u ( t ) = ( u 1 ( t ) u 2 ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq5_HTML.gif. Since we are assuming that the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1) is given (fixed), system (1.1) may be considered as a linear nonautonomous system with coefficients A ( t ) = A ( t , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq6_HTML.gif depending only on a time variable.

Here and further, C k ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq7_HTML.gif is the set of k times differentiable functions on ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq8_HTML.gif, L 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq9_HTML.gif is the set of Lebesgue absolutely integrable functions on ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq8_HTML.gif, and BV ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq10_HTML.gif is the set of functions of bounded variation on ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq8_HTML.gif.

Dynamic system (1.1) is said to be stable if for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq11_HTML.gif and for any solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1) there exists δ ( T , ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq12_HTML.gif such that u ( t ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq13_HTML.gif for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif, whenever u ( T ) = | u 1 ( T ) | 2 + | u 2 ( T ) | 2 < δ ( T , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq15_HTML.gif. Dynamic system (1.1) is said to be attractive (to the origin) if for every solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1)
lim t u ( t ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ2_HTML.gif
(1.2)

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

A solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1) is stable if for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq11_HTML.gif there exists δ ( T , ε ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq12_HTML.gif such that u ( t ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq13_HTML.gif for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif, whenever u ( T ) = | u 1 ( T ) | 2 + | u 2 ( T ) | 2 < δ ( T , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq15_HTML.gif.

A solution of (1.1) u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq16_HTML.gif, j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq17_HTML.gif, the classical Routh-Hurvitz condition of stability Re [ λ j ] 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq18_HTML.gif, j = 1 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq17_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ3_HTML.gif
(1.3)
is unstable if Re [ μ ] > Re [ λ 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq19_HTML.gif, 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 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ4_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ5_HTML.gif
(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 [13].

The asymptotic representation of solutions and error estimates in terms of the characteristic function was used in [46] 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq1_HTML.gif (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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ6_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ7_HTML.gif
(1.7)

where C L j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq20_HTML.gif is said to be the characteristic function, and χ 1 , 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq21_HTML.gif are the phase functions. In Section 6 (see Lemma 6.1) the following lemma is proved.

Lemma 1.1 Assume that every solution v ( t ) C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq22_HTML.gif of (1.6) approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif, then
lim t T t [ 2 P ( s ) + χ 1 ( s ) χ 2 ( s ) χ 1 ( s ) χ 2 ( s ) ] d s = , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ8_HTML.gif
(1.8)

where χ 1 , 2 ( t ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq24_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ9_HTML.gif
(1.9)

If the asymptotic behavior of χ 1 ( t ) χ 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq25_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq25_HTML.gif depending on the behavior of P ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq26_HTML.gif, Q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq27_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif. Anyway, under some restrictions, one can obtain stability theorems for (1.6) by considering different asymptotic expansions of χ 1 ( t ) χ 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq25_HTML.gif.

Assume that for some positive constants P 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq28_HTML.gif, Q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq29_HTML.gif, Q m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq30_HTML.gif,
| Q ( t ) | + | P ( t ) | P 1 , Q 0 Q ( t ) Q m . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ10_HTML.gif
(1.10)

Theorem 1.2 (Ignatyev [10])

Suppose that the functions P ( t ) C ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq31_HTML.gif, Q ( t ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq32_HTML.gif are real, and they satisfy conditions (1.10) and
2 P ( t ) + Q ( t ) 2 Q ( t ) m > 0 for some m = const . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ11_HTML.gif
(1.11)

Then linear equation (1.6) is asymptotically stable.

Condition that | Q ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq33_HTML.gif 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 , ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ12_HTML.gif
(1.12)
then condition (1.8) turns to
T [ 2 P ( s ) + Q ( t ) 2 Q ( t ) ] d s = , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ13_HTML.gif
(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 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq31_HTML.gif, Q ( t ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq32_HTML.gif are real, and there exists a non-negative continuous function k ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq34_HTML.gif of bounded variation on ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq8_HTML.gif such that
v f ( v ) > 0 for v 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ14_HTML.gif
(1.14)
2 P ( t ) + Q ( t ) 2 Q ( t ) k ( t ) Q ( t ) , t T , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ15_HTML.gif
(1.15)
T k ( t ) Q ( t ) d t = , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ16_HTML.gif
(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 ) < , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ17_HTML.gif
(1.17)
then every bounded solution of the nonlinear equation
v ( t ) + 2 P ( t ) v ( t ) + Q ( t ) f ( v ) = 0 , t T , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ18_HTML.gif
(1.18)

tends to zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif.

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq35_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ19_HTML.gif
(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 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ20_HTML.gif
(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 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ21_HTML.gif
(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 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ22_HTML.gif
(2.4)

Remark 2.1 Note that using equation (1.1), one can eliminate dependence a 12 ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq36_HTML.gif on u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq37_HTML.gif. 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq38_HTML.gif. Similar calculations show that a 12 ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq39_HTML.gif depends only on t, u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif, coefficients a k j ( t , u ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq3_HTML.gif, and their derivatives.

Here and further, often we suppress the dependence on t and u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif for simplicity.

Introduce the characteristic function of (2.1) that depends on an unknown phase function θ j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq40_HTML.gif:
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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ23_HTML.gif
(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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ24_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ25_HTML.gif
(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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ26_HTML.gif
(2.8)

Equation C A j ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq41_HTML.gif 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ27_HTML.gif
(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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ28_HTML.gif
(2.10)
To explain the motivation for the choice of an energy function for system (1.1) (assuming a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq42_HTML.gif), 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ29_HTML.gif
(2.11)
where χ j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq43_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq44_HTML.gif, are exact solutions of the characteristic equation C A j ( χ j ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq45_HTML.gif, e χ j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq46_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ30_HTML.gif
(2.12)
For the case of linear system (1.1), representation (2.11) gives the general solution of (1.1), where C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq47_HTML.gif, C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq48_HTML.gif are constants. For a nonlinear system, C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq47_HTML.gif, C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq48_HTML.gif depend on a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif. Solving equations (2.11) for C 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq49_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ31_HTML.gif
(2.13)
Replacing χ j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq43_HTML.gif by arbitrary differentiable functions θ j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq50_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ32_HTML.gif
(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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq40_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ33_HTML.gif
(2.15)

From (2.15) it follows that if for any given solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1) the phase functions θ j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq40_HTML.gif satisfy characteristic equation, that is, θ j ( t ) = χ j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq51_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq44_HTML.gif, then energy conservation laws E j ( t ) = const https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq52_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq44_HTML.gif 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ34_HTML.gif
(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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ35_HTML.gif
(2.17)
where θ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq53_HTML.gif is an arbitrary differentiable function, then
H A ( t ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ36_HTML.gif
(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 3.1 Suppose that for a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1), we have A ( t , u ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq54_HTML.gif, and there exist the complex-valued functions p 1 ( t ) , p 2 ( t ) ; θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq55_HTML.gif and the real numbers c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq56_HTML.gif, α such that for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif we have a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ37_HTML.gif
(3.1)
( 1 + | e s ( t ) | 2 ) [ J 0 ( t ) + H A ( t ) ] > ( 1 | e s ( t ) | 2 ) [ p 1 ( t ) p 2 ( t ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ38_HTML.gif
(3.2)
T J ( t , u ( t ) ) d t c < , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ39_HTML.gif
(3.3)
where s ( t ) = θ 1 ( t ) θ 2 ( t ) p 1 ( t ) + p 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq58_HTML.gif, J ( t , u ( t ) ) = J 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq59_HTML.gif, 2 θ ( t ) θ 1 ( t ) θ 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq60_HTML.gif
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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ40_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ41_HTML.gif
(3.5)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq61_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq44_HTML.gif, [ p 2 ( t ) p 1 ( t ) ] 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq62_HTML.gif, the error function J 0 ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq63_HTML.gif and conditions (3.1), (3.3) are close to the necessary and sufficient condition of the stability.

Theorem 3.2 Suppose that for a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1) A ( t , u ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq54_HTML.gif, there exist the complex-valued functions p 1 ( t ) , p 2 ( t ) ; θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq55_HTML.gif, and the real numbers c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq56_HTML.gif, α such that for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif, a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif and conditions (3.1), (3.2),
T J ( t , u ( t ) ) d t = https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ42_HTML.gif
(3.6)

are satisfied with J ( t , u ( t ) ) = J 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq59_HTML.gif as in (3.4), (3.5).

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1) is asymptotically stable.

Theorem 3.3 Suppose that for a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1), we have A ( t , u ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq54_HTML.gif, and there exist the complex-valued functions p 1 ( t ) , p 2 ( t ) ; θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq55_HTML.gif such that for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif we have a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif,
[ H A ( t ) ] < J 0 ( t ) | [ p 1 ( t ) p 2 ( t ) ] | , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ43_HTML.gif
(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 ) = , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ44_HTML.gif
(3.8)
where J 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq64_HTML.gif 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ45_HTML.gif
(3.9)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ46_HTML.gif
(3.10)

is unstable.

Remark 3.3 If
Re [ θ 1 ( t ) θ 2 ( t ) ] Re [ p 1 ( t ) p 2 ( t ) ] 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ47_HTML.gif
(3.11)

then Re [ s ] 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq65_HTML.gif, | e s ( t ) | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq66_HTML.gif and condition (3.2) is satisfied if Re [ H A ( t ) + J 0 ( t ) ] > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq67_HTML.gif.

Otherwise (3.2) is satisfied if J 0 ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq68_HTML.gif, Re [ H A ] | Re [ p 1 ( t ) p 2 ( t ) ] | https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq69_HTML.gif.

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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equa_HTML.gif
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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equb_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ48_HTML.gif
(3.12)
Sometimes it is convenient to use other than (3.4) formula for J 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq70_HTML.gif:
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ49_HTML.gif
(3.13)
Remark 3.4 If p 1 ( t ) p 2 ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq71_HTML.gif, and there exists a function θ ( t ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq72_HTML.gif such that
T | e ± 2 θ ( t ) C A j ( t ) θ ( t ) | d t < , j = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ50_HTML.gif
(3.14)
then H A ( t ) L 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq73_HTML.gif, J 0 ( t ) L 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq74_HTML.gif. 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ51_HTML.gif
(3.15)
and we get Re [ J 0 + H A ] 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq75_HTML.gif . From Theorem 3.1 it follows that in this case the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ52_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq76_HTML.gif are chosen by formula (2.17), then H A ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq77_HTML.gif, 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 ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ53_HTML.gif
(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 3.4 Suppose that for a given solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), there exist the complex-valued functions p 1 , p 2 ; θ 1 , 2 C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq79_HTML.gif such that conditions (3.2), (3.6) are satisfied with J ( t , u ) = J 3 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq80_HTML.gif 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ54_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ55_HTML.gif
(3.19)

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1) approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif.

Choosing
p 1 ( t ) 2 θ ( t ) , p 2 ( t ) 0 , α = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ56_HTML.gif
(3.20)

from Theorem 3.1 (in view of s = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq81_HTML.gif), we obtain the following theorem.

Theorem 3.5 Suppose that for a given solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1), A ( t , u ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq54_HTML.gif, and there exist complex-valued functions θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq82_HTML.gif such that for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif we have a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif,
| 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ57_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ58_HTML.gif
(3.22)
and (3.6) are satisfied, where J ( t , u ) = J 4 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq83_HTML.gif,
J 4 ( t ) = ( θ ( t ) θ ( t ) + Tr ( A ( t ) ) + a 12 ( t , u ) a 12 ( t , u ) + J 0 ( t ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ59_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ60_HTML.gif
(3.24)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1) is asymptotically stable.

By choosing
α = 0 , p 1 ( t ) = p 2 ( t ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ61_HTML.gif
(3.25)

we have s ( t ) = 2 θ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq84_HTML.gif, and assuming (3.11) we get | e s ( t ) | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq66_HTML.gif. From Theorem 3.2 we deduce the following theorem.

Theorem 3.6 Suppose that for a given solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1), A ( t , u ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq54_HTML.gif, and there exist complex-valued functions θ 1 ( t ) , θ 2 ( t ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq82_HTML.gif such that for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif we have a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif,
[ 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ62_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ63_HTML.gif
(3.27)
and (3.6) are satisfied with J 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq64_HTML.gif is as in (3.5), and J ( t , u ) = J 5 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq85_HTML.gif:
J 5 ( t ) = J 0 ( t ) + [ 2 θ 1 ( t ) H A ( t ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ64_HTML.gif
(3.28)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1) is asymptotically stable.

Theorem 3.7 Suppose that for a given solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1), A ( t , u ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq54_HTML.gif, there exist complex-valued function θ 2 ( t ) L 1 ( T , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq86_HTML.gif and the real numbers c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq56_HTML.gif, α such that for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif we have a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ65_HTML.gif
(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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ66_HTML.gif
(3.30)
equation (3.3) (or (3.6)) are satisfied, where J ( t , u ( t ) ) = J 6 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq87_HTML.gif,
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 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ67_HTML.gif
(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 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ68_HTML.gif
(3.32)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1) is stable (or asymptotically stable).

Theorem 3.8 Suppose that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), P ( t , u ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq88_HTML.gif, Q ( t , u ) C ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq89_HTML.gif, there exist the real numbers c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq56_HTML.gif, α and the complex-valued function θ 2 ( t ) L 1 ( T , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq90_HTML.gif such that for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif, 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 = https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ69_HTML.gif
(3.33)

are satisfied, where θ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq53_HTML.gif, J 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq64_HTML.gif are given by (3.29), (3.32).

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of equation (2.1) approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq91_HTML.gif.

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 4.1 Suppose that for a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1), we have A ( t , u ) C 3 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq92_HTML.gif, and for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif the conditions
[ s ( t ) ] 0 , [ p 1 ( t ) p 2 ( t ) ] 0 , s ( t ) = 2 R A ( t ) p 1 ( t ) + p 2 ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ70_HTML.gif
(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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ71_HTML.gif
(4.2)
and (3.3) (or (3.6)) are satisfied, where J ( t , u ( t ) ) = J 7 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq93_HTML.gif,
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 ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ72_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ73_HTML.gif
(4.4)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1) is stable (or asymptotically stable).

Theorem 4.2 Suppose that for a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1), we have A ( t , u ) C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq94_HTML.gif, and for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif we have a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ74_HTML.gif
(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 = . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ75_HTML.gif
(4.6)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1) is asymptotically stable.

Theorem 4.3 Suppose that for a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1), A ( t , u ) C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq94_HTML.gif, for some numbers c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq56_HTML.gif, α, and for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif, we have a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif,
[ 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ76_HTML.gif
(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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ77_HTML.gif
(4.8)
and (3.3) (or (3.6)) are satisfied with J ( t , u ( t ) ) = J 8 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq95_HTML.gif, 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 ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ78_HTML.gif
(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 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ79_HTML.gif
(4.10)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equc_HTML.gif

(small damping) is asymptotically stable.

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

Theorem 4.4 Suppose that for a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of (1.1) A ( t , u ) C 4 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq96_HTML.gif, for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif, the conditions a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif, (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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ80_HTML.gif
(4.11)
and (3.3) (or (3.6)) are satisfied, where J ( t , u ( t ) ) = J 9 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq97_HTML.gif,
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 ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ81_HTML.gif
(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 ) | ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ82_HTML.gif
(4.13)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1) is stable (or asymptotically stable).

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

Theorem 4.5 Suppose for a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1), A ( t , u ) C 3 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq92_HTML.gif, there exist the constants c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq56_HTML.gif, α such that and for t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif, we have a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq57_HTML.gif,
[ 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ83_HTML.gif
(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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ84_HTML.gif
(4.15)
and (3.3) (or (3.6)) are satisfied, where J ( t , u ( t ) ) = J 10 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq98_HTML.gif,
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 ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ85_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ86_HTML.gif
(4.17)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1) is stable (or asymptotically stable).

Remark 4.1 Note that if R A ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq99_HTML.gif and r 2 ( t ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq100_HTML.gif, then | e s ( t ) | = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq101_HTML.gif,
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equd_HTML.gif
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 = . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ87_HTML.gif
(4.18)

Remark 4.2 For the Euler equation u ( t ) + R A ( t ) u ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq102_HTML.gif with R A ( t ) = 1 4 t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq103_HTML.gif, we have r ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq104_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq105_HTML.gif with the other phase function θ 1 = 1 2 t + 1 t ln ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq106_HTML.gif that could be found by solving the equation H A ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq107_HTML.gif (see (6.56)).

The following theorem is deduced from Theorem 4.1 by taking p 1 = θ 1 θ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq108_HTML.gif, p 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq109_HTML.gif, α = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq110_HTML.gif, s = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq81_HTML.gif.

Theorem 4.6 Suppose that for a solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1), A ( t , u ) C 3 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq92_HTML.gif and for t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif, we have a 12 ( t , u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq111_HTML.gif and
[ R A ( t ) ] 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ88_HTML.gif
(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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ89_HTML.gif
(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 = , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ90_HTML.gif
(4.21)
where
J 0 ( t ) = 4 ( [ R A ( t ) ] ) 2 + ( [ R A ( t ) 2 R A ( t ) ] ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ91_HTML.gif
(4.22)

Then the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq2_HTML.gif of system (1.1) is asymptotically stable.

5 Stability theorems for the equations with real coefficients

Theorem 5.1 Assume that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the coefficients P ( t , u 1 ) C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq112_HTML.gif, Q ( t , u 1 ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq113_HTML.gif are real-valued, for some positive constants c j https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq114_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq44_HTML.gif, 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 = https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ92_HTML.gif
(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 = https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ93_HTML.gif
(5.2)

are satisfied.

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ94_HTML.gif
(5.3)
is asymptotically stable if one of the following conditions is satisfied:
  1. (i)

    0 < γ < β https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq115_HTML.gif,

     
  2. (ii)

    β = γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq116_HTML.gif, c b 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq117_HTML.gif,

     
  3. (iii)

    γ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq118_HTML.gif, b > β 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq119_HTML.gif,

     
  4. (iiii)

    γ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq118_HTML.gif, b = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq120_HTML.gif, 0 < β < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq121_HTML.gif.

     
A region of asymptotic stability of equation (5.3) described in Example 5.1 may be extended to
0 < γ 2 β https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ95_HTML.gif
(5.4)

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

Theorem 5.2 Assume that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the coefficients P ( t , u 1 ) C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq122_HTML.gif, Q ( t , u 1 ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq113_HTML.gif are real-valued, and for t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif,
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 = . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ96_HTML.gif
(5.5)

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of equation (2.1) approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq91_HTML.gif.

Theorem 5.3 Assume that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq123_HTML.gif, Q ( t , u 1 ) C ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq124_HTML.gif are real-valued, and for t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif,
| P ( t , u 1 ) | C , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ97_HTML.gif
(5.6)
T ( 2 P ( t , u 1 ) | R ( t , u 1 ( t ) ) k k | ) d t = for some positive number k . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ98_HTML.gif
(5.7)

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of equation (2.1) is asymptotically stable.

Theorem 5.4 Suppose that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq125_HTML.gif, Q ( t , u 1 ) C ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq124_HTML.gif are real functions, and condition (5.7) is satisfied. Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif.

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ99_HTML.gif
(5.8)

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

Theorem 5.5 Assume that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq125_HTML.gif, Q ( t , u 1 ) C ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq124_HTML.gif are real functions and
| P ( t , u 1 ) | C , t T > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ100_HTML.gif
(5.9)
T ( 2 P ( t , u 1 ) 2 t 2 t 2 | R ( t , u 1 ) | T ) d t = , t T > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ101_HTML.gif
(5.10)

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif is asymptotically stable.

Theorem 5.6 Suppose that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq123_HTML.gif, Q ( t , u 1 ) C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq126_HTML.gif are real and condition (5.10) is satisfied. Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif.

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ102_HTML.gif
(5.11)

is asymptotically stable.

Theorem 5.7 Assume that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the coefficients P ( t , u 1 ) C 3 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq127_HTML.gif, Q ( t , u 1 ) C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq126_HTML.gif are real functions, and for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq14_HTML.gif,
R ( t , u 1 ) = Q ( t , u 1 ) P 2 ( t , u 1 ) P ( t , u 1 ( t ) ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ103_HTML.gif
(5.12)
1 + | P ( t , u 1 ) + R ( t , u 1 ) 4 R ( t , u 1 ) | 2 c | R ( t , u 1 ) | , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ104_HTML.gif
(5.13)
T ( 2 P R 2 R | ( ( R ) 1 / 4 ) ( R ) 1 / 4 2 R | ) d t = . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ105_HTML.gif
(5.14)

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ106_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ107_HTML.gif
(5.16)

is asymptotically stable.

Theorem 5.8 Suppose that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq125_HTML.gif, Q ( t , u 1 ) C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq126_HTML.gif are real functions, and the conditions
R ( t , u 1 ( t ) ) ( 1 r 2 ( t ) ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ108_HTML.gif
(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 = https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ109_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ110_HTML.gif
(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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ111_HTML.gif
(5.20)

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1) approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif.

Remark 5.1 By taking r ( t ) = R ( t ) 4 R 3 / 2 ( t ) BV ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq128_HTML.gif, r ( t ) β < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq129_HTML.gif, we get J 0 ( t ) , ξ ( t ) L 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq130_HTML.gif, 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 ) | ) = . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ112_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Eque_HTML.gif
approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equf_HTML.gif
Theorem 5.9 Suppose that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the coefficients P ( t , u 1 ) C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq125_HTML.gif, Q ( t , u 1 ) C 2 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq126_HTML.gif are real functions, and for some constant ξ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq131_HTML.gif, 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ113_HTML.gif
(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 = , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ114_HTML.gif
(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 ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ115_HTML.gif
(5.24)

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif.

Theorem 5.10 Suppose that for a solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1), the functions P ( t , u 1 ) C ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq132_HTML.gif, Q ( t , u 1 ) C ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq124_HTML.gif 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 = , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ116_HTML.gif
(5.25)
S ( t , u 1 ) T t Q ( s , u 1 ) e t s 2 P ( y , u 1 ) d y d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ117_HTML.gif
(5.26)

Then the solution u 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq78_HTML.gif of (2.1) approaches zero as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif.

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 , ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ118_HTML.gif
(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 = . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ119_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq133_HTML.gif, P ( t ) ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq134_HTML.gif .

Theorems 5.1-5.10 are new versions of the stability theorem proved in [15, 913, 1721] by a different technique of construction of the energy function.

6 Proofs

Lemma 6.1 Assume that all the solutions of linear system (1.1) are attractive to the origin, and functions χ 1 , χ 2 C 1 ( T , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq135_HTML.gif are solutions of C A ( χ j ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq136_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq44_HTML.gif. Then
lim t | a 12 ( t ) | exp { T t [ Tr ( A ( s ) ) ] d s } | χ 1 ( t ) χ 2 ( t ) | = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ120_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq137_HTML.gif 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 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ121_HTML.gif
(6.2)

To eliminate u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq137_HTML.gif, 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq138_HTML.gif, so we get (2.1): L [ u 1 ] = u 1 ( t ) + 2 P u 1 ( t ) + Q u 1 ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq139_HTML.gif, 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq140_HTML.gif is proved similarly by elimination of u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq141_HTML.gif.

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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ122_HTML.gif
(6.3)
where χ j https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq142_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq44_HTML.gif are solutions of C A j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq143_HTML.gif. From H A ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq77_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ123_HTML.gif
(6.4)
Since we are assuming that the solutions e χ j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq46_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq44_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ124_HTML.gif
(6.5)
as t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq23_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq91_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equg_HTML.gif
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 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ125_HTML.gif
(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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ126_HTML.gif
(6.7)

Then (1.8) follows from (6.1). □

Lemma 6.2 If K ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq144_HTML.gif is a Hermitian 2 × 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq145_HTML.gif matrix with the entries k i j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq146_HTML.gif such that
det ( K ( t ) ) k 11 ( t ) k 22 ( t ) | k 12 | 2 0 , k 22 ( t ) > 0 , t T , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ127_HTML.gif
(6.8)
then the matrix K ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq144_HTML.gif is non-negative ( K ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq147_HTML.gif), and for any 2-vector u
u K ( t ) u det ( K ( t ) ) Tr ( K ( t ) ) | u | 2 , Tr ( K ( t ) ) k 11 ( t ) + k 22 ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ128_HTML.gif
(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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ129_HTML.gif
(6.10)
then k 12 ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq148_HTML.gif, and
u K ( t ) u = k 11 | u 1 | 2 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ130_HTML.gif
(6.11)
Proof of Lemma 6.2 From the quadratic equation for the real eigenvalues of K ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq144_HTML.gif
λ 2 λ Tr ( K ( t ) ) + det ( K ( t ) ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ131_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ132_HTML.gif
(6.13)
From det ( K ( t ) ) = k 11 ( t ) k 22 ( t ) | k 12 ( t ) | 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq149_HTML.gif, we have k 11 ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq150_HTML.gif and
Tr ( K ( t ) ) = k 11 ( t ) + k 22 ( t ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ133_HTML.gif
(6.14)
Further from
0 λ 1 Tr ( K ( t ) ) , λ 1 λ 2 det ( K ( t ) ) Tr ( K ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ134_HTML.gif
(6.15)
we get
u K ( t ) u λ 2 | u | 2 det ( K ( t ) ) Tr ( K ( t ) ) | u | 2 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ135_HTML.gif
(6.16)

 □

Lemma 6.3 If there exist the complex-valued functions p 1 ( t ) , p 2 ( t ) , θ 1 , 2 L 1 ( T , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq151_HTML.gif, and a real-valued function β ( t ) L 1 ( T , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq152_HTML.gif such that
β ( t ) + 2 [ H A ( t ) ] + 2 [ p 1 ( t ) | e s ( t ) | 2 + p 2 ( t ) ] | e s ( t ) | 2 + 1 > 0 , s ( t ) = θ 1 ( t ) θ 2 ( t ) p 1 ( t ) + p 2 ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ136_HTML.gif
(6.17)
β ( t ) J 0 ( t ) [ H A + p 1 + p 2 ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ137_HTML.gif
(6.18)
where J 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq64_HTML.gif is defined in (3.5), then the energy inequality
V 1 ( t ) + V 2 ( t ) C e β ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ138_HTML.gif
(6.19)
is satisfied, where the energy functions are defined in a more general form than in (2.14):
V j ( t ) = | ( θ j ( t ) a 11 ( t ) ) u 1 ( t ) a 12 ( t ) u 2 ( t ) | 2 | θ ( t ) e p j + θ 3 j ( t ) | 2 , j = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ139_HTML.gif
(6.20)
Proof of Lemma 6.3 Denoting
Z j = ( | d j | 2 d j ¯ a 12 ( t , u ( t ) ) d j a 12 ( t , u ( t ) ) ¯ | a 12 ( t , u ( t ) ) | 2 ) , d j = θ j ( t ) a 11 ( t , u ( t ) ) , j = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ140_HTML.gif
(6.21)
we can rewrite energy formula (6.20) in the form
V j ( t ) = u Z j u | θ e p j + θ 3 j | 2 , j = 1 , 2 , V 1 ( t ) = u Z 1 u | θ e p 1 + θ 2 | 2 = u Z 1 u | θ e p 2 + θ 1 | 2 | e s | 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ141_HTML.gif
(6.22)
By differentiation, we get
V j ( t ) = u Y j u | θ e p j + θ 3 j | 2 , Y j = Z j + A Z j + Z j A 2 Z j [ p j + θ 3 j + θ / θ ] , j = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ142_HTML.gif
(6.23)
V 1 ( t ) + V 2 ( t ) = u [ Z 1 | e s ( t ) | 2 + Z 2 ] u | θ e p 2 + θ 1 ( t ) | 2 , β ( V 1 + V 2 ) V 1 V 2 = u [ ( β Z 1 Y 1 ) | e s | 2 + β Z 2 Y 2 ] u | θ e p 2 + θ 1 ( t ) | 2 = u N u | θ e p 2 + θ 1 ( t ) | 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ143_HTML.gif
(6.24)
where
N = ( β Z 1 Y 1 ) | e s ( t ) | 2 + β Z 2 Y 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ144_HTML.gif
(6.25)
By direct calculations
N = ( A 1 + A 1 ¯ + β 1 A 0 a 12 ( A 2 ¯ β 1 A 3 ¯ ) a 12 ¯ ( A 2 β 1 A 3 ) | a 12 | 2 ( A 4 + β 1 A 5 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ145_HTML.gif
(6.26)
where
A 0 = | d 1 e s | 2 + | d 2 | 2 , A 1 = ( p 1 C A 1 d 1 ) | d 1 e s | 2 + ( p 2 C A 2 d 2 ) | d 2 | 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ146_HTML.gif
(6.27)
A 2 = ( C A 1 d 1 p 1 p 1 ¯ ) d 1 | e s | 2 + C A 2 d 2 ( p 2 + p 2 ¯ ) , A 3 = d 1 | e s | 2 + d 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ147_HTML.gif
(6.28)
A 4 = ( p 1 + p 1 ¯ ) | e s | 2 + p 2 + p 2 ¯ , A 5 = 1 + | e s | 2 , A 0 A 5 | A 3 | 2 = | e s θ 12 | 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ148_HTML.gif
(6.29)
β 1 = β + 2 Re [ H A ] , H A 1 = H A p 1 p 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ149_HTML.gif
(6.30)
Further
det [ N ] = n 11 n 22 n 12 n 21 = | a 12 e s θ 12 | 2 [ β 1 2 2 β 1 ( H A 1 ) F ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ150_HTML.gif
(6.31)
where
F = | A 2 | 2 ( A 1 + A 1 ¯ ) A 4 | e s θ 12 | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equh_HTML.gif
or
F = | C A 1 | e s | + C A 2 | e s | θ 1 θ 2 | 2 ( p 1 + p 1 ¯ ) ( C A 2 θ 1 θ 2 + C A 2 ¯ θ 1 θ 2 ¯ + p 2 + p 2 ¯ ) + ( p 2 + p 2 ¯ ) ( C A 1 θ 1 θ 2 + C A 1 ¯ θ 1 θ 2 ¯ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ151_HTML.gif
(6.32)
or
F ( 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 ( Re [ H A ( t ) p 1 ( t ) p 2 ( t ) ] ) 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equi_HTML.gif
or using notation (3.5), we get
F ( t ) = J 0 2 ( t ) [ ( H A 1 ( t ) ) ] 2 , J 0 = F 2 + ( [ H A 1 ] ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ152_HTML.gif
(6.33)
By Lemma 6.2 to have the non-negativity of the matrix N (with the entries n k j https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq153_HTML.gif), it is sufficient to show that
n 22 = | a 12 | 2 [ β 1 ( 1 + | e s | 2 ) + 2 Re ( p 1 | e s | 2 + p 2 ) ] > 0 , det [ N ] 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equj_HTML.gif
The first condition is condition (6.17), and the second condition follows from (6.18) and (6.31):
β 1 = β + 2 [ H A ] [ H A p 1 p 2 ] + J 0 = [ H A 1 ] + ( [ H A 1 ] ) 2 + F . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equk_HTML.gif
So, from conditions (6.17), (6.18) it follows N = ( β Z 1 Y 1 ) | e s ( t ) | 2 + β Z 2 Y 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq154_HTML.gif,
V 1 ( t ) + V 2 ( t ) = u ( Y 1 | e s | 2 + Y 2 ) u | θ e θ 1 + p 2 | 2 β u ( Z 1 | e s | 2 + Z 2 ) u | θ e θ 1 + p 2 | 2 = β ( t ) ( V 1 + V 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ153_HTML.gif
(6.34)

or (6.19) by integration. □

Lemma 6.4 If the phase functions θ j https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq155_HTML.gif are such that (3.1) is satisfied, then
V 1 ( t ) + V 2 ( t ) c | u ( t ) | 2 | θ 1 θ 2 | 2 α | e θ 1 + p 2 | 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_Equ154_HTML.gif
(6.35)
Proof of Lemma 6.4 Introducing the Hermitian matrix K ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq144_HTML.gif with the entries k i j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-144/MediaObjects/13662_2013_Article_485_IEq146_HTML.gif
K ( t ) = Z 1 | e s ( t ) | 2 + Z 2 = ( | d 1 e s ( t ) | 2 + | d 2 | 2 a 12 ( d 1 | e s ( t ) | 2 + d 2 ¯ ) a 12 ¯ ( d 1 | e s ( t ) | 2 + d 2 ) | a 12 | 2 ( 1 + |