ON THE DIRECT LEFSCHETZ STABILITY CRITERION FOR A SYSTEM OF NON-HOMOGENEOUS LINEAR FIRST ORDER ODEs, WITH VARIABLE COEFFICIENTS

(1)

Published Online: May 2, 2011

This paper is available online at http://pphmj.com/journals/adecp.htm

: tion Classifica ject

Sub s Mathematic

2010 35F35, 35F15, 34A12, 34A30, 34D20, 34K20.

Keywords and phrases: Lefschetz direct stability criterion, system of ODEs, variable coefficients.

Received March 23, 2011

ON THE DIRECT LEFSCHETZ STABILITY CRITERION FOR A SYSTEM OF NON-HOMOGENEOUS LINEAR FIRST

ORDER ODEs, WITH VARIABLE COEFFICIENTS

PAUL SUNNYBOY MAKHABANE and JOE HLOMUKA Department of Mathematics and Applied Mathematics

University of Venda

Thohoyandou 0950, South Africa e-mail: [email protected] [email protected]

Abstract

We study the stability of a system of first order linear ODEs with variable coefficients. Having selected the Lefschetz direct stability method (Meyer [6]), we modify it to suit our non-dynamic problem. The direct method requires the construction of a suitable Lyapunov function; not easy for a non-dynamic problem. For a dynamic problem it is common to associate the energy thereof with a Lyapunov function. For a non-dynamic problem it is harder to construct a Lyapunov function as there are no rules for that purpose. The Lefschetz method afforded us the construction of a credible Lyapunov function which enabled us to confirm the stability of the null solution to our problem.

1. Statement of the Problem We consider the following boundary value problem:

( ) ( ) ( ) ( )

⎪⎩

( )

⎪⎨

⎧

= +

=

, :

to Subject

, c 0 Y

b Y Y A

x x dx x

x d

(1)

(2)

where

( )

( ) ( )

( )

( ) ( ) ( )

( ) ( )

( )

^⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

x b

x a x

a x a

x a x

a x a

x a x

a x a

x

x x x

x

n nn

n n

n

2 1

2 22

21

1 12

11

2 1

;

; A b

y y y

Y

and

2 ,

1

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

cn

c c

c

in an open bounded domain Ω⊂ R¹; with c as a constant n×1 vector (Coddington and Levinsion [3]).

2. Basic Assumptions and Preliminary Notions

In the analysis of the problem, we assume the following holds:

2.1. Basic assumptions

(a) The matrix A( ) ( ( ))x = a_ij x is symmetric for all x∈Ω.

(b) There exists a unique, multi-valued, bounded and differentiable map,

: R1

g Ω→ defined by x a_ij

( )

x , where i, j =1,2,3,...,n.

(c) There exists a unique single-valued map h:Ω→R¹defined by x b_i

( )

x , where i =1,2,3,...,n.

(d) For problem (1), it therefore makes sense to choose

( ) ( )

.

:= ⊂Ω

∈S Dom g Dom h

x ∩

(e) All the norms in this paper are Rⁿ and Mⁿ_R^×ⁿ-norms, where Mⁿ_R^×ⁿ is assumed to be a vector space for some x∈S.

(3)

2.2. The norm of the matrix A

( )

x

An n× matrixn A( ) ( ( ))x = a_ij x ∈Mⁿ_R^×ⁿ with a_ij( )x ∈R defines a linear map .

:Rⁿ Rⁿ

T →

Let

( )

( ) ( ) ( )

. ,

2 1

2 22

21

1 12

11

Ω

∈

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

= x

x a x

a x a

x a x

a x a

x a x

a x a x A

nn n

n

n n

We define the norm of A( )x as follows:

( ) ∑ ( )

≤ =

≤ ∈Ω

=

×

n

j n ij

i a x x

x

A n n

R

1max 1 ; .

M

Since a_ij

( )

x;i, j =1,2,..., n are bounded, by Assumption 2.1(b), the norm is well- defined.

3. Existence and Uniqueness for the Solution to the Problem

We confirm the existence and uniqueness of the solution to the problem through the following theorem:

Theorem 3.1 (Existence and Uniqueness). The solution to (1) exists and is unique in S.

Proof. Let y′

( )

x =G

(

x, y

( )

x

)

, with G

(

x, y

( )

x

)

= A

( ) ( )

x y x +b

( )

x . Integrating both sides, we have

( ) ( ( ) )

∫

0^x^y^′^x ^dx ⁼

∫

0^x^G ^ξ ^y^ξ ^d^ξ

, ,

which implies that

( )

^x ⁻ ^y

( )

⁼

∫

^x^G

(

^ξ ^y

( )

^ξ

)

^d^ξ y

0 ,

0

( )

⁼ ⁺

∫ (

^ξ

( )

^ξ

)

^ξ

⇒ y x ^xG y d

0

. ,

c

(4)

Then

(

x y

)

G

(

x y

)

A

( ) ( )

x y x b

( )

x A

( ) ( )

x y x b

( )

x

G , ₁ − , ₂ = ₁ + − ₂ +

( ) ( )x y x b( )x A( ) ( )x y x b( )x

A + − −

= ₁ ₂

( ) ( )x y x A( ) ( )x y x

A ₁ − ₂

=

( ) ( )x

[

y x y ( )x

]

A ₁ − ₂

=

(

x, y₁

)

G

(

x, y₂

)

A

( )

x y₁

( )

x y₂

( )

x ,

G ⁿ ⁿ

R −

≤

− M ×

implying that G is Lipschitz on y

( )

x .

According to the Picard-Lindelof Theorem (Birkhoff and Rota [2]) and (Coddington and Levinsion [3]), the solution y( )x is unique if and only if

( ) (x y x)

G , is Lipschitz on y

( )

x. ~

To study the stability of the solution to (1), we propose the direct method as presented by Lefschetz (Meyer [6]). However, to apply the Lefschetz direct stability criterion, we have to establish the stability of the matrix A

( )

x.

4. The Stability of the Matrix A

( )

x

To establish the spectrum for A

( )

x , we formulate the following lemma:

Lemma 4.1. A( )x is stable for some D ⊂ in which S λ_max( )x <0.

Proof. Since A( )x is symmetric, we consider its Rayleigh quotient, where λ( )x is the eigenvalue of A( )x for some x∈ S.

We define the Rayleigh quotient for A( )x as follows:

( ) ( ) ( ) ( ) ( )

x

( )

x

x x x x A

T T

w w

⋅

= ⋅

λ (where w( )x is a corresponding eigenvector to λ(x)).

Hence

( ) ( )

1

( ) ( ) ( )

.

2 A x x x

x

x ^T

Rⁿ

w w w

⋅

= λ

(5)

The application of the Cauchy-Schwartz inequality and the boundedness of A( )x lead to

( ) ( ) ( )

( )

A

( )

x max a

( )

x ;

x x x

A

x _ij

R x

R n n

n R n n

n R

Ω

= ∈

=

≤

λ ×

×

M M

w w

. ..., , 3 , 2 , 1

, j n

i =

This implies that the continuous spectrum for A( )x is bounded. The symmetry of ( )x

A implies that λ( )x ∈R¹. Hence, there exists λ_max( )x such that λ( )x ≤λ_max( )x for all λ( )x and x∈S.

Selecting x∈D:=

{

x∈S :λ_max

( )

x < 0

}

, then A( )x is stable in D. ~ Remarks 4.2. (a) It can easily be shown that A( )x is not positively definite for all x∈Ω.

(b) Since A( )x is symmetric, by Assumption 2.1(a), its continuous spectrum is real and hence, ordered.

(c) In view of the preceding remarks, D:=

{

x∈S :λ_max

( )

x < 0

}

does exist.

5. The Lefschetz System for the Stability Analysis of the Problem

5.1. The original Lefschetz system

We present the following Lefschetz system (see Meyer [6]) on which the direct stability criterion would be based:

( ) ( ) ( ( ) )

( ) ( )

( )

( ) ( ( ) ) ( ( ) )

^{( )}

( ( ) ) ( )

⎪⎪

⎩

⎪⎪

⎨

⎧

−

= +

σ σ φ +

=

= σ

σ φ +

=

∫

^σ

definite.

positive

and symmetric arbitrary;

; :

equation Sylvester

the

from obtained is

where

; :

: function Lyapunov

ing correspond the

with

; :

where

,

0 /

C C BA B A

B t

d t t

x B t x t x V

t x c t

t b t Ax t x

T T t

T

(2)

(6)

The basic assumptions associated with this system are:

(a) x, b, c are real vectors; with n = 2 in our case;

(b) A is a real n× matrix with eigenvalues whose real parts are negative; n (c) B is a positive definite symmetric matrix satisfying condition to be specified later;

(d) φ is a continuous function on σ;

(e) σ( )0 = 0, for σ≠ 0,σφ( )σ >0.

Provided A( )x is stable, it is confirmed (see Meyer [6]), that −V(x( )t )> 0 implies that

2 . 1 2

1 2

1 ₁

⎥⎦⎤

⎢⎣⎡ + + α

⎥⎦⎤

⎢⎣⎡ + + α

>

−c^Tq Bb A^Tc c^T ^TC⁻ Bb A^Tc c^T

Next, we present our modified Lefschetz system:

5.2. The modified Lefschetz system

We consider the Lefschetz system (Meyer [6]) and for the purpose of our study, we modify it as follows:

( ) ( ) ( ) ( ( ) )

( ) ( )

( )

( ) ( ) ( )

⎪⎪

⎩

⎪⎪

⎨

⎧

×

>

σ σ

= σ φ

×

= σ

∈ σ

φ +

=

vector.

constant an

:

choice, arbitrary our

; 0 ,

vector, constant 1

an

; where

,

/ ,

n n q

x x x

n c x y c x

D x x q x y x A x y

T

T (3)

Remarks 5.3. (a) φ is continuous on σ

( )

x, provided σ x( )> 0 in D.

(b) In terms of (1) and (3), we have qφ

(

σ

( )

x

)

=b

( )

x . (c) The solution ^y( )^x exists and is unique.

(d) It is now clear that σ( )x φ(σ( )x )>0, which is one of the requirements by the Lefschetz direct stability criterion (Meyer [6]).

(7)

In terms of the Lefschetz direct stability criterion, our Lyapunov function is defined as follows:

( )

(

^y ^x

)

^y^T

( ) ( ) ( )

^x ^B ^x ^y ^x ⁺

∫

^σ^{( )}^x ^φ

(

^σ

( )

^x

)

^d^σ

V

0

,

: for x∈D, (4)

where we will later show that B( )x is uniquely determined as the solution to the ODE:

( )

A

( ) ( )

x B x B

( ) ( )

x A x C

( )

x, dx

x

dB + T + = −

with −C( )x assumed positive definite and symmetric. We will later show that under certain conditions, B( )x is also positive definite.

6. Confirmation of the Lefschetz Stability Criterion Using the Modified System

From (4), we have

( )

⁼

( ) ( ) ( )

^′ ⁺

( ) ( ) ( )

⁺

∫

0^σ^φ

(

^σ

( ) ) ( )

^σ

/ 2 y x x d x

dx x x dB y x y x y x B x y

V ^T

( ) ( ) ( )

^′ ⁺

( ) ( ) ( )

⁺

∫

^σ^φ

(

^σ

( ) ) ( )

^σ

=

0

2 x d x

dx x d dx y

x x dB y x y x y x

B ^T

( ) ( ) ( ) ( ) ( ) ( ) ( ( )) ( ) _⎟

⎠

⎜ ⎞

⎝

⎛ σ ⋅ σ

φ +

′ +

= dx

dy dy

x x d x

dx y x x dB y x y x y x

B ^T

2

( ) ( ) ( ) ( ) ( ) ( )^y ^x ( ( )^x )^c ^y( )^x dx

x x dB y x y x B x

y^T ′ + ^T +φσ ^T ′

= 2

( ) ( ) ( ) ( )x B x

[

A x y x q ( ( )x)

]

Y^T + φσ

= 2

( ) ( ) ( )y x ( ( )x )c

[

A( ) ( )x y x q ( ( )x )

]

dx x x dB

y^T +φσ ^T + φσ

+

( ) ( ) ( ) ( )x B x A x y x y( ) ( )x B x q ( ( )x)

y^T + φσ

= 2 2

( ) ( ) ( )y x ( ( )x )c A( ) ( )x y x qc

[

( ( )x)

]

²

dx x x dB

y^T +φσ ^T + ^T φσ

+

(8)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

y x dx

x x dB y x y x A x B x y x y x B x A x

y^T ^T + ^T + ^T

=

( ) ( )x B x q ( ( )x) ( ( )x)c A( ) ( )x y x

y^T φσ +φσ ^T

+ 2

( )

[

x

]

²

qc^T φσ +

( ) ( ) ( ) ( ) ( ) ( ) ( )y x B( )x q ( ( )x) dx

x x dB A x B x B x A x

y^T ^T ⎥⎦⎤ + φσ

⎢⎣⎡ + +

= 2

( )

(σ x )c^TA( ) ( )x y x +qc^T

[

φ(σ( )x )

]

²

φ

+ .

( ) ( ) ( ) ( ) ( ) ^C( )^x^; dx

x x dB A x B x B x

A^T + + = (5)

with ^C( )^x to be an arbitrarily chosen. Selecting α >0, we add and subtract ( )σ >0

ασφ (as in Aizerman and Gantmacher [1]):

Hence

( )

( ) [ ( ) ] ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) [ ( ) ] ( ) ( ) ( ) ( ) ( )

( ) ( )

⎢⎢

⎣

⎡

σ φ

− σ ασφ +

σ

⎥⎦ φ

⎢⎣ ⎤

⎡ + + α

−

=

σ φ α

− σ φ α

+

σ φ

− σ

⎥⎦ φ

⎢⎣ ⎤

⎡ +

−

=

−

.

2 1 2

2 1

2 2 1

2

2 /

q c

x y c c x A q x B x y x C x y

x y c x

y c

q c x

y c x A q x B x y x C x y

x y V

T

T T

T

T T

T T T T

(6)

From the requirement that −V^/

(

y

( )

x

)

be positive definite, by using the Lyapunov stability criterion (Lyapunov [4] and Massera [5]).

Since the right hand side of (6) (without ασφ( )σ > 0) is a positive quadratic expression in φ

(

σ

( )

x

)

, b²− ac4 < 0 implies that

( ) ( ) ( ) ( ) ( )

,

2 1 2

1 2

1 ₁

⎥⎦⎤

⎢⎣⎡ + + α

⎥⎦⎤

⎢⎣⎡ + + α

>

−c^Tq B x q A^T x c c^T ^TC⁻ x B x q A^T x c c^T (7)

like before. ~

(9)

7. The Solution for B

( )

x

We solve for B( )x from the first order linear differential equation (5), with ( )x

A and B( )x assumed symmetric. The ODE (5) associated with our modified system (3) may be compared to the Sylvester equation in (2). The only difference is the term ( )

dx . x dB

We use the integrating factor e²^xA^{( )}^x to determine the analytical solution to the following equation, in view of (6):

( )

A

( ) ( )

x B x B

( ) ( )

x A x C

( )

x . dx

x

dB + _T + = − (8)

Multiplying throughout by e²^xA^{( )}^x , we obtain

( )

^x ⁼ ^e⁻² ^{( )}

∫

^e² ^{( )}

(

⁻^C

( )

^x

)

^dx⁺^e⁻² ^{( )}^k^,

B ^xA^x ^xA^x ^xA^x

where k is a constant.

7.1. Positive definiteness and symmetry of ^B( )^x and −C

( )

x We select −C

( )

x = e⁻²^xA^{( )}^x .

Then

( )

^x ⁼^e⁻² ^{( )}

∫

^e² ^{( )}

(

⁻^C

( )

^x

)

^dx⁺^e⁻² ^{( )}^k^, ^k ^≥⁰^.

B ^xA^x ^xA^x ^xA^x

( )

∫

( )

(

⁻ ^{( )}

)

⁻ ^{( )}

− +

= e ²^xA^x e²^xA^x e ²^xA^x dx e ²^xA^xk

( )

∫

⁻ ( )

− +

= e ²^xA^x dx e ²^xA^x k

( ) ² ( ) ,

2 x e k

e⁻ ^xA^x + ⁻ ^xA^x

= k is a constant.

Therefore,

( ) (x x k)e ²^xA^{( )}^x .

B = + ⁻ (9)

We impose the following conditions on e⁻²^xA^{( )}^x:

(10)

We put: 0 x A

( )

x n n 1; x D.

R < ∈

≤ M ×

This, in turn, implies that, 0 ≤ x a_ij( )x <1. That is:

For x ≤0, we have 0 ≤ −x a_ij( )x <1;and for x≥ 0,0≤ x a_ij( )x <1. Therefore,

( )

1,

0≤ n×n ≤

x R

A

x _M (10)

implies that

( )

⎪⎩

( )

⎪⎨

⎧

<

≤

<

−

≤

. 1 0

b) (

or 1 0

) a (

x a x

ij ij

(11)

From the exponential expansion, we have

( )

( ) ( ) ( ) ( )

! . 4 16

! 3 8

! 2 2 4

4 4 3

3 2

2 = − + 2 − + −

− x A x x A x x A x

x xA I e ^xA^x

By (10), we then define the following matrix exponentials for A

( )

x:

( )

⎪⎩

⎪⎨

⎧

+

=

−

− =

. 2

, 2

2 2

x xA I e

x xA

(12)

Finally, we have

( )

( ) ( ) ( )

^⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−

− =

x xa x

xa x

xa

x xa x

xa x

xa

x xa x

xa x

xa

e

nn n

n

n n x

xA

2 1 2

2

2 2

1 2

2 2

2 1

2 22

21

1 12

11

2 (13)

and

( )

( ) ( ) ( )

. 2

1 2

2

2 2

1 2

2 2

2 1

2 22

21

1 12

11 2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+ +

−

=

x xa x

xa x

xa

x xa x

xa x

xa

x xa x

xa x

xa e

nn n

n

n n x

xA (14)

(11)

Main Theorem 7.2. For the inequality (10), and x∈D, the following are positive definite and symmetric:

(a) e⁻²^xA^{( )}^x ; (b) −C

( )

x ; (c) B

( )

x. Proof.

(a) y^T( )x e⁻²^xA^{( )}^x y( )x

( ) ( ) ( )

(

y₁ x, y₂ x,..., y_n x

)

=

( ) ( ) ( )

( ) ( )

( )

. 2

1 2

2

2 2

1 2

2 2

2 1

2 22

21

1 12

11

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⋅

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

⋅

x y

x xa x

xa x

xa

x xa x

xa x

xa

x xa x

xa x

xa

n nn

n n

We have

( ) ( ) ( )

(

y₁ x, y₂ x,..., y_n x

)

( ) ( ) ( )

( ) ( )

( )

^⎟

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⋅

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

⋅

x y

x xa x

xa x

xa

x xa x

xa x

xa

x xa x

xa x

xa

n nn n

n

n n

2 1

2 22

21

1 12

11

2 1 2

2

2 2

1 2

2 2

2 1

,

≥ 0

if and only if 11(a) holds.

The symmetry of e⁻²^xA^{( )}^x follows from the symmetry of A

( )

x .

(b) Having chosen −C

( )

x =e⁻²^xA^{( )}^x , the symmetry and positive definiteness of ( )x

−C follow.

(12)

(c) We have B( ) (x = x+k)e⁻²^xA^{( )}^x . Provided 0≤ x+k <1, B( )x is both

symmetric and positive definite. ~

8. Conclusion

We observe that Main Theorem 7.2 validates our choice of the Lyapunov function (4) and the solution to equation (8) and hence, the criterion (7). What is remarkable is that a criterion based on a dynamic Lefschetz system could still be applied to a non-dynamic system (1), thus affording us the confirmation of stability for a system of linear first order ODEs with variable coefficients.

References

[1] M. A. Aizerman and F. R. Gantmacher, Absolute Stability of Regular Systems, Academy of Sciences USSR, Moscow, 1963.

[2] Garrett Birkhoff and Gian-Carlo Rota, Ordinary Differential Equations, 2nd ed., pp. 151-179, Harvard University, Xerox College Publishing, 1969.

[3] F. A. Coddington and N. Levinsion, Theory of Ordinary Differential Equations, McGraw Hill, New York, 1955.

[4] A. M. Lyapunov, The general problem of the stability of motion, Inter. J. Control 55(3) (1992), 531-534.

[5] J. Massera, On Lyapunov condition of stability, Ann. Math. 50 (1949), 705-721.

[6] K. R. Meyer, A remark on a result of Lefschetz, Contrib. Diff. Equa. 3 (1964), 435-437.