- Email: [email protected]
Comments on “On stability of switched homogeneous nonlinear systems” by Lijun Zhang, Sheng Liu, and Hai Lan [J. Math. Anal. Appl. 334 (1) (2007) 414–430]
J. Math. Anal. Appl. 373 (2011) 343–344
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Note
Comments on “On stability of switched homogeneous nonlinear systems” by Lijun Zhang, Sheng Liu, and Hai Lan [J. Math. Anal. Appl. 334 (1) (2007) 414–430] Alexander O. Ignatyev Institute of Applied Mathematics and Mechanics of National Academy of Sciences of Ukraine, R. Luxemburg Street, 74, Donetsk 83114, Ukraine
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 22 February 2010 Available online 2 June 2010 Submitted by R.O. Popovych
We show that the proof of the results obtained in the above paper is wrong since it is based on lemma which is not true. Therefore the results of the paper cannot be considered as true ones. © 2010 Elsevier Inc. All rights reserved.
Keywords: Stability Lyapunov function Quadratic form
In the above paper [1], a sufficient condition is obtained for stability of switched nonlinear homogeneous systems. The obtained results are essentially based on the following lemma and remark. Consider the system
x˙ = f (x),
x ∈ Rn
(1)
where f (x) is a homogeneous polynomial vector field of odd degree k. Lemma 1. If system (1) is asymptotically stable, then there is a positive matrix P such that its Lyapunov function V (x) = x T P x. Remark 1. According to Lemma 1, the existence of the quadratic Lyapunov function is a sufficient and necessary condition for stable homogeneous systems. To show that this lemma and remark are false, all we need to do is to construct a system (1) such that its zero solution is asymptotically stable but there is no quadratic form which can be a Lyapunov function of system (1). Consider the system
x˙ = −100 y 3 − x3 ,
y˙ = 100x3 .
(2)
Let us show that its trivial solution
x = 0,
y=0
(3)
is asymptotically stable. Consider the function
V1 =
1 4
x4 + y 4 .
DOI of original article: 10.1016/j.jmaa.2006.12.065. E-mail address: [email protected]. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.05.046
©
2010 Elsevier Inc. All rights reserved.
344
A.O. Ignatyev / J. Math. Anal. Appl. 373 (2011) 343–344
Its derivative along solutions of system (2) is equal to
V˙ 1 = x3 −100 y 3 − x3 + 100x3 y 3 = −x6 0. There is the unique solution of system (2) belonging to the set x = 0. This solution is (3). Therefore, according to Barbashin– Krasovskii Theorem [2], the trivial solution of system (2) is asymptotically stable. Let us show that there exists no positive definite quadratic form
V =
1 2
x2 + 2axy + by 2
such that its derivative along solutions of system (2) is negative definite. V˙ has the following form
V˙ = x −100 y 3 − x3 + ay −100 y 3 − x3 + 100ax4 + 100bx3 y
= (100a − 1)x4 + (100b − a)x3 y − 100xy 3 − 100ay 4 . Let us show that there are no a ∈ R and b ∈ R such that V˙ < 0. To do this, let us consider next cases. 1. For x = 0 we have V˙ = −100ay 4 , whence we obtain that a must satisfy the following condition
a > 0.
(4)
2. For y = x we have V˙ = (100a − 1 + 100b − a − 100 − 100a)x4 = (−a + 100b − 101)x4 . The condition V˙ < 0 should be satisfied, then necessary
a > 100b − 101.
(5)
3. For y = 0 we have 100a − 1 < 0, whence
a<
1 100
(6)
.
4. For y = −2x we have V˙ = [100a − 1 − 2(100b − a) + 800 − 1600a]x4 = (−1498a − 200b + 799)x4 , whence 1498a > 799 − 200b, and
a>
799 1498
−
200 1498
b.
(7)
The set of real numbers a and b, satisfying inequalities (4)–(7), is empty (we can use the plane O ba to check this), i.e. there do not exist a ∈ R, b ∈ R such that V˙ < 0. References [1] L. Zhang, S. Liu, H. Lan, On stability of switched homogeneous nonlinear systems, J. Math. Anal. Appl. 334 (2007) 414–430. [2] Y.A. Barbashin, N.N. Krasovskii, The stability of motion as a whole, Dokl. Akad. Nauk SSSR 86 (3) (1952) 453–456.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Note
Comments on “On stability of switched homogeneous nonlinear systems” by Lijun Zhang, Sheng Liu, and Hai Lan [J. Math. Anal. Appl. 334 (1) (2007) 414–430] Alexander O. Ignatyev Institute of Applied Mathematics and Mechanics of National Academy of Sciences of Ukraine, R. Luxemburg Street, 74, Donetsk 83114, Ukraine
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 22 February 2010 Available online 2 June 2010 Submitted by R.O. Popovych
We show that the proof of the results obtained in the above paper is wrong since it is based on lemma which is not true. Therefore the results of the paper cannot be considered as true ones. © 2010 Elsevier Inc. All rights reserved.
Keywords: Stability Lyapunov function Quadratic form
In the above paper [1], a sufficient condition is obtained for stability of switched nonlinear homogeneous systems. The obtained results are essentially based on the following lemma and remark. Consider the system
x˙ = f (x),
x ∈ Rn
(1)
where f (x) is a homogeneous polynomial vector field of odd degree k. Lemma 1. If system (1) is asymptotically stable, then there is a positive matrix P such that its Lyapunov function V (x) = x T P x. Remark 1. According to Lemma 1, the existence of the quadratic Lyapunov function is a sufficient and necessary condition for stable homogeneous systems. To show that this lemma and remark are false, all we need to do is to construct a system (1) such that its zero solution is asymptotically stable but there is no quadratic form which can be a Lyapunov function of system (1). Consider the system
x˙ = −100 y 3 − x3 ,
y˙ = 100x3 .
(2)
Let us show that its trivial solution
x = 0,
y=0
(3)
is asymptotically stable. Consider the function
V1 =
1 4
x4 + y 4 .
DOI of original article: 10.1016/j.jmaa.2006.12.065. E-mail address: [email protected]. 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2010.05.046
©
2010 Elsevier Inc. All rights reserved.
344
A.O. Ignatyev / J. Math. Anal. Appl. 373 (2011) 343–344
Its derivative along solutions of system (2) is equal to
V˙ 1 = x3 −100 y 3 − x3 + 100x3 y 3 = −x6 0. There is the unique solution of system (2) belonging to the set x = 0. This solution is (3). Therefore, according to Barbashin– Krasovskii Theorem [2], the trivial solution of system (2) is asymptotically stable. Let us show that there exists no positive definite quadratic form
V =
1 2
x2 + 2axy + by 2
such that its derivative along solutions of system (2) is negative definite. V˙ has the following form
V˙ = x −100 y 3 − x3 + ay −100 y 3 − x3 + 100ax4 + 100bx3 y
= (100a − 1)x4 + (100b − a)x3 y − 100xy 3 − 100ay 4 . Let us show that there are no a ∈ R and b ∈ R such that V˙ < 0. To do this, let us consider next cases. 1. For x = 0 we have V˙ = −100ay 4 , whence we obtain that a must satisfy the following condition
a > 0.
(4)
2. For y = x we have V˙ = (100a − 1 + 100b − a − 100 − 100a)x4 = (−a + 100b − 101)x4 . The condition V˙ < 0 should be satisfied, then necessary
a > 100b − 101.
(5)
3. For y = 0 we have 100a − 1 < 0, whence
a<
1 100
(6)
.
4. For y = −2x we have V˙ = [100a − 1 − 2(100b − a) + 800 − 1600a]x4 = (−1498a − 200b + 799)x4 , whence 1498a > 799 − 200b, and
a>
799 1498
−
200 1498
b.
(7)
The set of real numbers a and b, satisfying inequalities (4)–(7), is empty (we can use the plane O ba to check this), i.e. there do not exist a ∈ R, b ∈ R such that V˙ < 0. References [1] L. Zhang, S. Liu, H. Lan, On stability of switched homogeneous nonlinear systems, J. Math. Anal. Appl. 334 (2007) 414–430. [2] Y.A. Barbashin, N.N. Krasovskii, The stability of motion as a whole, Dokl. Akad. Nauk SSSR 86 (3) (1952) 453–456.