Journal of Applied Mathematics and Stochastic Analysis, 15:2 (2002), 141-150.
STABILITY ANALYSIS OF NONLINEAR LYAPUNOV
SYSTEMS ASSOCIATED WITH AN nTH ORDER
SYSTEM OF MATRIX DIFFERENTIAL EQUATIONS
K.N. MURTY1 and MICHAEL D. SHAW
Florida Institute of Technology
Department of Mathematical Sciences
Melbourne, FL 32901 USA
(Received March, 2000; Revised October, 2000)
This paper introduces the notion of Lipschitz stability for nonlinear 8th order
matrix Lyapunov differential systems and gives sufficient conditions for Lipschitz
stability. We develop variation of parameters formula for the solution of the
nonhomogeneous nonlinear 8th order matrix Lyapunov differential system. We
study observability and controllability of a special system of 8th order nonlinear
Lyapunov systems.
Key words:
Controllability, Lyapunov Systems, Matrix Differential
Equations, Observability, Stability Analysis, Variation of Parameters.
AMS subject classifications: 34D10, 34D99.
1. Introduction
Stability analysis of nonlinear systems is an important area of current research and many
concepts of stability analysis have been recently developed for first order nonlinear Lyapunov
systems. In this paper, we develop a variation of parameters formula for 8th order nonlinear
Lyapunov systems and use it as a tool to study the various concepts of stability. Our paper is
organized as follows: In Section 2, we develop the nonlinear variation of parameters formula
for nonhomogeneous systems. In Section 3, we study various concepts of stability for
homogeneous systems. In Section 4, we introduce the notion of Lipschitz stability and
investigate sufficient conditions for the 8th order nonlinear system to be Lipschitz stable and
uniformly Lipschitz stable. In Section 5, we study the controllability and observability of the
8th order nonlinear Lyapunov system.
2. General Solution of Homogeneous Systems, Nonlinear Variation of
Parameters
In this section, we develop the general solution of the nonlinear matrix Lyapunov system
X Ð8Ñ œ E8 X Ð>Ñ  8G" E8" X Ð>ÑF  8G# E8# X Ð>ÑF #  á
1Permanent
address: Andhra University, Department of Applied Mathematics, Viskhapatnam, India.
Printed in the U.S.A. ©2002 by North Atlantic Science Publishing Company
141
142
K.N. MURTY and MICHAEL D. SHAW
 8G< E8< X Ð>ÑF <  á  X Ð>ÑF 8
Ð#Þ"Ñ
in terms of two fundamental matrix solutions of X w œ EX Ð>Ñ and X w œ F ‡ X Ð>Ñ where F ‡ is
the transpose of the complex conjugate matrix of F . In (2.1), E and F are constant square
matrices of order 8 ‚ 8 and X − GÒ‘ ß ‘8 ‚ ‘8 Ó is a variable matrix. Throughout this
paper, ] Ð>Ñ and ^Ð>Ñ stand for the fundamental matrix solutions of X w œ EX and X w œ F ‡ X .
Theorem 2.1: If ] is a fundamental matrix solution of X w œ EX , then ] Ð8Ñ is a
fundamental matrix solution of X w œ E8 X where 8 is a positive integer.
Proof: ] is a fundamental matrix solution of X w œ EX if and only if ] w Ð>Ñ œ E] Ð>Ñ.
This implies that ] ww Ð>Ñ œ E] w Ð>Ñ œ EE] œ E# ] . This further implies that ] ww Ð>Ñ œ E$ ]
and so on. Hence, ] Ð8Ñ Ð>Ñ œ E8 ] Ð>Ñ. Thus, ] is also a fundamental matrix solution of
X w œ E8 X .
Theorem 2.2: Let G be a constant square matrix of order 8 ‚ 8. Then any solution of
X Ð8Ñ œ
8
<œ!
ˆ 8< ‰E8< X Ð>ÑF <
Ð#Þ#Ñ
with E! œ F ! œ M , is of the form X Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ, where ] Ð>Ñ is a fundamental matrix
solution of X w œ EX and ^Ð>Ñ is a fundamental matrix solution of X w œ F ‡ X .
Proof: It can be easily verified that X defined by X Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ is a solution of
(2.2). For
Ð] Ð>ÑG^ ‡ Ð>ÑÑÐ8Ñ œ E8 ] Ð>ÑG^ ‡ Ð>Ñ  8G" E8" ] Ð>ÑG^ ‡ Ð>ÑF
 8G# E8# ] Ð>ÑG^ ‡ Ð>ÑF #  á  8G< E8< ] Ð>ÑG^ ‡ Ð>ÑF <
 á  ] Ð>ÑG^ ‡ Ð>ÑF 8 Þ
By the Leibnitz theorem, we have
w
Ð] Ð>ÑG^ ‡ Ð>ÑÑÐ8Ñ œ ] Ð8Ñ Ð>ÑG^ ‡ Ð>Ñ  8G" ] Ð8"Ñ Ð>ÑG^ ‡ Ð>Ñ
ww
Ð<Ñ
Ð8Ñ
 8G# ] Ð8#Ñ Ð>ÑG^ ‡ Ð>Ñ  á  8G< ] Ð8<Ñ Ð>ÑG^ ‡ Ð>Ñ  á  ] Ð>ÑG^ ‡ Ð>ÑÞ
Ð5Ñ
The two equations are the same since ] Ð5Ñ Ð>Ñ œ E5 ] Ð>Ñ and ^ ‡ Ð>Ñ œ ^ ‡ Ð>ÑF 5 for
5 œ "ß #ß á ß 8. Thus, X Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ is a solution of (2.1). To prove that every
solution of the equation is of this form, let X Ð>Ñ be a solution of (2.2) and let O be a square
matrix of order 8 ‚ 8 defined by OÐ>Ñ œ ] " Ð>ÑX Ð>Ñ. Then X Ð>Ñ œ ] Ð>ÑOÐ>Ñ. Now,
X Ð>Ñ œ ] Ð>ÑOÐ>Ñ if and only if
] Ð8Ñ Ð>ÑOÐ>Ñ  8G" ] Ð>Ñ8" O w Ð>Ñ  8G# ] Ð>Ñ8# O ww Ð>Ñ  á
 8G< ] Ð>Ñ8< O Ð<Ñ Ð>Ñ  á  ] Ð>ÑO Ð8Ñ Ð>Ñ
œ E8 ] Ð>ÑOÐ>Ñ  8G" E8" ] Ð>ÑO w Ð>Ñ  8G# E8# ] ЇÑÐ>ÑO ww Ð>Ñ  á
 8G< E8< ] Ð>ÑO Ð<Ñ Ð>Ñ  á  ] Ð>ÑO Ð8Ñ Ð>ÑÞ
Since ] Ð>Ñ, ] w Ð>Ñ, á , and ] Ð8Ñ Ð>Ñ are linearly independent, equation (2.3) can hold if
Ð5Ñ
and only if O Ð5Ñ œ OF 5 if and only if O ‡ œ F ‡5 O ‡ for 5 œ "ß #ß á ß 8. Since ^Ð>Ñ
w
is a fundamental matrix solution of X œ F ‡ X , it follows that ^ also a fundamental
Ð#Þ$Ñ
Stability Analysis of Nonlinear Lyapunov Systems
143
8
matrix solution of X Ð8Ñ œ F ‡ X . Therefore, there exists a constant matrix G such that
O ‡ œ ^G ‡ , then X œ ] O œ ] G^ ‡ and the proof is complete.
We shall now develop the variation of parameters formula for the solution of the
nonhomogeneous system
X Ð8Ñ œ
8
<œ!
ˆ 8< ‰E8< X Ð>ÑF <  J Ð>ß X Ð>ÑÑÞ
Ð#Þ%Ñ
Theorem 2.3: Any solution X Ð>Ñ of Ð#Þ%Ñ is of the form X Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ  X: Ð>Ñ,
where X: Ð>Ñ is a particular solution of Ð#Þ%Ñ.
Proof: It can easily be verified that X Ð>Ñ defined by X Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ  X: Ð>Ñ is a
solution of (2.4). To prove that every solution is of this form, let X be any solution of (2.4)
and X: Ð>Ñ be any particular solution of (2.4). Then X Ð>Ñ  X: Ð>Ñ is a solution of the linear
homogeneous system (2.2) and has the form ] Ð>ÑG^ ‡ Ð>Ñ.
Hence,
X Ð>Ñ  X: Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ and X Ð>Ñ œ ] Ð>ÑG^ ‡Ð>Ñ  X: Ð>Ñ. The proof is complete.
Theorem 2.4: Let ] Ð>Ñ be a fundamental matrix solution of X w œ EX and ^Ð>Ñ be a
fundamental matrix solution of X w œ F ‡ X .
Furthermore, suppose that
G − G 8 Ò‘ ß ‘8 ‚ ‘8 Ó such that
w
Ð] w G w ^ ‡  ] G w ^ ‡ ÑÐ3#Ñ œ ! for 3 œ "ß #ß á ß 8Þ
Ð#Þ&Ñ
Then a particular solution X: Ð>Ñ of Ð#Þ%Ñ is given by
X: Ð>Ñ œ
' ' ' á'
7"
>
] Ð>Ñ
+ +
7#
+
78"
"
] " Ð=ÑJ Ð=ß X Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7# . 7" ^ ‡ Ð>Ñ.
+
Proof: Any solution of the homogeneous system (2.1) is of the form X Ð>Ñ œ
] Ð>ÑG^ ‡ Ð>Ñ, where G is an 8 ‚ 8 constant matrix. Such a solution cannot be a solution of
the nonhomogeneous system (2.4) unless J œ !. Assume G −
G 8 Ò‘ ß ‘8 ‚ ‘8 Ó and seek a particular solution X: Ð>Ñ of the form X: Ð>Ñ œ
] Ð>ÑGÐ>Ñ^ ‡ Ð>Ñ. Substituting X: Ð>Ñ into (2.3) and using (2.5) yields ] Ð>ÑG Ð8ÑÐ>Ñ
"
^ ‡ Ð>Ñ œ J Ð>ß X Ð>ÑÑ which in turn yields G Ð8Ñ Ð>Ñ œ ] " Ð>ÑJ Ð>ß X Ð>ÑÑ^ ‡ Ð>Ñ .
After
integrating 8 times, we have
GÐ>Ñ œ ' ' ' á '
>
+
7" 7#
+
+
78"
"
] " Ð=ÑJ Ð=ß X Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7# . 7" Þ
+
Hence,
X: Ð>Ñ œ
] Ð>Ñ ' ' ' á '
>
+
+
7" 7#
+
78"
"
] " Ð=ÑJ Ð=ß X Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7# . 7" ^ ‡ Ð>Ñ
+
Ð#Þ'Ñ
and X: Ð>Ñ is easily verified as a solution of (2.4).
Theorem 2.5: Any solution X Ð>Ñ of the nonhomogeneous Lyapunov system Ð#Þ%Ñ is of the
form X Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ  X: Ð>Ñ where X: Ð>Ñ is given by Ð#Þ'Ñ.
144
K.N. MURTY and MICHAEL D. SHAW
Proof: For 8 œ ", the homogeneous system (2.2) reduces to X w œ EX  X F . The
general solution in terms of the variation of parameters formula for X w œ EÐ>ÑX Ð>Ñ 
X Ð>ÑFÐ>Ñ  J Ð>ß X Ð>ÑÑ has been established in [4].
And for 8  ", we have if X Ð>Ñ is any solution of (2.4) and X: Ð>Ñ is a particular solution of
(2.4), then X Ð>Ñ  X: Ð>Ñ is a solution of the homogeneous system (2.1). Hence,
X Ð>Ñ  X: Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ and X Ð>Ñ œ ] Ð>ÑG^ ‡Ð>Ñ  X: Ð>Ñ.
3. Stability Analysis of the Homogeneous System
In this section, we shall be concerned with the various concepts of stability of the
homogeneous system
X Ð8Ñ œ
8
<œ!
ˆ 8< ‰E8< X Ð>ÑF < Þ
Ð$Þ"Ñ
The proofs of the next theorem in each case are simple and hence omitted. For basic
results on stability, see [1].
Theorem 3.1: Let ] Ð>Ñ be a fundamental matrix solution of X w œ EX and ^Ð>Ñ be a
fundamental matrix solution of X w œ F ‡ X . Then the matrix system Ð$Þ"Ñ
Ð+Ñ
is stable if and only if there exists a positive constant O such that
² ] Ð>Ñ ² ² ^Ð>Ñ ² Ÿ O for all > >! !ß
Ð,Ñ
is asymptotically stable if and only if it is stable and
² ] Ð>Ñ ² ² ^Ð>Ñ ² Ä ! as > Ä ∞ß
Ð-Ñ
is uniformly stable if and only if there exists a positive constant O such that
² ] Ð>Ñ] " Ð=Ñ ² ² ^Ð>Ñ^ " Ð=Ñ ² Ÿ O for all > = >! !ß
Ð.Ñ
is strongly stable if and only if there exists a positive constant O such that
² ] Ð>Ñ] " Ð>Ñ ² ² ^Ð>Ñ^ " Ð>Ñ ² Ÿ O for all > >! !, and
Ð/Ñ
is uniformly asymptotically stable if and only if there exist positive constants
7" ß 7# ß α, and " such that
² ] Ð>Ñ] " Ð=Ñ ² Ÿ 7" /αÐ>=Ñ
and
² ^Ð>Ñ^ " Ð=Ñ ² Ÿ 7# /" Ð>=Ñ for >! Ÿ = Ÿ >  ∞Þ
Theorem 3.2:
Suppose there exist positive constants P" and P# such that
² ] Ð>Ñ] " Ð=Ñ ² Ÿ P" and ² ^Ð>Ñ^ " Ð=Ñ ² Ÿ P# for all ∞  > = >! and J satisfies
the condition ² J Ð>ß X Ð>ÑÑ ² Ÿ # Ð>Ñ ² X Ð>Ñ ² where # is a positive continuous
∞ ∞
∞
function such that '>! '>! á '>! # Ð=Ñ.=  ∞. Then there exists a positive constant P
such that for any >" >! and ² X Ð>" Ñ ² Ÿ -ÎP, we have ² X Ð>Ñ ² Ÿ P ² X Ð>" Ñ ² for all
> >" > ! .
Stability Analysis of Nonlinear Lyapunov Systems
145
Proof: Any solution X Ð>Ñ of the nonhomogeneous matrix Lyapunov system (2.4)
satisfying X Ð>" Ñ œ X" has the form
"
X Ð>Ñ œ ] Ð>Ñ] " Ð>" ÑX Ð>" Ñ^ ‡ Ð>" Ñ^ ‡ Ð>Ñ
 ] Ð>ÑÐ' ' á '
> 7"
>" >"
78"
"
] " Ð=ÑJ Ð=ß X Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7" Ñ^ ‡ Ð>ÑÞ
>"
Hence,
"
² X Ð>Ñ ² Ÿ ² ] Ð>Ñ" Ð>" Ñ ² ² X Ð>" Ñ ² ² ^ ‡ Ð>" Ñ^ ‡ Ð>Ñ ²
 ' ' á'
> 7"
>" >"
78"
"
² ] Ð>Ñ] " Ð=Ñ ² ² J Ð=ß X Ð=ÑÑ ² ² ^ ‡ Ð=Ñ^ ‡ Ð>Ñ ² .=. 78" á . 7"
>"
Ÿ P" P# ² X Ð>" Ñ ²  ' ' á '
> 7"
>" >"
78"
P" P# # Ð=Ñ ² X Ð=Ñ ² .=. 78" á . 7"
>"
Ÿ P" P# ² X Ð>" Ñ ²  P" P# ' ' á '
> 7"
>" >"
78"
P" P# # Ð=Ñ ² X Ð=Ñ ² .=. 78" á . 7" Þ
>"
By applying the Reid-Bellman inequality [2], we obtain
² X Ð>Ñ ² Ÿ P" P# ² X Ð>" Ñ ² expÐ' ' á '
> 7"
>" >"
78"
# Ð=Ñ.=. 78" á . 7" Ñ
>"
Ÿ P ² X Ð>" Ñ ² for >"  >  ∞Þ
Theorem 3.3: Suppose there exists an 7 œ maxÖ7" ß 7# ×  ! such that
² ] Ð>Ñ] " Ð=Ñ ² Ÿ 7" /αÐ>=Ñ and ² ^Ð>Ñ^ " Ð=Ñ ² Ÿ 7# /" Ð>=Ñ for >! Ÿ = Ÿ >  ∞
where α and " are constants such that α  "  ! and J satisfies a Lipschitz condition with
the Lipschitz constant P  GÎ7. Then there exists a positive constant O such that
² X Ð>" Ñ ² Ÿ Ðα  " ÑÎO implies
² X Ð>Ñ ² Ÿ 7" 7# expÐ  Ðα  " Ñ>" Ñ ² X Ð>" Ñ ²
‚ expÐ7" 7# ' ' á '
> 7"
>" >"
78"
# Ð=Ñ.=. 78" á . 7" ÑÞ
>"
Proof: The proof utilizes the variation of parameters formula given in Theorem 2.4.
4. Lipschitz Stability of the Nonhomogeneous Lyapunov System
In this section, we give sufficient conditions for Lipschitz stability on nonlinear 8th order
matrix Lyapunov differential systems and study Lipschitz stability properties of the zero
solution of perturbed matrix Lyapunov system with the variation of parameters formula
developed in Section 2. Before we proceed to give our new results, we shall give the
following definitions [2, 3].
146
K.N. MURTY and MICHAEL D. SHAW
Definition 4.1: The zero solution of the 8th order Lyapunov system (2.4) is said to be
uniformly Lipschitz stable if there exists a constant 7  ! and a $  ! such that
² X Ð>ß >! ß X! Ñ ² Ÿ 7 ² X! ² whenever ² X! ² Ÿ $ and > >! !Þ
Definition 4.2: The zero solution of the 8th order Lyapunov system (2.4) is said to be
asymptotically stable if there exist positive constants 7 and α such that
² X Ð>ß >! ß X! Ñ ² Ÿ 7 ² X! ² /αÐ>>! Ñ whenever ² X! ² Ÿ $ and > >! !Þ
Let 0 Ð>ß X Ð>ÑÑ œ
8
8
8<
X Ð>ÑF <
<œ! Ð < ÑE
with F ! œ E! œ " and the linear system be
X w œ 0 Ð>ß X Ð>ÑÑ
Ð%Þ#Ñ
[ w œ 0 Ð>ß [ Ð>ÑÑ  J Ð>ß [ Ð>ÑÑ
Ð%Þ#Ñ
and its perturbed system be
satisfying the initial data X Ð>! Ñ œ X! , where >! − ‘ and 0 ß J − GÒ‘ ‚ ‘8 ‚ ‘8 ß
‘8 ‚ ‘8 Ó. Assume that ] Ð>Ñ is a fundamental matrix solution of X w œ EX and ^Ð>Ñ be a
fundamental matrix solution of X w œ F ‡ X . Obviously, system (4.1) admits the zero solution.
We now have the following theorems.
Theorem 4.1: Suppose that
Ð3Ñ
² ] " Ð>ß >! ß X! ÑJ Ð>ß X Ð>ß >! ß X! Ñ ² Ÿ 1Ð>ß ² X! ² Ñ or
² J Ð>ß X Ð>ß >! ß X! ÑÑ^ " Ð>ß >! ß X! Ñ ² Ÿ 1Ð>ß ² X! ² Ñ where
1 − GÒ‘ ‚ ‘ ß ‘ Ó, 1Ð>ß !Ñ œ !, and the trivial solution of
?w œ 1Ð>ß ?Ñß ?Ð>! Ñ œ ?! !
Ð%Þ$Ñ
is uniformly stable, and
Ð33Ñ
the trivial solution of Ð%Þ"Ñ is uniformly stable.
Then the trivial solution of Ð%Þ#Ñ is uniformly stable.
Proof: Any solution [ Ð>ß >! ß X! Ñ of (4.2) is of the form
"
[ Ð>Ñ œ ] Ð>Ñ] " Ð>! ÑX! ^ ‡ Ð>! Ñ^ ‡ Ð>Ñ
 ] Ð>ÑÐ' ' á '
> 7"
>! >!
78"
"
] " Ð=ÑJ Ð=ß [ Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7" Ñ^ ‡ Ð>ÑÞ
>!
If we set 7Ð>Ñ œ ² ] Ð>Ñ^Ð>Ñ ² and ² X! ² œ ?! with H 7Ð>Ñ Ÿ 1Ð>ß 7Ð>ÑÑ, then by the
comparison theorem [2] we have the estimate
7Ð>Ñ œ ² ] Ð>Ñ^Ð>Ñ ² Ÿ <Ð>ß >! ß ?! Ñ œ <Ð>ß >! ß ² X! ² Ñ
Ð%Þ%Ñ
the maximal solution of (4.3) with O œ ". Since [ Ð>! ß >! ß X! Ñ œ X Ð>! ß >! ß X! Ñ, it
follows that ² [ Ð>ß >! ß X! Ñ ² œ ² X Ð>ß >! ß ] Ð>Ñß ^Ð>Ñ ² œ ² X Ð>ß >!ß FÐ>ÑÑ ² where F is a
"
solution of FÐ8Ñ œ ] " Ð>ÑJ Ð>ß FÐ>ÑÑ^ ‡ Ð>Ñ.
By assumption Ð33Ñ, given any %  ! and >! in ‘ , there exists a $ Ð%Ñ  ! such that
² X Ð>ß >! ß X! Ñ ²  % whenever > >! and ² X! ²  $ Ð%Ñ.
Ð%Þ&Ñ
Stability Analysis of Nonlinear Lyapunov Systems
147
Since the null solution of (4.3) is uniformly stable, we conclude that if $ Ð%Ñ  ! for
>! − ‘ , there exists a (Ð%Ñ  ! such that <Ð>ß >! ß ?! Ñ  (Ð%Ñ whenever > >! if >! is
sufficiently small. From (4.4) and (4.5), it follows that ² [ Ð>ß >! ß X! Ñ ²  % whenever
² X! ²  (Ð%Ñ for > >! . The proof is complete.
Theorem 4.2: Suppose that
Ð3Ñ
² ] Ð>ß >! ß X! Ñ ² Ÿ O" and ² ] " Ð>ß >! ß X! Ñ ² Ÿ O" for > >! and
² ^Ð>ß >! ß X! Ñ ² Ÿ O# and ² ^ " Ð>ß >! ß X! Ñ ² Ÿ O# for > >! and ² X! ²  # .
Ð33Ñ
² J Ð>ß [ Ð>ÑÑ ² Ÿ 1Ð>ß ² [ ² Ñ where 1 − GÒ‘ ‚ ‘ ß ‘ Ó, 1Ð>ß !Ñ œ !, 1Ð>ß ?Ñ
is nonincreasing and the trivial solution of
?w œ O # 1Ð>ß ?Ñß ?Ð>! Ñ œ ?! !
Ð%Þ'Ñ
where O œ O" O# is stable. Then the trivial solution of Ð%Þ#Ñ is stable.
Proof:
Any solution of the perturbed system X w œ 8<œ! Ð 8< ÑE8< X Ð>ÑF < with
F ! œ E! œ M is of the form X Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ. This solution satisfies the initial condition
X Ð>! Ñ œ X! if and only if X Ð>Ñ œ ] Ð>Ñ] " Ð>! ÑX! ^ " Ð>! Ñ^ ‡ Ð>Ñ. Assumption Ð3Ñ yields
² X Ð>Ñ ² Ÿ O"# O## ² X! ² for ² X! ²  # . Suppose there exists a >" >! such that
for
and
² FÐ>Ñ ² œ ² ] Ð>Ñ] " Ð>! Ñ ²  #
>! Ÿ >  > "
² FÐ>" Ñ ² œ
² ] Ð>" Ñ] " Ð>! Ñ ² œ #. Then any solution of (4.2) is of the form
"
[ Ð>Ñ œ ] Ð>Ñ] " Ð>! ÑX! ^ ‡ Ð>! Ñ^ ‡ Ð>Ñ
 ] Ð>ÑÐ' ' á '
> 7"
>! >!
78"
"
] " Ð=ÑJ Ð=ß [ Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7" Ñ^ ‡ Ð>ÑÞ
>!
Using the bounds on ] Ð>Ñ, ^Ð>Ñ, ] " Ð>Ñ, and ^ " Ð>Ñ along with assumption Ð33Ñ, we get
² [ Ð>Ñ ² Ÿ O"# O## ² X! ²  O"# O## ' ' á '
> 7"
>! >!
78"
1Ð=ß ² [ Ð=Ñ ² Ñ.=. 78" á . 7" Þ
>!
Hence,
² [ Ð>Ñ ² Ÿ O # ² X! ²  O # ' ' á '
> 7"
>! >!
78"
1Ð=ß ² [ Ð=Ñ ² Ñ.=. 78" á . 7" Þ
>!
This fact leads to
² [ Ð>Ñ ² Ÿ <Ð>ß >! ß ?! Ñ œ <Ð>ß >! ß ² X! ² Ñ,
for >! Ÿ >  >" where <Ð>ß >! ß ?! Ñ is the maximal solution of ?w œ O # 1Ð>ß ?Ñ, ?Ð>! Ñ œ ?! such
that ² X! ² Ÿ ?! . By using (33Ñ, given (  !, there exists a $"  ! such that <Ð>ß >! ß ?! Ñ  (
for > >! whenever ?!  $" . Thus, if ² X! ² Ÿ ?!  $" , then ² [ Ð>Ñ ²  ( for all > >! .
This contradicts the fact that ² FÐ>" Ñ ² œ ² ] Ð>" Ñ] " Ð>! Ñ ² œ (. Thus, ² FÐ>" Ñ ²  (
for all > >! . Now,
² [ Ð>Ñ ² Ÿ O"# O## ² X! ²  O"# O## ' ' á '
> 7"
>! >!
78"
1Ð=ß ² [ Ð=Ñ ² Ñ.=. 78" á . 7" Þ
>!
Thus, ² [ Ð>ß >! ß X! Ñ ² Ÿ <Ð>ß >! ß ?! Ñ where <Ð>Ñ is the maximal solution of (4.6). Given any
%  !, there exists a $ œ $ Ð%ß >! Ñ  ! such that ² X! ²  $ implies
² [ Ð>Ñ ² Ÿ <Ð>ß >! ß ?! Ñ œ <Ð>ß >! ß ² X! ² Ñ  %
148
K.N. MURTY and MICHAEL D. SHAW
for all > >! . Thus, the null solution of (4.2) is stable.
The following theorem gives a set of sufficient conditions for the uniform Lipschitz
stability of the perturbed matrix differential system (4.2).
Theorem 4.3: Suppose that
Ð3Ñ
the zero solution of the unperturbed system Ð%Þ"Ñ is uniformly Lipschitz
stable and that hypothesis Ð3Ñ of Theorem %Þ# holds.
Ð33Ñ
² J Ð>ß [ Ð>Ñ ² Ÿ # Ð>Ñ ² [ ² where # − GÒ‘ ß ‘Ó and
' ' á'
> 7"
>! >!
78"
# Ð=Ñ.=. 78" á . 7"  ∞ for all > >! Þ
>!
Then the zero solution of Ð%Þ#Ñ is uniformly Lipschitz stable.
Proof: Let X Ð>ß >! ß X! Ñ be the solution of (4.1) and [ Ð>ß >! ß X! Ñ be the solution of the
perturbed system (4.2). Then clearly
"
² X Ð>ß >! ß X! Ñ ² Ÿ ² [ Ð>ß >! ß X! Ñ ² œ ² ] Ð>Ñ] " Ð>! ÑX! ^ ‡ Ð>!Ñ^ ‡Ð>Ñ
 ] Ð>ÑÐ' ' á '
> 7"
>! >!
78"
"
] " Ð=ÑJ Ð=ß [ Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7" Ñ^ ‡ Ð>Ñ ²
>!
Ÿ O"# O## ² X! ²  O"# O## ' ' á '
> 7"
>! >!
78"
# Ð=Ñ ² [ Ð=Ñ ² .=. 78" á . 7" Þ
>!
By the Gronwall-Reid-Bellman inequality
² [ Ð>ß >! ß X! Ñ ² Ÿ O"# O## ² X! ² expÐO"# O## Ñ' ' á '
> 7"
>! >!
Ÿ O"# O## ² X! ² expÐO"# O## Ñ' ' á '
∞ 7"
>!
>!
78"
# Ð=Ñ.=. 78" á . 7"
>!
78"
# Ð=Ñ.=. 78" á . 7" Ÿ P ² X! ² ,
>!
∞ 7
7
where P œ O"# O## expÐO"# O## Ñ'>! '>!" á '>!8" # Ð=Ñ.=. 78" á . 7" .
complete.
Thus, the proof is
5. Controllability and Observability
Consider the Lyapunov system
X Ð8Ñ œ E8 X Ð>Ñ  8G" E8" X Ð>ÑF  8G# E8# X Ð>ÑF #  á  8G< E8< X Ð>ÑF <  á
 X Ð>ÑF 8  GÐ>ÑY Ð>ÑH‡ Ð>Ñ
Ð&Þ"Ñ
[ Ð>Ñ œ KÐ>ÑX Ð>ÑL ‡ Ð>Ñ where Eß Fß X are all square matrices of order 8 ‚ 8. G and H are
8 ‚ 7 matrices and Y is an 7 ‚ 7 matrix. We call Y the control and [ the observation.
Definition 5.1: The system (5.1) is said to be completely controllable if for any given
initial condition X Ð>! Ñ œ X! and for any >" >! , there is some Y Ð>Ñ defined on >! Ÿ > Ÿ >"
such that X Ð>Ñ œ !.
Definition 5.2: The system (5.1) is said to be observable if for any given >" >! we can
determine X Ð>Ñ for >! Ÿ > Ÿ >" from [ Ð>Ñ.
Stability Analysis of Nonlinear Lyapunov Systems
149
From the variation of parameters formula, the solution of system (2.4) has the form
X Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ
 ] Ð>ÑÐ' ' ' á '
> 7" 7#
+ +
+
78"
"
] " Ð=ÑJ Ð=ß X Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7# . 7" Ñ^ ‡ Ð>Ñ.
+
Then,
"
] " Ð>ÑX Ð>Ñ^ ‡ Ð>Ñ
œ G  Ð' ' ' á '
> 7" 7#
+ +
+
78"
"
] " Ð=ÑJ Ð=ß X Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7# . 7" Ñ
+
and
"
G œ ] " Ð>ÑX Ð>Ñ^ ‡ Ð>Ñ
 Ð' ' ' á '
78"
> 7" 7#
+ +
+
"
] " Ð=ÑJ Ð=ß X Ð=ÑÑ^ ‡ Ð=Ñ.=. 78" á . 7# . 7" Ñ.
+
From this, we can determine G and apply the variation of parameters formula to determine
X Ð>Ñ.
Theorem 5.1: Suppose for some >" >! the matrices K‡ ] ‡ ] and L^^ ‡ L ‡ are positive
definite. Then the system Ð&Þ"Ñ is observable.
Proof: Let [ Ð>Ñ œ KÐ>ÑX Ð>ÑLÐ>Ñ. Then
] Ð>" Ñ[ Ð>" Ñ^ ‡ Ð>" Ñ œ ] Ð>" ÑKÐ>" ÑX Ð>" ÑLÐ>" Ñ^ ‡ Ð>"Ñ
and hence
K‡ Ð>" Ñ] ‡ Ð>" Ñ] Ð>" Ñ[ Ð>" Ñ^ ‡ Ð>" Ñ^Ð>" ÑLÐ>" Ñ
œ K‡ Ð>" Ñ] ‡ Ð>" Ñ] Ð>" ÑKÐ>" ÑX Ð>" ÑLÐ>" Ñ^ ‡ Ð>" Ñ^Ð>"ÑLÐ>"ÑÞ
Since the matrices K‡ ] ‡ ] and L^^ ‡ L ‡ are positive definite, whenever K and L
are nonsingular, we can solve for X Ð>" Ñ and hence for X Ð>Ñ for > >! .
"
"
Theorem 5.2: If ] " Ð>ÑGÐ>ÑG ‡ Ð>Ñ] ‡ Ð>Ñ and ] " Ð>ÑHÐ>Ñ H‡ Ð>Ñ^ ‡ Ð>Ñ are
all positive definite for all >, then the matrix system is completely controllable.
Proof: Any solution X Ð>Ñ of (5.1) is of the form
X Ð>Ñ œ ] Ð>ÑG^ ‡ Ð>Ñ
 ] Ð>ÑÐ' ' ' á '
> 7" 7#
+ +
+
78"
"
] " Ð=ÑGÐ=ÑY Ð=ÑH‡ Ð=Ñ^ ‡ Ð=Ñ.=. 78" á . 7# . 7" Ñ^ ‡ Ð>Ñ.If we
+
choose
G œ  ' ' ' á'
> 7"
+ +
+
7#
78"
+
then X Ð>Ñ œ !. Choose Y as
"
] " Ð=ÑGÐ=ÑY Ð=ÑH‡ Ð=Ñ^ ‡ Ð=Ñ.=. 78" á . 7# . 7" Ñ^ ‡ Ð>Ñß
150
K.N. MURTY and MICHAEL D. SHAW
Y Ð>Ñ œ 
"
8x
‡
‡"
Ð>ÑÖ] " Ð>ÑGÐ>ÑG ‡ Ð>Ñ] ‡ Ð>Ñ×" G
Ð>" >! Ñ8 G Ð>Ñ]
"
‚ Ö^ " Ð>ÑHÐ>ÑH‡ Ð>Ñ^ ‡ Ð>Ñ×" ^ " Ð>ÑHÐ>ÑÞ
If we substitute this expression into (5.1), we find that the equality holds. Thus, the system
(5.1) is completely controllable.
References
[1]
[2]
[3]
[4]
Bellman, R., Stability Theory of Differential Equations, McGraw-Hill, New York 1953.
Fausett, D.F. and Köksal, S., Variation of parameters formula and Lipschitz stability of
nonlinear matrix differential equations, Proc. of the First World Congress of Nonlinear
Analysts, Walter deGruyter, Berlin 1996.
Lakshmikantham, V. and Deo, S.G., Method of Variation of Parameters for Dynamic
Systems, Gordon and Breach Science Publishers, UK 1998.
Murty, K.N., Howell, G. and Sivasundaram, S., Two (multi) point nonlinear Lyapunov
systems existence and uniqueness, JMAA 167 (1992), 505-512.