V i etnam Jour nal of M athem ati cs 34:1 (2006) 1–15
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T h e C e nt ral E x p o n e nt an d A sy m p t o t ic
S t a b ilit y o f Lin e ar D iffe re nt ia l A lg e b ra ic
E q u at io n s of In d e x 1
H o an g N a m
Hong Duc University, Le Lai S tr., T hanh Hoa Province, V ietn am
Received Oct ob er 29, 2003
Revised J une 20, 2005
A b st ra ct . In t his pap er, we int rodu ce a concept of t he cent ral exp onent of linear
different ial algebraic equat ions (DAEs) similar t o t he one of linear ordin ary d ifferent ial
equ at ion s (ODE s), and use it for invest igat ion of asympt ot ic st ability of t he DAE s.
1 . In t ro d u c t io n
Different ial algebraic equat ions (DAEs) have b een developed a s a highly t opical
sub ject of applied mat hemat ics. T he research on t his t opic has been carried out
by many mat hem at icians in t he world (see [1, 5, 7] a nd t he references t herein)
foralinearDAE
A(t)x′+B(t)x= 0,
where A(t) is singular for all t∈R+. Under cert a in condit ions we are able t o
transform it int o a system consisting of a system of ordinary different ial equat ions
(ODEs) and a syst em of algebraic equat ions so that we can use met hods and
result s of t he t heory of ODEs. Many result s on st abilit y propert ies of DAEs were
obt ained: asympt ot ical and exp onent ial st ability of DAE s which a re of index 1
and 2 [6], Floquet t heory of p eriodic DAEs, cr it eria for t he t rivia l solut ion of
DAEs wit h sma ll nonlinear it ies t o b e asympt ot ically st able. Similar result s for
aut onomous qua silinear syst ems a re given in [7].
In t his pap er we are int erst ed in st abilit y and asympt ot ica l proper t ies of t he
DAE
A(t)x′+B(t)x+f(t , x )= 0,
2H oang N am
which can be considered as a linear DAE A x ′+B x = 0 perturbed by t he term
f.
For t his aim we int roduce a concept of cent ral exponent of linear DAEs
similar t o that of ODEs (see [2]).
T he paper is organized as follows. In Sec. 2 we introduce t he notion of
the central exponent and some propert ies of central exponent s of linear DAEs
of index 1. In Sec. 3 we invest igat e exponent ial asympt ot ic st ability of linear
DAE s wit h respect t o small linear as well as nonlinear pert urbat ion.
2 . T h e C e n t ra l E x p o n e nt o f Lin e ar D A E o f In d e x 1 an d It s P ro p e rt ie s
In t his paper we will consider a linear DAE
A(t)x′+B(t)x= 0,(2.1)
where A , B :R+= (0,+∞)→L(Rm,Rm) are b ounded cont inuous (m×m)ma-
t rix funct ions, rank A(t)= r < m,N(t):= kerA(t) is of t he const ant dimension
m−rfor all t∈R+and N(t) is smoot h, i.e t here exist s a cont inuously differ-
ent iable ma t rix funct ion Q∈C1(R+,L(Rm,Rm)) such t hat Q(t) is a pro ject ion
ont o N(t). We shall use the not at ion P=I−Q. We will always assume that (2.1)
is of index 1, i.e t here exist s a C1-smoot h project or Q∈C1(R+,L(Rm,Rm)) ont o
ker A(t) such t hat t he m at rix
A1(t):= A(t)+ (B(t)−A(t)P′(t))Q(t)
(or, equivalent ly, t he m at rix G(t):= A(t)+ B(t)Q(t)) has bounded inverse on
each int erval [t0,T]⊂R+(see [5, 6]).
For definit ion of a solut ion x(t) of the DAE (2.1) one does not require x(t)
t o be C1-smooth but only a part of it s coordinat es be smoot h. Namely, we
int roduce t he spa ce
C1
A(0,∞)= {x(t):R+→Rm,x(t) is continuous and P(t)x(t)∈C1}.
A funct ion x∈C1
A(0,∞)issaidtobeasolution of (2.1) on R+if t he ident it y
A(t)
(P(t)x(t))′−P′(t)x(t)
+B(t)x(t)= 0
holds for all t∈R+.NotethatC1
A(0,∞) does not d epend neit her on t he choice
of P, nor on the definit ion of a solut ion of (2.1) above, as solution of DAEs of
index 1.
D e fi n it io n 2 .1 . A m eas u rable bou n ded fu n ct i on R(·)on R+is called C-function
of system (2.1) if for any ε> 0there exists a positive num ber DR , ε >0such
that the following estim ate
x(t)DR , ε x(t0)e
t
t0
(R(τ) + ε)d τ
(2.2)
holds for all t≥t0≥0and any solution x(·)of (2.1).
T he C ent ral E xponent and A sym pt ot i c Stabi li t y 3
T he set RA , B of all C-functions of (2.1) is called C-class of (2.1).
For any funct ion f:R+→Rwe denot e it s upper mean value by f , i.e.
f:= lim sup
T→ ∞
1
T
T
0
f(t)dt .
D e fi n it io n 2 .2 . T he num ber
Ω := inf
R∈R A , B
R
is called the central exponent of system (2.1).
Let V(dim V(t)= d= const ant ) be an invariant subspace of the solut ion
space of syst em of (2.1), i.e. Vis a linear space spa nned by solut ions of (2.1),
V(t) is t h e sect ion of Vat time t. Not ice t ha t like a linear ODE , t he solut ions
of t he DAE (2.1) form a finit e-dimensional linear subspace of t he space of con-
t inuous Rm-valued funct ions on R+, underst ood also as a subspa ce of t he linea r
(funct ion) space of solut ions.
D e fi n it io n 2 .3 . A functionRis called C-function of ( 2.1) with respect to V
if for any ε> 0,thereexistsDR , ε >0such that for any solution x(t)∈V,we
have
x(t)DR , ε x(t0)e
t
t0
(R(τ) + ε)d τ
,for all t≥t0≥0.
Denote by RVthe collection of all C-functions of V.Thenumber
ΩV:= inf
RV∈R V
RV
is called cen t ral expon en t of (2.1) with respect to V.
Rem ark 2.1. If V1⊂V2t hen RV2⊂R
V1, hence Ω V1ΩV2.Inparticular,
ΩVΩA , B .
Let X(t)= [x1(t), ..., x m(t)] be a ma xim al fundament al solut ion mat rix
(F SM) of (2.1), i.e x1(t), ..., x m(t)aresolutionsof(2.1) a nd t hey span t he solu-
tion space imPs(t) of (2.1) (see [5]). Here Ps(t)= I−QA −1
1Bis the canonical
project ion of (2.1). Denot e by X(t , t 0) t he maxima l F SM of (2.1) normalized
at t0,t0∈R+(see [1]), i.e. X(·,t0) is a maximal F SM sat isfying t he init ial
condit ion
A(t0)(X(t0,t0)−I)= 0.
Such a F SM exist s and is t he solut ion of t he init ial value pr oblem p osed wit h
init ial value X(t0,t0)= Ps(t0). Not e t ha t t he normalized maxim al F SMs play
t he r ole of t he Cauchy ma t rix for t he DAEs.
Le m m a 2 .1 . Suppose that (2.1) is a DA E of index 1and the coeffi cient m atrices
A(t),B(t)are contin uous and bounded on R+. Suppose further that the m atrices
4H oang N am
A−1
1and P′are bounded on R+. T hen the central expon ent Ωof ( 2.1) satifies
the following equality
Ω = lim
T→ ∞ lim sup
n→ ∞
1
nT
n
i= 1
ln X(i T , (i−1)T)
= inf
T > 0
lim sup
n→ ∞
1
nT
n
i= 1
ln X(i T , (i−1)T).(2.3)
Proof. T he pr oof is a simple a nalogue of t he O DE case given in [2] (t he idea is
t o use boundedness of A , B , A −1
1,P′and a property of t he mat rix norm).
Not e t hat formula (2.3) can serve as a definit ion of the central exponent (as
for t he cent ral exponent Ω Vwe can use t he rest r ict ion of X(t , τ )toVinst ead
of Xin t he ab ove formula). Now we will derive some pr opert ies of t he cent ral
exponent of linear DAE of index 1 and of it s corresponding ODE.
T h e o re m 2 . 2 . Suppose that (2.1) is a linear DAE of in dex 1 an d the m atrices
A(t),B(t),A−1
1,P′(t)are bounded on R+. T hen the central expon ent Ωxof
(2.1) is sm aller than or equal to the central exponent Ωuof the correspon ding
ODE of (2.1) under P∈C1,i.e of the ODE
u′= (P′−PA−1
1B0)u. (2.4)
Proof. Denot e by X(t , s) t h e maxim al fundament al solut ion mat r ix of (2.1)
normalized at sand by U(t , s) t he Ca uchy m at rix of (2.4). T hen X(t , s)and
U(t , s) are relat ed by t he following equality (see [1], p.18)
X(t , s)= Ps(t)U(t , s)P(s),∀t≥s≥0,
hence
X(t , s)Ps(t) U(t , s) P(s).
Since t he mat r ices A(t), B(t), A−1
1(t) are bounded on R+, t he pro ject ors P=
A−1
1A,Q=I−P,Qs=QA −1
1Band Ps=I−Qsare bounded on R+,too. Let
Psb1,Pb2,wehave
X(t , s)b1b2U(t , s).
T herefore
lnX(t , s)ln(b1b2)+ lnU(t , s).
T his implies t ha t
n
j= 1
ln X(j T, (j−1)T)nln(b1b2)+
n
j= 1
lnU(j T, (j−1)T).
Hence, by (2.3)
T he C ent ral E xponent and A sym pt oti c Stabi li t y 5
Ωx= lim
T→ ∞ lim sup
n→ ∞
1
nT
n
j= 1
lnX(j T, (j−1)T)
lim
T→ ∞ lim sup
n→ ∞
1
nT
n
j= 1
lnU(j T, (j−1)T)+nln(b1b2)
lim
T→ ∞ lim sup
n→ ∞
1
nT
n
j= 1
lnU(j T, (j−1)T)= Ω u.
Hence Ω xΩu. T he t heorem is proved.
In Definition 2.3 we introduced t he not ion of central exponent of a DAE
wit h respect t o an invariant subspace of t he solut ion space. T his can cer t ainly
be done for ODEs.
Not e t hat t he corresponding ODE (2.4) of t he DAE (2.1) under some pro jec-
t or P(t) is defined on t he whole phase space Rm. T he funct ion spa ce spanned by
solut ions u(t) of (2.4) sat isfying u(t)∈im P(t)fort≥0,is an invariant subspace
of t he solut ion space of that ODE. Wit h an abuse of language we denot e t hat
funt ion spa ce by imP. We show t hat t he cent ral exponent of t his ODE wit h
respect t o t he funct ion spa ce imPis closely relat ed t o t he cent ral exp onent of
t he DAE (2.1) (in t he sense t ha t it char act erizes b et t er t he cent r al exp onent of
t he DAE (2.1)).
Let us consid er t he corr esp ondin g ODE of (2.1) under a pr oject or P
u′= (P′Ps−PG−1B)u. (2.5)
Similar ly t o Definit ion 2.3 we call t he numb er
ΩUi m P:= inf
R∈R i m P
R,
where Ri m Pis t he cla ss of C-funct ions of t he invar iant subspace im Pof the
solut ion space of (2.5), cen t ral expon en t of ODE (2.5) with respect t o im P.
Clearly, Ω Ui m PΩU.
Denot e by U(t , t 0) t he Cauchy mat r ix of (2.5).
P ut
Ui m P(t , t 0):= P(t)U(t , t 0)P(t0)= U(t , t 0)P(t0).(2.6)
Onecanseethat
ΩUi m P= lim
T→ ∞ lim sup
n→ ∞
1
nT
n−1
i= 0
ln
Ui m P
(i+ 1)T, iT
.
Denot e by Ω x,Ωuand Ω Ui m Pt he cent ral exponent s of (2.1), (2.5) and of
(2.5) wit h resp ect t o im P.
T h e o re m 2 .3 . Suppose that (2.1) is a lin ear DA E of in dex 1 with the coeffi cient
m atrices A(t),B(t)bein g continuous and boun ded on R+. T hen the following
assertions are true