HIGHER-ORDER LINEAR MATRIX DESCRIPTOR DIFFERENTIAL EQUATIONS OF APOSTOL

13 Pages·2009·0.24 MB·English

Electronic Journal of Differential Equations, Vol. 2009(2009), No. 25, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
HIGHER-ORDER LINEAR MATRIX DESCRIPTOR
DIFFERENTIAL EQUATIONS OF APOSTOL-KOLODNER TYPE
GRIGORISI.KALOGEROPOULOS,ATHANASIOSD.KARAGEORGOS,
ATHANASIOSA.PANTELOUS
Abstract. In this article, we study a class of linear rectangular matrix de-
scriptordifferentialequationsofhigher-orderwhosecoefficientsaresquarecon-
stantmatrices. UsingtheWeierstrasscanonicalform,theanalyticalformulas
forthesolutionofthisgeneralclassisanalyticallyderived,forconsistentand
non-consistentinitialconditions.
1. Introduction
LinearDescriptorMatrixDifferentialEquations(LDMDEs)areinherentinmany
physical,engineering,mechanical,andfinancial/actuarialmodels. LDMDEs,which
are also known in the literature as functional matrix differential equations, are a
special and commonly used class of matrix differential equations. Having in mind
the applications of LDMDEs, for instance in finance, we provide the well-known
input-output Leondief model and its several important extensions, advice [3]. In
this article, our long-term purpose is to study the solution of LDMDES of higher
order (1.1) into the mainstream of matrix pencil theory. This effort is significant,
since there are numerous applications. Thus, we consider
FX(r)(t)=GX(t) (1.1)
where F,G ∈ M(n×n;F), (i.e. the algebra of square matrices with elements in
the field F) with detF = 0 (0 is the zero element of M(n=1,F)), and X ∈
C∞(F,M(n×m;F)). For the sake of simplicity we set M = M(n×n;F) and
n
M =M(n×m;F).
nm
MatrixpenciltheoryhasbeenextensivelyusedforthestudyofLinearDescriptor
Differential Equations (LDDEs) with time invariant coefficients, see for instance
[3], [7]-[9]. Systems of type (1.1) are more general, including the special case when
F = I , where I is the identity matrix of M , since the well-known class of
n n n
higher-orderlinearmatrixdifferentialequationsofApostol-Kolodnertypeisderived
straightforwardly, see [1] for r =2, [2] and [10].
Thepaperisorganizedasfollows: InSection2somenotationsandthenecessary
preliminary concepts from matrix pencil theory are presented. Section 3 contains
2000 Mathematics Subject Classification. 34A30,34A05,93C05,15A21,15A22.
Key words and phrases. Matrixpenciltheory;Weierstrasscanonicalform;
linearmatrixregulardescriptordifferentialequations.
(cid:13)c2009TexasStateUniversity-SanMarcos.
SubmittedNovember4,2008. PublishedFebruary3,2009.
1
2 G.I.KALOGEROPOULOS,A.D.KARAGEORGOS,A.A.PANTELOUS EJDE-2009/25
the case that system (1.1) has consistent initial conditions. In Section 4 the non
consistentinitialconditioncaseisfullydiscussed. Inthiscase,thearbitrarilychosen
initial conditions which have physical meaning for descriptor (regular) systems, in
some sense, can be created or structurally changed at a fixed time t = t . Hence,
0
it is derived that (1.1) should adopt a generalized solution, in the sense of Dirac
δ-solutions.
2. Mathematical Background and Notation
Thisbriefsectionintroducessomepreliminaryconceptsanddefinitionsfromma-
trix pencil theory, which are being used throughout the paper. Descriptor systems
of type (1.1) are closely related to matrix pencil theory, since the algebraic geo-
metric, and dynamic properties stem from the structure by the associated pencil
sF −G.
Definition 2.1. GivenF,G∈M andanindeterminates∈F,thematrixpencil
nm
sF −G is called regular when m=n and det(sF −G)6=0. In any other case, the
pencil will be called singular.
Definition 2.2. The pencil sF −G is said to be strictly equivalent to the pencil
sF˜−G˜ if and only if there exist nonsingular P ∈M and Q∈M such as
n m
P(sF −G)Q=sF˜−G˜.
In this article, we consider the case that pencil is regular. Thus, the strict
equivalencerelationcanbedefinedrigorouslyonthesetofregularpencilsasfollows.
Here, we regard (2.1) as the set of pair of nonsingular elements of M
n
g :={(P,Q):P,Q∈M , P,Q nonsingular} (2.1)
n
and a composition rule ∗ defined on g as follows:
∗:g×g such that (P ,Q )∗(P ,Q ):=(P ·P ,Q ·Q ). (2.2)
1 1 2 2 1 2 2 1
It can be easily verified that (g,∗) forms a non-abelian group. Furthermore, an
action ◦ of the group (g,∗) on the set of regular matrix pencils Lreg is defined as
n
◦:g×Lreg →Lreg such that
n n
((P,Q),sF −G)→(P,Q)◦(sF −G):=P(sF −G)Q.
This group has the following properties:
(a) (P ,Q )◦[(P ,Q )◦(sF −G)] = (P ,Q )∗(P ,Q )◦(sF −G) for every
1 1 2 2 1 1 2 2
nonsingular P ,P ∈M and Q ,Q ∈M .
1 2 n 1 2 n
(b) e ◦(sF −G)=sF −G, sF −G∈Lreg where e =(I ,I ) is the identity
g n g n n
element ofthegroup(g,∗)onthesetofLreg definesatransformationgroup
n
denoted by N, see [6].
For sF −G∈Lreg, the subset
n
g◦(sF −G):={(P,Q)◦(sF −G):(P,Q)∈g}⊆Lreg
n
will be called the orbit of sF −G at g. Also N defines an equivalence relation on
Lreg which is called a strict-equivalence relation and is denoted by E .
n s−e
So, (sF −G)E (sF˜−G˜) if and only if P(sF −G)Q=sF˜−G˜ , where P,Q∈
s−e
M are nonsingular elements of algebra M .
n n
EJDE-2009/25 HIGHER-ORDER LINEAR MATRIX DESCRIPTOR 3
TheclassofE (sF−G)ischaracterizedbyauniquelydefinedelement,known
s−e
asacomplexWeierstrasscanonicalform,sF −Q ,see[6],specifiedbythecomplete
w w
set of invariants of E (sF −G).
s−e
Thisisthesetofelementary divisors (e.d.) obtainedbyfactorizingtheinvariant
polynomials f (s,sˆ) into powers of homogeneous polynomials irreducible over field
i
F. In the case where sF −G is a regular, we have e.d. of the following type:
• e.d. of the type sp are called zero finite elementary divisors (z. f.e.d.)
• e.d. ofthetype(s−a)π,a6=0are called nonzero finite elementary divisors
(nz. f.e.d.)
• e.d. of the type sˆq are called infinite elementary divisors (i.e.d.).
Let B ,B ,...,B be elements of M . The direct sum of them denoted by B ⊕
1 2 n n 1
B ⊕···⊕B is the block diag{B ,B ,...,B }.
2 n 1 2 n
Then, the complex Weierstrass form sF −Q of the regular pencil sF −G is
w w
defined by sF −Q := sI −J ⊕sH −I , where the first normal Jordan type
w w p p q q
element is uniquely defined by the set of f.e.d.
ν
X
(s−a )p1,...,(s−a )pν, p =p (2.3)
1 ν j
j=1
of sF −G and has the form
sI −J :=sI −J (a )⊕···⊕sI −J (a ). (2.4)
p p p1 p1 1 pν pν ν
And also the q blocks of the second uniquely defined block sH −I correspond to
q q
the i.e.d.
σ
X
sˆq1,...,sˆqσ, q =q (2.5)
j
j=1
of sF −G and has the form
sH −I :=sH −I ⊕···⊕sH −I . (2.6)
q q q1 q1 qσ qσ
Thus, H is a nilpotent element of M with index q˜= max{q :j =1,2,...,σ},
q n j
where
Hq˜=O,
q
and I ,J (a ),H are defined as
pj pj j qj
 
1 0 ... 0 aj 1 0 ... 0
Ipj =0... 1... ...... 0...∈Mpj, Jpj(aj)=0... a...j .1.. ...... 0... ∈Mpj
0 0 ... 1 0 0 0 aj 1
0 0 0 0 a
j
 
0 1 0 ... 0
0 0 1 ... 0
Hqj =... ... ... ... ...∈Mqj.
 
0 0 0 0 1
0 0 0 0 0
(2.7)
In the last part of this section, some elements for the analytic computation of
eA(t−t0), t ∈ [t0,∞) are provided. To perform this computation, many theoretical
and numerical methods have been developed. Thus, the interesting readers might
4 G.I.KALOGEROPOULOS,A.D.KARAGEORGOS,A.A.PANTELOUS EJDE-2009/25
consult papers [2, 4, 10, 11, 13] and the references therein. In order to have com-
putationalformulas,seethefollowingSections3and4,thefollowingknownresults
should firstly be mentioned.
Lemma 2.3 ([4]). eJpj(aj)(t−t0) =(dk1k2)pj, where
d =eaj(t−t0)(t(−k t−0)kk2−)!k1, 1≤k1 ≤k2 ≤pj
k1k2 2 1
0, otherwise
Another expression for the exponential matrix of Jordan block, see (2.7), is
provided by the following Lemma.
Lemma 2.4 ([13]).
pj−1
X
eJpj(aj)(t−t0) = fi(t−t0)[Jpj(aj)]i (2.8)
i=0
where the f (t−t )’s are given analytically by the following p equations:
i 0 j
f (t−t )=eaj(t−t0)Xk b ak−i(t−t0)pj−1−i, k =0,1,2,...,p −1 (2.9)
pj−1−k 0 k,i j (p −1−i)! j
j
i=0
where
k−i(cid:18) (cid:19)(cid:18) (cid:19)
b =X pj k−l (−1)l
k,i l i
l=0
and
[J (a )]i =(c(i) ) , for 1≤k ,k ≤p (2.10)
pj j k1k2 pj 1 2 j
where
(cid:18) (cid:19)
i
c(i) = ai−(k2−k1).
k1k2 k2−k1 j
3. Solution space form of consistent initial conditions
In this section, the main results for consistent initial conditions are analytically
presented for the regular case. The whole discussion extends the existing litera-
ture; see for instance [2]. Moreover, it should be stressed out that these results
offer the necessary mathematical framework for interesting applications, see also
introduction. Now, in order to obtain a unique solution, we deal with consistent
initial value problem. More analytically, we consider the system
FX(r)(t)=GX(t), (3.1)
with known initial conditions
X(t ),X0(t ),...,X(r−1)(t ). (3.2)
0 0 0
FromtheregularityofsF−G,thereexistnonsingularM(n×n,F)matricesP and
Q such that (see also section 2), such as
PFQ=F =I ⊕H , (3.3)
w p q
PGQ=G =J ⊕I , (3.4)
w p q
EJDE-2009/25 HIGHER-ORDER LINEAR MATRIX DESCRIPTOR 5
where I ,J ,H and I are given by (2.7) where
p p q q
I =I ⊕...⊕I ,
p p1 pν
J =J (a )⊕...⊕J (a ),
p p1 1 pν ν
H =H ⊕...⊕H ,
q q1 qσ
I =I ⊕...⊕I .
q q1 qσ
Note that Pν p =p and Pσ q =q, where p+q =n.
j=1 j j=1 j
Lemma 3.1. System (3.1) is divided into two subsystems: The so-called slow sub-
system
Y(r)(t)=J Y (t), (3.5)
p p p
and the relative fast subsystem
H Y(r)(t)=Y (t). (3.6)
q q q
Proof. Consider the transformation
X(t)=QY(t). (3.7)
Substituting the previous expression into (3.1) we obtain
FQY(r)(t)=GQY(t).
Whereby, multiplying by P, we arrive at
F Y(r)(t)=G Y(t).
w w
(cid:20) (cid:21)
Y (t)
Moreover, we can write Y(t) as Y(t) = p ∈ M . Taking into account the
Y (t) nm
q
above expressions, we arrive easily at (3.5) and (3.6). (cid:3)
Remark 3.2. System (3.5) is the standard form of higher-order linear matrix
differential equations of Apostol-Kolodner type, which may be treated by classical
methods, see for instance [1, 2], [4] and [10] and references therein. Moreover, it
shouldbealsomentionedthatsection5of[13]describesamethodforsolvinghigher-
order equations of the form q(D)X(t) = AX(t) where q is a scalar polynomial, D
isdifferentiationwithrespecttotandAisasquarematrix. Suchequationsclearly
include the standard form equations of the Apostol-Kolodner type. Thus, it is
convenient to define new variables as
Z (t)=Y (t),
1 p
Z (t)=Y0(t),
2 p
...
Z (t)=Y(r−1)(t).
r p
Then, we have the system of ordinary differential equations
Z0(t)=Z (t),
1 2
Z0(t)=Z (t),
2 3
(3.8)
...
Z0(t)=J Z (t).
r p 1
Now, (3.8) can be expressed using vector-matrix equations,
Z0(t)=AZ(t) (3.9)
6 G.I.KALOGEROPOULOS,A.D.KARAGEORGOS,A.A.PANTELOUS EJDE-2009/25
where Z(t)=[ZT(t)ZT(t)...ZT(t)]T (where ()T is the transpose tensor) and the
1 2 r
coefficient matrix A is given by
O I O ... O
p
O O Ip ... O
A= ... ... ... ... ...  (3.10)
O O O ... Ip
J O O ... O
p
with corresponding dimension of A and Z(t), pr×pr and pr×m, respectively.
Equation (3.9) is a linear ordinary differential system and has a unique solution
for any initial condition
Z1(t0)  Yp(t0) 
Z(t0)=Z2.(.t.0)= Yp.0(.t.0) ∈Mpr×m. (3.11)
Zr(t0) Yp(r−1)(t0)
It is well-known that the solution of (3.11) has the form
Z(t)=eA(t−t0)Z(t ). (3.12)
0
Then Y (t)=Z (t)=LZ(t), where
p 1
L=[I O...O]∈M . (3.13)
p p×pr
Finally, by combining (3.11)-(3.12) and (3.13), we obtain
Y (t)=LeA(t−t0)Z(t ) (3.14)
p 0
To obtain a more analytic formula for the solution of (3.12), we should compute
analytically the matrix eA(t−t0) ∈Mpr, see [2], [4], [10], [11] and [13].
First, considering the Jordan canonical form, there exists a nonsingular matrix
R∈M such that J=R−1AR, where J∈M is the Jordan Canonical form of
pr pr
matrix A. Afterwards, defining
Z(t)=RΘ(t)
then, combining (3.2) and (3.9), we obtain
RΘ0(t)=ARΘ(t).
Finally, multiplying the above expression by R−1, we take
Θ0(t)=JΘ(t).
It is well-known that the solution of (3.2) is given by
Θ(t)=eJ(t−t0)Θ(t ),
0
where
 Y (t ) 
p 0
Θ(t0)=R−1Z(t0)=R−1 Yp.0(.t.0) ∈Mpr×m
Y(r−1)(t )
p 0
EJDE-2009/25 HIGHER-ORDER LINEAR MATRIX DESCRIPTOR 7
Proposition 3.3. The characteristic polynomial matrix A is given by
v
Y
ϕ(λ)=det(λI −A)= (λr−a )pj, (3.15)
pr j
i=1
where Pv p =p.
j=1 j
Proof. We obtain the characteristic polynomial of matrix A
λI −I O ... O 
p p
 O λIp −Ip ... O 
ϕ(λ)=det(λI −A)=det O O λIp ... O .
pr  ... ... ... ... ... 
−J O O ... λI
p p
Afterwards, we consider some simple transformations. Thus, we multiply the first
block by λ and we add it to the second one. Moreover, we multiply the second
block by λ and we add it to the third one. Continuing as above, we finally obtain
that
λI −I O ... O   λI −I O ... O
p p p p
 O λIp −Ip ... O   λ2Ip O −Ip ... O
 O O λIp ... O ∼ λ3Ip O O ... O.
 ... ... ... ... ...   ... ... ... ... ...
−J O O ... λI λrI −J O O ... O
p p p p
Now we make p row transformations to the determinant, as follows
λrI −J O O ... O
p p
 λ2Ip O −Ip ... O
(−1)pdet λ3Ip O O ... O.
 ... ... ... ... ...
λI −I O ... O
p p
Continuing as above, we transfer the above determinant into the form
λrI −J O O ... O 
p p
 λIp −Ip O ... O 
(−1)p(−1)p...(−1)pdet λ2Ip O −Ip ... O 
| (r−1){−ztimes }  ... ... ... ... ... 
λr−1I O O ... −I
p p
=(−1)(r−1)p|λrI −J ||−I |...|−I |
p p p p
| {z }
(r−1)−times
=(−1)2(r−1)p|λrI −J |=|λrI −J |.
p p p p
Thus, we obtain the expression
ϕ(λ)=det(λI −A)=|λrI −J |.
pr p p
Moreover, we recall that J =J (a )⊕···⊕J (a ). Thus
p p1 1 pv v
v
Y
|λrI −J |= |λrI −J (a )|.
p p pj pj j
j=1
8 G.I.KALOGEROPOULOS,A.D.KARAGEORGOS,A.A.PANTELOUS EJDE-2009/25
Note also that
λr−a −1 0 ... 0 
j
 0 λr−aj −1 ... 0 
|λrIpj −Jpj(aj)|=det ... ... ... ... ... =(λr−aj)pj,
 
 0 0 0 ... −1 
0 0 0 ... λr−a
j
for j = 1,2,...,ν. Consequently, the characteristic polynomial (3.15) is derived.
(cid:3)
Remark 3.4. The eigenvalues of matrix A are given by (3.16)
q (cid:16) 2kπ+ϕ 2kπ+ϕ (cid:17)
λ = r |a | cos j +zsin j , (3.16)
jk j r r
where a = |a |(cosϕ +zsinϕ ) and z2 = −1 for every j = 1,2,...,ν and k =
j j j j
0,1,2,...,r−1.
Remark 3.5. The characteristic polynomial is ϕ(λ) = Qνj=1(λr−aj)pj, with
a 6=a for i6=j and Pν p =p. Without loss of generality, we define that
i j j=1 j
d =τ ,d =τ ,...,d =τ , and d <τ ,...,d <τ
1 1 2 2 l l l+1 l+1 ν ν
where d ,τ , j =1,2,...,ν is the geometric and algebraic multiplicity of the given
j j
eigenvalues a , respectively.
j
• Consequently, when d =τ ,
j j
 
λ
jk
 λjk 
Jjk(λjκ)= ... ∈Mτjk,
λ
jk
isalsoadiagonalmatrixwithdiagonalelementstheeigenvalueλ ,forj =1,...,l.
jk
• When d <τ ,
j j
 
λ 1
jk
 λjk 1 
 
Jjk,zj = λjk ... ∈Mzj
 ... 1 
λ
jk
for j =l+1,l+2,...,ν, and z =1,2,...,d .
j j
Proposition 3.6. The fast subsystem (3.6) has only the zero solution.
Proof. By successively taking r-th derivatives with respect to t on both sides of
(3.6)andmultiplyingbyleftbythematrixH , q∗−1, times(whereq∗ istheindex
q
EJDE-2009/25 HIGHER-ORDER LINEAR MATRIX DESCRIPTOR 9
of the nilpotent matrix H , i.e. Hq∗ =O), we obtain the following equations
q q
H Y(r)(t)=Y (t),
q q q
H2Y(2r)(t)=H Y(r)(t),
q q q q
H3Y(3r)(t)=H2Y(2r)(t),
q q q q
...
Hq∗Y(q∗r)(t)=Hq∗−1Y((q∗−1)r)(t).
q q q q
The conclusion, i.e. Y (t) = O, is obtained by repetitively substitution of each
q
equation in the next one, and using the fact that Hq∗ =O. (cid:3)
q
Hence, the set of consistent initial conditions for system F Y(r)(t) = G Y(t)
w w
has the form
n (cid:20)Y(k)(t )(cid:21) o
Y(k)(t )= p 0 , k =0,1,...,r−1 . (3.17)
0 O
q
Theorem 3.7. The analytic solution of (3.1) is given by
ν r−1
X(t)=Qn,pLR ⊕ ⊕ eJjk(λjk)(t−t0)R−1Z(t0). (3.18)
j=1k=0
where L=[I O...O]∈M ; R∈M such that J=R−1AR, where J∈M
p p×pr pr pr
is the Jordan Canonical form of matrix A; and
Z(t )=(cid:2)YT(t ) Y0T(t )...Y(r−1)T(t )(cid:3)T ∈M .
0 q 0 q 0 p 0 m×pr
Proof. Combining (3.2), (3.2) and the above discussion, the solution is
ν r−1
Θ(t)= ⊕ ⊕ eJjk(λjk)(t−t0)Θ(t0).
j=1k=0
Then, multiplying by R and bearing in mind that Θ(t )=R−1Z(t ), we obtain
0 0
ν r−1 ν r−1
Z(t)=RΘ(t)=R ⊕ ⊕ eJjk(λjk)(t−t0)Θ(t0)=R ⊕ ⊕ eJjk(λjk)(t−t0)R−1Z(t0).
j=1k=0 j=1k=0
Now, consider (3.13), then we obtain
ν r−1
Yp(t)=LZ(t)⇔Yp(t)=LR ⊕ ⊕ eJjk(λjk)(t−t0)R−1Z(t0).
j=1k=0
Using the results of Proposition 2; i.e., that the second (fast) sub-system (3.6) has
only the zero solution, we obtain
(cid:20) (cid:21)
Y (t)
X(t)=QY(t)=[Q Q ] p =Q Y (t).
n,p n,q O n,p p
Finally, (3.18) is obtained. (cid:3)
The next remark presents the set of consistent initial condition for system (3.1).
Remark 3.8. Combining (3.7) and (3.17), we obtain
(cid:20) (cid:21)
Y (t )
X(t )=QY(t )=[Q Q ][ p 0 =Q Y (t ).
0 0 n,p n,q O n,p p 0
Then, the set of consistent initial conditions for (3.1) is given by
(cid:8)Q Y (t ) Q Y0(t ) ... Q Y(r−1)(t )(cid:9). (3.19)
n,p p 0 n,p p 0 n,p p 0
10 G.I.KALOGEROPOULOS,A.D.KARAGEORGOS,A.A.PANTELOUS EJDE-2009/25
Now, taking into consideration (3.2) and (3.19), we conclude
X(t )=Q Y (t ),
0 n,p p 0
X0(t )=Q Y0(t ),
0 n,p p 0
...
X(r−1)(t )=Q Y(r−1)(t ).
0 n,p p 0
Remark 3.9. If Q˜ is the existing left inverse of Q , then considering (3.11),
n,p n,p
we have
 Y (t )   Q˜ X (t ) 
p 0 p,n p 0
 Yp0(t0)   Q˜p,nXp0(t0) 
Z(t0)= .. = .. 
 .   . 
Yp(r−1)(t0) Q˜p,nXp(r−1)(t0)
Q˜  X (t ) 
p,n p 0
 Q˜p,n  Xp0(t0) 
= ...  ... 
Q˜ X(r−1)(t )
p,n p 0
=Q˜Ψ(t ).
0
Finally, the solution (3.18) is given by
X(t)=Qn,pLR ⊕ν r⊕−1eJjk(λjk)(t−t0)R−1Q˜Ψ(t0). (3.20)
j=l+1k=0
where Ψ(t ) = [XT(t ) X0T(t ) ... X(r−1)T(t )]T ∈ M and Q˜ is the
0 p 0 p 0 p 0 m×pr n,p
existing left inverse of Q .
n,p
The following two expressions, i.e. (3.21) and (3.22) are based on Lemma 1 and
2,respectively. Thus,twonewanalyticalformulasarederivedwhicharepractically
veryinteresting. TheirproofsarestraightforwardexerciseofLemma1,2and(3.19)
Lemma 3.10. Considering the results of Lemma 1, we obtain the expression
h(cid:16) (cid:17)
X(t)=Qn,pLR ⊕lj=0⊕rk−=10eλjk(t−t0)Iτjk
(3.21)
(cid:16) (cid:17)i
⊕ ⊕ν ⊕r−1 ⊕dj (d ) R−1Q˜Ψ(t ).
j=l+1 k=0 zj=1 k1k2 zj 0
where
d =eλjk(t−t0)(t(−k t−0)kk2−)!k1, 1≤k1 ≤k2 ≤zj
k1k2 2 1
0, otherwise
for j =l+1,l+2,...,ν and z =1,2,...,d .
j j
Lemma 3.11. Considering the results of Lemma 2.4, we obtain the expression
h(cid:16) (cid:17)
X(t)=Qn,pLR ⊕lj=0⊕rk−=10eλjk(t−t0)Iτjk
⊕(cid:16)⊕ν ⊕r−1 ⊕dj zXj−1f (t−t )[J (λ )]i(cid:17)iR−1Q˜Ψ(t ). (3.22)
j=l+1 k=0 zj=1 i 0 zj jk 0
i=0