# Transferring Instantly the State of Higher-Order Linear Descriptor (Regular) Differential Systems Using Impulsive Inputs

Authors: Athanasios D. Karageorgos; Athanasios A. Pantelous; Grigoris I. Kalogeropoulos
Journal: Journal of Control Science and Engineering
 (2009)

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## Abstract

In many applications, and generally speaking in many dynamical differential systems, the problem of transferring the initial state of the system to a desired state in (almost) zero-time time is desirable but difficult to achieve. Theoretically, this can be achieved by using a linear combination of Diracδ-function and its derivatives. Obviously, such an input is physically unrealizable. However, we can think of it approximately as a combination of small pulses of very high magnitude and infinitely small duration. In this paper, the approximation process of the distributional behaviour of higher-order linear descriptor (regular) differential systems is presented. Thus, new analytical formulae based on linear algebra methods and generalized inverses theory are provided. Our approach is quite general and some significant conditions are derived. Finally, a numerical example is presented and discussed.


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## Body

## 1. Introduction

From the point of view of several important applications, in several fields of research, see for instance [1, 2], taking a given state of a linear system to a desired state in minimum time is very desirable, though it is a challenging problem in control and system theory.Significant attention has been given to this problem in the case of linear systems; see [1–3]. Recently, Kalogeropoulos et al. [4] have further enriched these first approaches, as they have relaxed some of the rather restricted assumptions that [1, 2] are considered. Afterwards, the method has been also applied to the more general class of linear descriptor (regular) systems; see [5].In this paper, a further extension of [5], in the class of linear descriptor (regular) equations, is provided. Comparing with the existing literature, see [1–5], we solve this problem(1)
for higher-order linear differential descriptor (regular) systems (compare with [5]),(2)
using higher-order consistent initial conditions (compare with [1–5]),(3)
obtaining more analytical formulas, that is, see AppendicesA and B, Theorems 20 and 23. (compare with [1–5]),(4)
without using the controllability matrix (compare with [3]),(5)
applying analytical methods for the exact determination of the generalized inverses of Vandermonde and relative to this matrices; see [6].To summarize, in this paper, we investigate how we can transfer the initial state of an open loop, linear higher-order descriptor (regular) differential system in (practically speaking, almost) zero-time, that is,(1)Fx̲(r)(t)=Gx̲(t)+b̲u(t)
with known initial conditions(2)x̲(to),x̲′(to),…,x̲(r-1)(to),
where F,G∈ℳ(n×n;𝔽), and b̲∈ℳ(n×1;𝔽) (i.e., ℳ is the algebra of n×m matrices with elements in the field 𝔽) with detM=0(0is the zero element of ℳ(n=1,𝔽)), x̲(t)∈𝒞∞(𝔽,ℳ(n×1;𝔽)), and u(t)∈𝒞∞(𝔽,ℳ(1;𝔽)). For the sake of simplicity, we set in the sequel ℳn=ℳ(n×n;𝔽) and ℳnm=ℳ(n×m;𝔽).In order to solve this problem, the appropriate input vector has to be made up as a linear combination of theδ-function of Dirac and its derivatives, see [1, 2], and for more details consult [7]. Obviously, such an input is very hard to imagine physically. However, we can think of it approximately as a combination of small pulses of very high magnitude and infinitely small duration.Linear descriptor (singular or regular) differential systems have been extensively used in control theory; see for instance [8–10], and for more details [11].A brief outline of the paper is as follows. Section2 provides the incentives and the typical modelling features of the problem. Moreover, a classical approximated expression for the controller, that is, a linear combination of the δ-function of Dirac and its derivatives that is based on the normal (Gaussian) probability density function, is used. Then, the need to determine the unknown coefficients is derived. Section 3 is divided into four extensive subsections. In Section 3.1, the reduction of the higher-order system to first-order is discussed. The first-order descriptor (regular) is divided into a fast and a slow subsystem, using the Weierstrass canonical form. Section 3.2 investigates and presents some analytical formulas based on the slow subsystem. In Section 3.3, the theory of the {1}-generalized inverses is used. Finally, some significant conditions for the solution of the slow subsystem are presented in Section 3.4. In Section 4, a necessary condition based on the fast subsystem is discussed and obtained. Section 5 provides an interesting numerical application from physics and Section 6 concludes the paper. Two appendices for the analytical calculations of two important integrals are also available.


## 2. Preliminary Results—Matrix Pencil Framework

In this section, some preliminary results for matrix pencil and system theory are briefly presented. First, we assume that for annth-order linear differential system, see (1), the input can be a linear combination of Dirac δ-function and its first (n-1)th-derivatives as follows:(3)uo(t)=aoδ(t)+a1δ(1)(t)+a2δ(2)(t)+⋯+an-1δ(n-1)(t)=∑k=1n-1akδ(k)(t),
where δ(k)(t) or dkδ(t)/dtk is the kth-derivative of the Dirac δ-function, and ai for i=0,1,…,n-1 are the magnitudes of the delta function and its derivatives. Furthermore, we assume that the state of the system at time 0- is(4)x̲(0-)=x̲′(0-)=⋯x̲(r-1)(0-)=[00⋯0]t,
and at time 0+, it achieves(5)x̲(0+)=[x10x20⋯xn0]t,x̲′(0+)=[x11x21⋯xn1]t,…,x̲(r-1)(0+)=[x1r-1x2r-1⋯xnr-1]t.With the following definitions, a brief presentation of the most important elements of matrix pencil theory is given.Definition 1.
GivenF,G∈ℳnm and an indeterminate s∈𝔽, the matrix pencil sF-G is called regular when m=n and det(sF-G)≠0. In any other case, the pencil will be called singular.Definition 2.
The pencilsF-G is said to be strictly equivalent to the pencil sF̃-G̃ if and only if there exist nonsingular P∈ℳm and Q∈ℳn such as
(6)P(sF-G)Q=sF̃-G̃.In this paper, we consider the case that pencil isregular. Thus, the strict equivalence relation can be defined rigorously on the set of regular pencils as follows.This is the set ofelementary divisors (e.d.s) obtained by factorizing the invariant polynomials fi(s,ŝ) into powers of homogeneous polynomials irreducible over field 𝔽.In the case wheresF-G is a regular, we have e.d. of the following types:(i)
e.d. of the typesp are called zero finite elementary divisors (z. f.e.d.),(ii)
e.d. of the type(s-a)π, a≠0 are called nonzero finite elementary divisors (nz. f.e.d.),(iii)
e.d. of the typeŝq are called infinite elementary divisors (i.e.d.).LetB1,B2,…,Bn be elements of ℳn. The direct sum of them denoted by B1⨁B2⨁⋯⨁Bn is the blockdiag{B1,B2,…,Bn}.Then, the complex Weierstrass formsFw-Qw of the regular pencil sF-G is defined by(7)sFw-Qw≜sIp-Jp⨁sHq-Iq.
Now, the Jordan type element, that is, Jp, is uniquely defined by the set of f.e.d.(8)(s-a1)p1,…,(s-aν)pν,∑j=1νpj=p,
of sF-G and has the form(9)sIp-Jp≜sIp1-Jp1(a1)⨁⋯⨁sIpν-Jpν(aν),
where(10)Ip≜Ip1⨁Ip2⨁⋯⨁Ipν,Jp≜Jp1(a1)⨁Jp2(a2)⨁⋯⨁Jpν(aν).
The q blocks of the second, that is, sHq-Iq, see (7), are uniquely defined by the set of i.e.d.(11)ŝq1,…,ŝqσ,∑j=1σqj=q
of sF-G and has the form(12)sHq-Iq≜sHq1-Iq1⨁⋯⨁sHqσ-Iqσ,
where(13)Iq≜Iq1⨁Iq2⨁⋯⨁Iqσ,Hq≜Hq1⨁Hq2⨁⋯⨁Hqσ.
Furthermore, Hq is a nilpotent element of ℳn with index q̃=max{qj:j=1,2,…,σ}, where
(14)Hqq̃=𝕆,
and Ipj,Jpj(aj),Hqj are defined as(15)Ipj=[10⋯001⋯0⋮⋮⋱⋮00⋯1]∈ℳpj,Jpj(aj)=[aj10⋯00aj1⋯0⋮⋮⋱⋮⋮000aj10000aj]∈ℳpj,Hqj=[010⋯0001⋯0⋮⋮⋱⋮⋮0000100000]∈ℳqj.
Moreover, for the matrices F and G, we have the parameterization(16)PFQ=Fw=Ip⨁Hq,(17)PGQ=Gw=Jp⨁Iq.
Since the state x̲(0+) which we wish to reach is specified, we need to determine the n unknown coefficients ai, for i=0,1,…,n-1.In practice, we cannot create an exact impulse function nor its derivatives. However, if we use one of the approximations of Diracδ-function, we will be able to change the state in some minimum practical time depending mainly upon how well we generate the approximations. Let the Dirac δ-function be viewed as the limit of sequence function(18)δ(t)=lima→0δa(t),
whereδa(t) is called a nascent delta function. This limit is in the sense that(19)lima→0∫-∞+∞δa(t)f(t)dt=f(0).Some wellknown and very useful in applicationsnascent delta functions are the Normal and Cauchy distributions, the rectangular function, the derivative of the sigmoid (or Fermi-Dirac) function, the Airy function, and so forth; see for instance [2, 5, 12–17]. The results given below are based on the normal function. Thus, by taking into consideration expression (18) and the normal (Gaussian) probability density distribution, we obtain(20)δ(t)=limσ→01σ2πe-t2/2σ2=limσ→01σΦ(tσ),
where Φ(x)=(1/2π)e-x2/2.So, the approximate expression for the controller (2) is given by(21)u(t)=1σΦ(tσ)ao+1σ2Φ(1)(tσ)a1+⋯+1σnΦ(n-1)(tσ)an-1=∑k=0n-11σk+1Φ(k)(tσ)ak.
Then, we take the limit(22)uo(t)=limσ→0u(t).
In the next section, the main results are presented.


## 3. The Main Results

### 3.1. Order Reduction of System (1)

The section begins with the following important lemma.Lemma 3.
System (1) is divided into the following two subsystems:
(23)y̲p(r)(t)=Jpy̲p(t)+b̲̃puo(t),(24)Hqy̲q(r)(t)=y̲q(t)+b̲̃quo(t).Proof.
Consider the transformation(25)x̲(t)=Qy̲(t).
Substituting the previous expression into (1) and considering also (3), we obtain
(26)FQy̲(r)(t)=GQy̲(t)+b̲uo(t).
Multiplying by P, we arrive at
(27)Fwy̲(r)(t)=Gwy̲(t)+Pb̲uo(t).
Now, we denote
(28)y̲(t)=[y̲p(t)y̲q(t)],Pb̲≜b̲̃=[b̲̃pb̲̃q],
where p+q=n.
Taking into account the following expressions, that is,
(29)[Ip𝕆p,q𝕆q,pHq][y̲p(r)(t)y̲q(r)(t)]=[Jp𝕆p,q𝕆q,pIq][y̲p(t)y̲q(t)]+[b̲̃pb̲̃q]uo(t),
we arrive easily at (23) and (24).System (23) is standard form of nonhomogeneous  higher-order linear differential equations of Apostol-Kolodner type, which may be treated by methods that are more classical; see for instance [18] and references therein.Thus, it is convenient to define new variables as(30)z̲1(t)=y̲p(t),z̲2(t)=y̲p′(t),⋮z̲r(t)=y̲p(r-1)(t).
Then, we have the following system of ordinary differential equations:(31)z̲1′(t)=z̲2(t),z̲2′(t)=z̲3(t),⋮z̲r′(t)=Jpz̲1(t)+b̲̃puo(t).
Finally, (31) can be expressed by using vector-matrix equations(32)z̲′(t)=Rz̲(t)+L̲uo(t),
where z̲(t)=[z̲1t(t)z̲2t(t)⋯z̲rt(t)]t (()t is the transpose tensor) and the coefficient matrices R and L̲ are given by(33)R=[𝕆pIp𝕆p⋯𝕆p𝕆p𝕆pIp⋯𝕆p⋮⋮⋮⋱⋮𝕆p𝕆p𝕆p⋯IpJp𝕆p𝕆p⋯𝕆p]∈ℳrp,L̲=[0̲p0̲p⋮b̲̃p]∈ℳ(rp)1,
with corresponding dimension of z̲(t), rp×1.The state of system (1) at time 0- is x̲(0-)=[00⋯0]t and at time 0+ it achieves x̲(0+)=[x1x2⋯xn]t.Now, considering (25), at time 0- we have(34)y̲(0-)=Q-1x̲(0-)=y̲′(0-)=Q-1x̲′(0-)=⋯=y̲(r-1)(0-)=Q-1x̲(r-1)(0-)=0̲t,
and at time 0+ we obtain(35)y̲(0+)=Q-1x̲(0+)=[χ10χ20⋯χn0]t,y̲′(0+)=Q-1x̲′(0+)=[χ11χ21⋯χn1]t,…,y̲(r-1)(0+)=Q-1x̲(r-1)(0+)=[χ1r-1χ2r-1⋯χnr-1]t.
Moreover,
(36)y̲p(i)(0-)=0̲pt∈ℳp,y̲p(i)(0+)=[χ1iχ2i⋯χpi]t∈ℳp,y̲q(i)(0-)=0̲qt∈ℳq,y̲q(i)(0+)=[χp+1iχp+2i⋯χni]t∈ℳq,
where χji∈𝔽 for i=0,1,2,…,r-1, and j=1,2,…,n.Furthermore,(37)z̲(0-)=[z̲1t(0-)z̲2t(0-)⋯z̲rt(0-)]t=[y̲pt(0-)y̲p′t(0-)⋯y̲p(r-1)t(0-)]t=0̲rpt,z̲(0+)=[z̲1t(0+)z̲2t(0+)⋯z̲rt(0+)]t=[y̲pt(0+)y̲p′t(0+)⋯y̲p(r-1)t(0+)]t=[χ10χ20⋯χp0χ11χ21⋯χp1⋯χ1r-1χ2r-1⋯χpr-1]t.


### 3.2. The Solution of Subsystem (32)

In order to solve subsystem (32), the following definitions should be provided.Definition 4.
The characteristic polynomial of matrixR is given by
(38)φ(λ)=∏i=1κ(λ-λ1)τi,
with λi≠λj for i≠j and ∑i=1κτi=rp. Without loss of generality, we assume that
(39)d1=τ1,d2=τ2,…,dl=τl,for1<l<κ,dl+1<τl+1,…,dk<τκ,
where di,τi,i=1,2,…,κ is the geometric and algebraic multiplicity of the given eigenvalues λi, respectively.Generally speaking, the matrixR is not diagonalizable. However, we can generate rp linearly independent vectors v̲1,v̲2,…,v̲rp and rp×rp similarity transformation C=[v̲1,v̲2,…,v̲rp] that takes R into the Jordan canonical form, as the following definition clarifies.Definition 5.
There exists an invertible matrixC∈ℳrp such as J=C-1RC, J∈ℳrp; is it Jordan canonical form of matrix R. Analytically,
(40)J=Jo⨁Jl+1⨁Jl+2⨁⋯⨁Jκ.(i)
The block diagonal matrixJo=J1⨁J2⨁⋯⨁Jl, where
(41)Ji=[λiλi⋱λi]∈ℳτi,
is also a diagonal matrix with diagonal elements the eigenvalue λi, for i=1,…,l. Consequently, the dimension of Jo is s×s,s≜∑i=1lτi.(ii)
Also, the block matrixJj=Jj,1⨁Jj,2⨁⋯⨁Jj,dj∈ℳτj, where(42)Jj,zj=[λj1λj1λj⋱⋱1λj]∈ℳzjforj=l+1,l+2,…,k,zj=1,2,…,dj.According to the classical theory of ordinary differential equations, the solution of system (32) is given by the following lemma.Lemma 6.
The solution of subsystem (32) is given by
(43)z̲(t)=C∫-∞teJ(t-τ)b̲uo(τ)dτ,
with initial condition z̲(0-)=0̲rpt.Proof.
Consider the transformation(44)z̲(t)=Cζ̲(t),
where nonsingular C∈ℳrp, and ζ̲(0-)=C-1z̲(0-)=0̲rpt.
Substituting (44) into (43), we obtain
(45)Cζ̲′(t)=RCζ̲(t)+L̲uo(t)⇔ζ̲′(t)=C-1RCζ̲(t)+C-1L̲uo(t).
Furthermore, we define b̲≜C-1L̲, such as the last equation is transformed into
(46)ζ̲′(t)=Jζ̲(t)+b̲uo(t).
Now, according to the relevant theory of first-order differential systems of (46) form, see for instant [16], using also (44), the solution is expressed  by (43).Definition 7.
The exponential matrixeJ(t-τ) is defined as
(47)eJ(t-τ)=eJo(t-τ)⨁eJl+1(t-τ)⨁eJl+2(t-τ)⨁⋯⨁eJκ(t-τ)∈ℳrp,
where
(48)eJo(t-τ)=eJ1(t-τ)⨁eJ2(t-τ)⨁⋯⨁eJl(t-τ)∈ℳs,eJi(t-τ)=eλi(t-τ)⨁eλi(t-τ)⨁⋯⨁eλi(t-τ)∈ℳτi,
for i=1,…,l.
Furthermore,
(49)eJj(t-τ)=eλj1(t-τ)⨁eλj2(t-τ)⨁⋯⨁eλjdj(t-τ)∈ℳτj,
where
(50)eJj,zj(t-τ)=eλj(t-τ)[1t-τ⋯(t-τ)μzj-1(μzj-1)!01⋯(t-τ)μzj-2(μzj-2)!⋮⋮⋱⋮00⋯1]∈ℳμzj,
where μzj are the Weyr characteristics via Ferrer diagrams, for j=l+1,l+2,…,k and zj=1,2,…,dj. Note that ρj=maxzi=1,2,…,djμzj is the index of annihilation for the eigenvalue λj.
Now, taking into consideration (21) the solution (22) is transposed into
(51)z̲(t)=limσ→0{CeJt∫-∞te-Jτb̲u(τ)dτ},
or equivalently
(52)z̲(t)=C[limσ→0{eJot∫-∞te-Joτb̲su(τ)dτ}limσ→0{eJl+1t∫-∞te-Jl+1τb̲τl+1u(τ)dτ}⋮limσ→0{eJkt∫-∞te-Jκτb̲τκu(τ)dτ}],
where b̲=[b̲stb̲τl+1t⋯b̲τκt]t.Remark 1.
[4, 5] Given the large number of terms involved, in order to make our calculations affordable, we consider the fact that Φ(t/σ) and its derivatives tend to zero very strongly with t→0 (note also that σ→0). Thus, by letting t=K·σ, where K is chosen large enough (i.e., K→∞) the assumption as stated above is valid, that is,
(53)Φ(tσ)=t=KσΦ(K)→0,
and its derivatives Φ(i)(t/σ)=t=KσΦ(i)(K)→K→∞0, for i=0,1,2,…,k-1.Now, using the Remark1, we obtain that limσ→0eJit=t=KσI, for i=1,2,…,l,l+1,l+2,…,k, and Φ-1(K+λiσ)→Φ-1(K), as well. Moreover, for the analytic determination of vector (3), we need to calculate the integral ∫-∞te-Jτb̲u(τ)dτ.In this part of the paper, two subsystems are derived; see the following lemmas, and AppendicesA and B.Lemma 8.
For the diagonal matrixJo, the following integral is given by (54):
(54)limσ→0∫-∞te-J0tb̲su(τ)dτ=Φ-1(K)[b̲τ1∑k=0n-1αkλ1k⋮b̲τi∑k=0n-1αkλik⋮b̲τl∑k=0n-1αkλlk]∈ℳs1.Proof.
See AppendixA.Lemma 9.
For the diagonal matrixJj=Jj1⨁Jj2⨁⋯⨁Jjdj∈ℳτj, the following integral is given by (55):
(55)limσ→0∫-∞te-Jjtb̲τju(τ)dτ=Φ-1(K)[Vj1Vj2⋮Vjzj⋮Vjdj]·a̲,
where
(56)Vjzj≜[∑c=1μzjbjzjc1(c-1)!·dc-1λj0dλjc-1∑c=1μzjbjzjc1(c-1)!·dc-1λjdλjc-1⋯∑c=1μzjbjzjc1(c-1)!·dc-1λjn-1dλjc-1∑c=2μzjbjzjc1(c-2)!·dc-2λj0dλjc-2∑c=2μzjbjzjc1(c-2)!·dc-2λjdλjc-2⋯∑c=2μzjbjzjc1(c-2)!·dc-2λjn-1dλjc-2⋮⋮⋱⋮∑c=μzjμzjbjzjc1(c-μzj)!·dc-μzjλj0dλjc-μzj∑c=μzjμzjbjzjc1(c-μzj)!·dc-μzjλjdλjc-μzj⋯∑c=μzjμzjbjzjc1(c-μzj)!·dc-μzjλjn-1dλjc-μzj],a̲=[α0α1α2⋮αn-1].Proof.
See AppendixB.Now, we revisit (54), thus
(57)Φ-1(K)[b̲τ1∑k=0n-1αkλ1k⋮b̲τi∑k=0n-1αkλik⋮b̲τl∑k=0n-1αkλlk]=Φ-1(K)(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl)[∑k=0n-1αkλ1k∑k=0n-1αkλ2k⋮∑k=0n-1αkλlk]=Φ-1(K)(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl)×[1λ1λ12⋯λ1n-11λ2λ22⋯λ2n-1⋮⋮⋮⋱⋮1λlλl2⋯λln-1]︸Vl[α0α1α2⋮αn-1]=Φ-1(K)(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl)Vla̲.
Note that Vl is the well-known Vandermonde matrix.Combining (52)–(57), we obtain the following system:(58)C-1z̲(0+)=Φ-1(K)×(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl︸l⨁Iτl+1⨁⋯⨁Iτκ)×[VlV(l+1)1⋮V(l+1)dl+1⋮Vj1⋮Vjdj⋮Vκ1⋮Vκdκ]a̲,
where Vl∈ℳsn and Vjzj∈ℳμzjn with j=l+1,l+2,…,κ and zj=1,2,…,dj. Furthermore, we obtain the initial condition(59)C-1z̲(0+)=[z̲lt(0+)z̲(l+1)1t(0+)⋯z̲(l+1)dl+1t(0+)⋯z̲j1t(0+)⋯z̲jdjt(0+)⋯z̲κ1t(0+)⋯z̲κdκt(0+)].The system (58) can now be divided into two types of subsystems:(S1)
(60)1Φ-1(K)z̲l(0+)=(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl)Vla̲,(S2)
(61)1Φ-1(K)z̲jzj(0+)=Vjzja̲,
for j=l+1,l+2,…,κ and zj=1,2,…,dj.Proposition 10.
System (60) is solvable if
(62)zi1bi1=zi2bi2=⋯=ziτibiτi,
for every nonzero element of vectors b̲τ1,b̲τ2,…,b̲τl, where z̲i(0+)=[zi1zi2⋯ziτi]t, for every i=1,2,…,l.
Moreover, if one of the elements of vectorsb̲τ1,b̲τ2,…,b̲τl is zero, then the relative element of the ith-row of the vector z̲τi(0+) should be also zero.Proof.
System (60) contains l-subsystems of the following type:
(63)(0̲τ1⨁⋯⨁b̲τi⨁⋯⨁0̲τl)Vlα̲=1Φ-1(K)(Iτ1⨁⋯⨁Iτi⨁⋯⨁Iτl)z̲l(0+),
or equivalently
(64)[bi1bi1λi⋯bi1λin-1bi2bi2λi⋯bi2λin-1⋮⋮⋱⋮biτibiτiλi⋯biτiλin-1][α0α1⋮αn-1]=1Φ-1(K)[zi1(0+)zi2(0+)⋮ziτi(0+)].(i)
If one of the coefficientsbiγ=0,for γ= 1, 2,…, τi, then the relative element of the ith row of the vector z̲τi(0+) should be also zero (in order to obtain solution).(ii)
If every of the coefficientsbiγ≠0, for γ= 1, 2,…, τi, then we have(65)bi1(α0+α1λi+⋯+αn-1λin-1)=zi11Φ-1(K)bi2(α0+α1λi+⋯+αn-1λin-1)=zi21Φ-1(K)⋮biτi(α0+α1λi+⋯+αn-1λin-1)=ziτi1Φ-1(K)}⇔α0+α1λi+⋯+αn-iλin-1=zi1bi11Φ-1(K)⋮α0+α1λi+⋯+αn-iλin-1=ziτibiτi1Φ-1(K)}⇒zi1bi1=zi2bi2=⋯=ziτibiτi
for every i=1,2,…,l.
Consequently, system (60) is solvable if (62) is satisfied for every i=1,2,…,l.Now, we will work with subsystems (61) which can be written as follows:(66)[∑c=1μzjbjzjc(c-1)!dc-1λj0dλjc-1∑c=1μzjbjzjc(c-1)!dc-1λjdλjc-1∑c=1μzjbjzjc(c-1)!dc-1λj2dλjc-1⋯∑c=1μzjbjzjc(c-1)!dc-1λjn-1dλjc-1⋮⋮⋮⋱⋮bjzjμzj-1bjzjμzj-1λj+bjzjμzjbjzjμzj-1λj2+2λjbjzjμzj⋯bjzjμzj-1λjn-1+(n-1)λjn-2bjzjμzjbjzjμzjbjzjμzjλj,bjzjμzjλj2⋯bjzjμzjλjn-1]α̲=1Φ-1(K)z̲jzj(0+).The coefficient matrixVjzjcan be transposed to the following equivalent matrix; see (67). Note that Vjzj is a μzj×n matrix. Moreover, from the μzj elements bjzjc, for c=1,2,…,μzj, we can assume that bjzjμzj=bjzjμzj-1=⋯=bjzjμzj-μ=0. Thus, we obtain(67)Vjzj*=[000⋯11(μzj-μ)!dμzj-μλjμzj+1-μdλjμzj-μ⋯1(μzj-μ)!dμzj-μλjn-1dλjμzj-μ⋮⋮⋮⋯⋮⋱⋮001⋯⋯12!d2λjn-1dλj2012λj⋯⋯(n-1)λjn-11λjλj2⋯⋯λjn-1000⋯00⋯0⋮⋮⋮⋱⋮⋮⋮⋮000⋯00⋯0].In order to be able to understand (67) better, the following example is helpful.Example 1.
We take the coefficient matrixV(68)[b3b3λ+b4b3λ2+2λb4+b5b3λ3+3λ2b4+3λb5+b6b3λ4+4λ3b4+6λ2b5+2λb6b4b4λ+b5b4λ2+2λb5+b6b4λ3+3λ2b5+3λb6b4λ4+4λ3b5+6λ2b6b5b5λ+b6b5λ2+2λb6b5λ3+3λ2b6b5λ4+4λ3b6b6b6λb6λ2b6λ3b6λ4].
We assume that b6=0 and b5≠0. Since b5≠0, we are not interested in b3, b4.
Thus, under our assumption, we obtain
(69)[b3b3λ+b4b3λ2+2λb4+b5b3λ3+3λ2b4+3λb5b3λ4+4λ3b4+6λ2b5b4b4λ+b5b4λ2+2λb5b4λ3+3λ2b5b4λ4+4λ3b5b5b5λb5λ2b5λ3b5λ400000].
Afterwards, we multiply the 3rd row with -b4/b5 and it is added to the 2nd row. Moreover, we multiply the 3rd row with -b3/b5 and it is added with the 1st row. Then, the following equivalent matrix is derived:
(70)[0b42λb4+b53λ2b4+3λb54λ3b4+6λ2b50b52λb53λ2b54λ3b5b5b5λb5λ2b5λ3b5λ400000].
Now, we multiply the 2nd row with -b4/b5 and it is added with the 1st row. Thus, we obtain
(71)[00b53λb56λ2b50b52λb53λ2b54λ3b5b5b5λb5λ2b5λ3b5λ400000].
Finally, we multiply the matrix with the element 1/b5and we conclude to obtain
(72)[0013λ6λ2012λ3λ24λ31λλ2λ3λ400000]=[(1)′′2!(λ)′′2!(λ2)′′2!(λ3)′′2!(λ4)′′2!(1)′1!(λ)′1!(λ2)′1!(λ3)′1!(λ4)′1!1λλ2λ3λ400000].Remark 2.
Considering the results already presented, it is clear that there exists a nonsingular matrixLjzj, μzj×μzj such that
(73)LjzjVjzj=Vjzj*.
Thus, system (61) can be transformed into
(74)1Φ-1(K)Ljzjz̲jzj(0+)=LjzjVjzja̲⇔Vjzj*a̲=z̲jzj*(0+),
where
(75)z̲jzj*(0+)=1Φ-1(K)Ljzjz̲jzj(0+),∀j=l+1,l+2,…,κ,∀zj=1,2,…,dj.Proposition 11.
Subsystems (74) are solvable when the elements of vectors
(76)z̲jzj*(0+),foreveryzj=1,2,…,dj,j=l+1,l+2,…,κ
are included into the vector with the greatest dimension, that is, ρj× 1 where
(77)ρj=maxμzj,zj=1,2,…,dj.
Equivalently, if one assumes that
(78)z̲j1*(0+)=[γ1γ2⋮γρj],
then each of the following vectors z̲j2*(0+), z̲j3*(0+),…, z̲jdj*(0+) should be vectors of [γ1γ2⋮γξ] type  with ξ≤ρj.Proof.
We take thedj subsystems; as follows:
(79)Vj1*a̲=z̲j1*(0+),⋯Vjdj*a̲=z̲jdj*(0+).
The matrixVj1* has μ1 rows,Vj2* has μ2 rows,⋮Vjdj* has μdjrows.
Without loss of generality, we assume that μ1≤μ2≤⋯≤μdj=ρj.
Looking carefully at the type of matrices
(80)Vj1*,Vj2*,…,Vjdj*,
we can easily verify that the μ1-first rows of them are identically same. Thus, it is necessary the relevant μ1-first rows of z̲j1*(0+),z̲j2*(0+),…,z̲jdj*(0+) to be also identically the same. Analogously, the μ2-first rows of Vj1*, Vj2*,…,Vjdj* should be identically the same with the relevant μ2-first rows of z̲j1*(0+),z̲j2*(0+),…,z̲jdj*(0+). And so on until the μdj=ρj row.(81)Φ-1(K)(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl︸l⨁1Φ-1(K)[Iτl+1⨁⋯⨁Iτκ]b̲τ1⨁b̲τ2⨁⋯⨁b̲τl︸l)×[VlV(l+1)1*⋮V(l+1)dl+1*⋮Vj1*⋮Vjdj*⋮Vκ1*⋮Vκdκ*]a̲=[z̲l(0+)z̲(l+1)1*(0+)⋮z̲(l+1)dl+1*(0+)⋮z̲j1*(0+)⋮z̲jdj*(0+)⋮z̲κ1*(0+)⋮z̲κdκ*(0+)].Consequently, it is time to use the results of Propositions10 and 11. Thus, system (81) is solvable if we obtain the following.(i)
The first nonzero elements of vector coefficientb̲τ1,b̲τ2,…,b̲τl (practically speaking, without loss of generality, we assume that the first elements are nonzero).(ii)
The matrixV(q+1)zq+1* with dimension ρl+1×n, where ρl+1=maxμzl+1,zl+1=1,2,…,dl+1.⋮(iii)
The matrixVjzj* with dimension ρj×n, whereρj=maxμzj,  zj=1,2,…,dj.⋮(iv)
The matrixVκzκ* with dimension ρκ×n, whereρκ=maxμzκ, zκ=1,2,…,dκ.Consequently, system (81) is transposed to the solvable system (83)(82)Φ-1(K)(bτ1⨁bτ2⨁⋯⨁bτl︸l⨁1Φ-1(K)[Iτl+1⨁⋯⨁Iτκ]bτ1⨁bτ2⨁⋯⨁bτl︸l)×[VlV(l+1)zl+1*V(l+2)zl+2*⋮Vκdκ*]a̲=[z̲l(0+)z̲(l+1)zl+1*(0+)z̲(l+2)z2*(0+)⋮z̲κzκ*(0+)]
or equivalently,(83)[VlV(l+1)zl+1*V(l+2)zl+2*⋮Vκdκ*]a̲=[z̲l*(0+)z̲(l+1)zl+1*(0+)z̲(l+2)z2*(0+)⋮z̲κzκ*(0+)],
where z̲l*(0+)=(1/Φ-1(K))(1/bτ1⨁1/bτ2⨁⋯⨁1/bτl︸l)z̲l(0+), since (bτ1⨁bτ2⨁⋯⨁bτl) is  nonsingular.Remark 3.
The matrices given by (67) with some row transformations can be transformed to the following:
(84)[1λjλj2λj3⋯⋯λjn-1012λj3λj2(n-1)λjn-20013λj(n-1)(n-2)2!λjn-3⋮011(μzj-μ)!dμzj-μλjμzj-μ+1dλjμzj-μ⋯1(μzj-μ)!dμzj-μλjn-1dλjμzj-μ000⋮⋮⋮000].
The matrix (84) is denoted by Vjzj**. Thus, there is a nonsingular matrix Λjzjsuch as
(85)ΛjzjVjzj*=Vjzj**.
Then, subsystems (74) are transformed to
(86)ΛjzjVjzj*a̲=Λjzjz̲jzj*(0+)⇔Vjzj**a̲=z̲jzj**(0+),
where Λjzjz̲jzj*(0+)=z̲jzj**(0+) for every zj=1,2,…,dj, and j=l+1,l+2,…,κ. Consequently, system (83) is transformed to
(87)[VlV(l+1)zl+1**V(l+2)zl+2**⋮Vκdκ**]a̲=[z̲l*(0+)z̲(l+1)zl+1**(0+)z̲(l+2)z2**(0+)⋮z̲κzκ**(0+)].Comment 1.
Without loss of generality, we can assume that the matricesVjdj** do not contain zero rows. On contrary, if there exist some zero rows, the solvability of system (87) is not affected; only the number of nonzero rows which are included in (87) is  changed.Remark 4.
System (87) can be further transposed to a more convenient system. Analytically, if we multiply the 1st row of Vandermonde matrix Vl, that is, [1λ1λ12⋯λ1n-1] with the number (−1) and it is added to the 1st row of each of V(l+1)zl+1**,V(l+2)zl+2**,…,Vκzκ**, then Vjzj** is given by
(88)[0λj-λ1λj2-λ12⋯λjn-1-λ1n-1012λj⋯(n-1)λjn-2⋮1⋮⋱00⋯01,1(ρj-1)!dρj-1λjρj+1dλjρj-1,…,1(ρj-1)!dρj-1λjn-1dλjρj-1].
(Note that we have assumed that the matrices Vjzj**, for j=l+1,l+2,…,κ, do not cotain zero rows; see also Comment)We can easily see that the 1st row of matrix (88) can be rewritten as what follows.The  elementλjκ+λjκ-1λ1+⋯+λjλ1κ-1+λ1κ=∑m1,m2=0,∑i=12mi=κκλjm1λ1m2.Thus, the first row is presented as(89)(λj-λ1)[01∑m1,m2=0∑i=12mi=11λjm1λ1m2∑m1,m2=0∑i=12mi=22λjm1λ1m2⋯∑m1,m2=0∑i=12mi=n-2n-2λjm1λ1m2].Since, the elementλj-λ1≠0, we can multiply by left the equation (88) with a properly chosen transformation matrix, so as to obtain(90)[01∑m1m2=0∑i=12mi=11λjm1λ1m2⋯∑m1m2=0∑i=12mi=n-2n-2λjm1λ1m2012λj⋯(n-1)λjn-2O0⋯11(ρj-1)!dρj-1λjρj+1dλjρj-1⋯1(ρj-1)!dρj-1λjn-1dλjρj-1].Finally, this subsection is concluded by considering system (91), where the matrices Ajzjfor j=l+1,l+2,…,κ are derived by taking into account a properly chosen transformation left-matrix Z as follows:(91)Z[VlV(l+1)zl+1**V(l+2)zl+2**⋮Vκdκ**]a̲=Z[z̲l*(0+)z̲(l+1)zl+1**(0+)z̲(l+2)z2**(0+)⋮z̲κzκ**(0+)]⇔[VlAl+1Al+2⋮Aκ]a̲=[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)],
where(92)[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)]=Z[z̲l*(0+)z̲(l+1)zl+1**(0+)z̲(l+2)z2**(0+)⋮z̲κzκ**(0+)],
with
(93)Vl∈ℳln,Vj∈ℳjnforj=l+1,l+2,…,κ,z̲l*(0+)∈ℳl1,d̲l+2(0+)∈ℳj1,forj=l+1,l+2,…,κ.ρj=maxzi=1,2,…,djμzj is the index of annihilation for the eigenvalue λj.Consequently, the system (91) contains the following subsystems(94)Vla̲=z̲l*(0+),Al+1a̲=d̲l+1(0+),⋯Aκa̲=d̲κ(0+)},
where Vl, Aj for j=l+1,l+2,…,κ are nonsquare matrices.In order to solve system (91), we should use the relevant theory of generalized inverses. Some basic elements are briefly presented in the next subsection.


### 3.3. Solving Subsystem (91)

This subsection is beginning with the brief presentation of the most basic results of{1}-generalized inverses.Definition 12.
For a matrixA∈ℳpq, the X∈ℳqp is called {1}-inverses of matrix A, if
(95)AXA=A.The following proposition provides the means for finding{1}-inverses; see [19, 20] for proofs and details.Proposition 13.
LetA∈ℳpq, with rankA=r<min{p,q} and P∈ℳp, Q∈ℳq, invertible such that
(96)PAQ=Ir⨁𝕆p-r,q-r.
Then, every X, {1}-inverse of matrixA, is written as follows:
(97)X=Q(Ir⨁𝕆p-r,q-r)P,
for arbitrary L∈ℳ(q-r)(p-r).Definition 14.
We define any{1}-inverse of matrixA byA(1).The following result is due to [6].Theorem 15.
LetA∈ℳpq, B∈ℳdν, D∈ℳpν. Then the matrix equation
(98)AXB=D,
is consistent if and only if for some A(1), B(1)(99)AA(1)DB(1)B=D
and the general solution is given by
(100)X=A(1)DB(1)+Y-A(1)AYBB(1)
for arbitrary Y∈ℳpd.The following characterization of the set ofA(1) is due to [20].Corollary 16.
LetA∈ℳpq and A(1)∈ℳqp, then any other {1}-inverse Ã(1)∈ℳqp of matrixA is given by
(101)Ã(1)=A(1)+Z-A(1)AZAA(1)
for some Z∈ℳqp.A special case of{1}-inverses is the left/right inverses of a full column/row rank matrix, respectively. The proofs and further details are presented extensively by [19, 20].Definition 17.
Consider the following matrices.(a)
LetPi(α) be a m×m-matrix(102)Pi(α)=[1𝕆1⋱α1⋱𝕆1].
(The nonzero element a is in the ith row and the ith column.)
Thus, whenever a matrixA is multiplied from the left by Pi(α), then its ith row is multiplied by the nonzero number a.
(b)
LetPi(j,a) be an m×m-matrix(103)Pi(j,α)=[101⋱1⋯α⋱⋮1⋱01].
(The nonzero element a is in the ith row and the jth column.)
Thus, whenever a matrixA is multiplied from the left by Pi(j,a), then its jth row is multiplied by the nonzero number a and it is added to thejth row of A.
(c)
LetQi(a) be an n×n-matrix(104)Qi(α)=[11⋱1α1⋱1].
(The nonzero element a is in the ith-row and the ith-column).
Thus, whenever a matrixA is multiplied from the right by Qi(α), then its ith column is multiplied by the nonzero number a.
(d)
LetQi(j,a) be an n×n-matrix(105)Qi(j,α)=[11⋱1⋮1⋱α⋯1⋱1].
(The nonzero element a is in the jth row and the ith column.)
Thus, whenever a matrixA is multiplied from the right by Qi(j,a), then its ith column is multiplied by the nonzero number a and it is added to thejth column of A.Now, in order to calculate the generalized inverses of Vandermonde and the relative to it matrices, we use the recent results of paper [6]. In this context, the following propositions are relevant.Proposition 18.
The{1}-inverse of the Vandermonde matrix is given by
(106)Vl(1)=Q[Il𝕆n-l,l]P,
where
(107)P=∏s=l-11∏j=ls+1Pj(1λj-λs)Pj(s,-1)∈ℳl,
(the multiplication counts backwards, starting from l→l-1→⋯ etc.)
(108)Q=∏s=1l∏r=s+1nQr(s,-∑m1,…,ms=0m1+⋯+ms=r-sr-s∏ζ=1sλζmζ)∈ℳn,
and the general solution of system Vla̲=z̲l*(0+) is given by
(109)a̲=Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲,
for an arbitrarily chosen vector ψ̲∈ℳn1.Proof.
The proof is a straightforward application of Theorem15. Analytically,
(110)PVlQ=[Il𝕆l,n-l]⇔Vl=P-1[Il𝕆l,n-l]Q-1.
In order for the matrix Q[Il𝕆n-l,l]P to be the {1}-inverse of Vl, we should prove the following equality
(111)VlQ[Il𝕆n-l,l]PVl=Vl.
Thus,
(112)VlQ[Il𝕆n-l,l]PVl=P-1[Il𝕆l,n-l]Q-1Q[Il𝕆n-l,l]PP-1[Il𝕆l,n-l]Q-1=P-1[Il𝕆l,n-l][Il𝕆n-l,l][Il𝕆l,n-l]Q-1=P-1(Il·Il+𝕆l,n-l·𝕆n-l,l)[Il𝕆l,n-l]Q-1=P-1[Il𝕆l,n-l]Q-1=Vl.
Hence, the solution of system Vla̲=z̲l*(0+) is given by (113).Proposition 19.
The{1}-inverse of the matrix Al for j=l+1,l+2,…,κ is given by
(113)Aj(1)=Qj[0̲pjtIpj𝕆n-pj-1,pj]Pj,
where
(114)Pj=∏s=2pjPpj-s+2(1λj-λ1)Ppj-s+2(pj-s+1,-1),Qj=∏s=2pj+1∏m=s+1nQm(s,-1(s-2)!ds-2dλjs-2(∑m1,m2=0m1+m2=m-2m-2λ1m1λjm2)).
for j=l+1,l+2,…,κ, and the general solution of systems Aja̲=d̲j(0+) is given by
(115)a̲=Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲j
for an arbitrarily chosen vector ψ̲j∈ℳn1 for j=l+1,l+2,…,κ.Proof.
The proof is also a straightforward application of Theorem15. Analytically,
(116)PjAjQj=[0̲pjIpj𝕆pj,n-pj-1]⇔Aj=Pj-1[0̲pjIpj𝕆pj,n-pj-1]Qj-1.
In order for the matrix
(117)Qj[0̲pjtIpj𝕆n-pj-1,pj]Pj
to be the {1}-inverse of Aj, we should prove the following equality:
(118)AjQj[0̲pjtIpj𝕆n-pj-1,pj]PjAj=Aj.
Thus,
(119)AjQj[0̲pjtIpj𝕆n-pj-1,pj]PjAj=Pj-1[0̲pjIpj𝕆pj,n-pj-1]Qj-1Qj[0̲pjtIpj𝕆n-pj-1,pj]PjPj-1×[0̲pjIpj𝕆pj,n-pj-1]Qj-1=Pj-1[0̲pjIpj𝕆pj,n-pj-1][0̲pjtIpj𝕆n-pj-1,pj]×[0̲pjIpj𝕆pj,n-pj-1]Qj-1=Pj-1(0̲pj·0̲pjt+Ipj·Ipj+𝕆pj,n-pj-1·𝕆n-pj-1,pj)×[0̲pjIpj𝕆pj,n-pj-1]Qj-1=Pj-1[0̲pjIpj𝕆pj,n-pj-1]Qj-1=Aj.
Hence, the solution of system Aja̲=d̲j(0+) is given by (115).


### 3.4. The Solution of System (91)

This subsection concludes the whole discussion. The following theorem holds.Theorem 20.
System (91) is solvable, that is,
(120)Aa̲=b̲,
where
(121)A≜[VlAl+1Al+2⋮Aκ],b̲≜[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)],
if the following conditions are satisfied:
(122)Vl[Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲]=d̲l,(123)Aj[Vl(1)z̲l(0+)+(In-Vl(1)Vl)ψ̲]=d̲j,(124)Ai[Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲]=d̲i
for i≠j
and i,j=l+1,l+2,…,κ.Proof.
Following Propositions18 and 19, we have      (i)
for systemVla̲=z̲l*(0+), the solution is given by a̲=Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲,(ii)
for systemsAja̲=d̲j(0+), the solution is given by a̲=Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲ for arbitrarily chosen vector ψ̲∈ℳn1 for j=l+1,l+2,…,κ.
There are infinitely many solutions, since the vectorψ̲∈ℳn1 is arbitrary. Moreover, each of the above subsystems also contains the solution of system Aa̲=b̲. Thus, we assume that the desired solution is given by a̲, then it should also satisfy the first system, that is, Vla̲=z̲l*(0+). Thus, we take
(125)a̲=Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲,Aa̲=b̲⇔A(Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲)=b̲⇔AVl(1)z̲l*(0+)+A(In-Vl(1)Vl)ψ̲=b̲⇔AVl(1)z̲l*(0+)+(A-AVl(1)Vl)ψ̲=b̲⇔[VlAl+1Al+2⋮Aκ]Vl(1)z̲l*(0+)+([VlAl+1Al+2⋮Aκ]-[VlAl+1Al+2⋮Aκ]Vl(1)Vl)ψ̲≜[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)]⇔[z̲l*(0+)Al+1Vl(1)z̲l*(0+)Al+2Vl(1)z̲l*(0+)⋮AκVl(1)z̲l*(0+)]+[𝕆lAl+1(In-Vl(1)Vl)Al+2(In-Vl(1)Vl)⋮Aκ(In-Vl(1)Vl)]ψ̲=[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)]⇔AjVl(1)z̲l*(0+)+Aj(In-Vl(1)Vl)ψ̲=d̲j(0+)⇔Aj[Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲]=d̲j(0+)
for every j=l+1,l+2…,κ. So (123) has been derived. Analogously, the system Aa̲=b̲ should also satisfy every subsystem Aja̲=d̲j(0+) forj=l+1,l+2,…,κ.
Consequently, we have that
(126)a̲=Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲,foreveryj=l+1,l+2,…,κ,Aa̲=b̲⇔A(Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲)=b̲⇔AAj(1)d̲j(0+)+A(In-Aj(1)Aj)ψ̲=b̲⇔[VlAl+1Al+2⋮Aκ]Aj(1)d̲j(0+)+([VlAl+1Al+2⋮Aκ]-[VlAl+1Al+2⋮Aκ]Aj(1)Aj)ψ̲≜[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)]⇔[VlAj(1)d̲j(0+)Al+1Aj(1)d̲j(0+)Al+2Aj(1)d̲j(0+)⋮AκAj(1)d̲j(0+)]+[Vl(In-Aj(1)Aj)Al+1(In-Aj(1)Aj)Al+2(In-Aj(1)Aj)⋮Aκ(In-Aj(1)Aj)]ψ̲=[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)].
From the above equalities, the expressions (122)–(124) are obtained.With this important theorem, the discussion of system (32) has finished. In the next section, we present the solution of system (24).


## 3.1. Order Reduction of System (1)

The section begins with the following important lemma.Lemma 3.
System (1) is divided into the following two subsystems:
(23)y̲p(r)(t)=Jpy̲p(t)+b̲̃puo(t),(24)Hqy̲q(r)(t)=y̲q(t)+b̲̃quo(t).Proof.
Consider the transformation(25)x̲(t)=Qy̲(t).
Substituting the previous expression into (1) and considering also (3), we obtain
(26)FQy̲(r)(t)=GQy̲(t)+b̲uo(t).
Multiplying by P, we arrive at
(27)Fwy̲(r)(t)=Gwy̲(t)+Pb̲uo(t).
Now, we denote
(28)y̲(t)=[y̲p(t)y̲q(t)],Pb̲≜b̲̃=[b̲̃pb̲̃q],
where p+q=n.
Taking into account the following expressions, that is,
(29)[Ip𝕆p,q𝕆q,pHq][y̲p(r)(t)y̲q(r)(t)]=[Jp𝕆p,q𝕆q,pIq][y̲p(t)y̲q(t)]+[b̲̃pb̲̃q]uo(t),
we arrive easily at (23) and (24).System (23) is standard form of nonhomogeneous  higher-order linear differential equations of Apostol-Kolodner type, which may be treated by methods that are more classical; see for instance [18] and references therein.Thus, it is convenient to define new variables as(30)z̲1(t)=y̲p(t),z̲2(t)=y̲p′(t),⋮z̲r(t)=y̲p(r-1)(t).
Then, we have the following system of ordinary differential equations:(31)z̲1′(t)=z̲2(t),z̲2′(t)=z̲3(t),⋮z̲r′(t)=Jpz̲1(t)+b̲̃puo(t).
Finally, (31) can be expressed by using vector-matrix equations(32)z̲′(t)=Rz̲(t)+L̲uo(t),
where z̲(t)=[z̲1t(t)z̲2t(t)⋯z̲rt(t)]t (()t is the transpose tensor) and the coefficient matrices R and L̲ are given by(33)R=[𝕆pIp𝕆p⋯𝕆p𝕆p𝕆pIp⋯𝕆p⋮⋮⋮⋱⋮𝕆p𝕆p𝕆p⋯IpJp𝕆p𝕆p⋯𝕆p]∈ℳrp,L̲=[0̲p0̲p⋮b̲̃p]∈ℳ(rp)1,
with corresponding dimension of z̲(t), rp×1.The state of system (1) at time 0- is x̲(0-)=[00⋯0]t and at time 0+ it achieves x̲(0+)=[x1x2⋯xn]t.Now, considering (25), at time 0- we have(34)y̲(0-)=Q-1x̲(0-)=y̲′(0-)=Q-1x̲′(0-)=⋯=y̲(r-1)(0-)=Q-1x̲(r-1)(0-)=0̲t,
and at time 0+ we obtain(35)y̲(0+)=Q-1x̲(0+)=[χ10χ20⋯χn0]t,y̲′(0+)=Q-1x̲′(0+)=[χ11χ21⋯χn1]t,…,y̲(r-1)(0+)=Q-1x̲(r-1)(0+)=[χ1r-1χ2r-1⋯χnr-1]t.
Moreover,
(36)y̲p(i)(0-)=0̲pt∈ℳp,y̲p(i)(0+)=[χ1iχ2i⋯χpi]t∈ℳp,y̲q(i)(0-)=0̲qt∈ℳq,y̲q(i)(0+)=[χp+1iχp+2i⋯χni]t∈ℳq,
where χji∈𝔽 for i=0,1,2,…,r-1, and j=1,2,…,n.Furthermore,(37)z̲(0-)=[z̲1t(0-)z̲2t(0-)⋯z̲rt(0-)]t=[y̲pt(0-)y̲p′t(0-)⋯y̲p(r-1)t(0-)]t=0̲rpt,z̲(0+)=[z̲1t(0+)z̲2t(0+)⋯z̲rt(0+)]t=[y̲pt(0+)y̲p′t(0+)⋯y̲p(r-1)t(0+)]t=[χ10χ20⋯χp0χ11χ21⋯χp1⋯χ1r-1χ2r-1⋯χpr-1]t.


## 3.2. The Solution of Subsystem (32)

In order to solve subsystem (32), the following definitions should be provided.Definition 4.
The characteristic polynomial of matrixR is given by
(38)φ(λ)=∏i=1κ(λ-λ1)τi,
with λi≠λj for i≠j and ∑i=1κτi=rp. Without loss of generality, we assume that
(39)d1=τ1,d2=τ2,…,dl=τl,for1<l<κ,dl+1<τl+1,…,dk<τκ,
where di,τi,i=1,2,…,κ is the geometric and algebraic multiplicity of the given eigenvalues λi, respectively.Generally speaking, the matrixR is not diagonalizable. However, we can generate rp linearly independent vectors v̲1,v̲2,…,v̲rp and rp×rp similarity transformation C=[v̲1,v̲2,…,v̲rp] that takes R into the Jordan canonical form, as the following definition clarifies.Definition 5.
There exists an invertible matrixC∈ℳrp such as J=C-1RC, J∈ℳrp; is it Jordan canonical form of matrix R. Analytically,
(40)J=Jo⨁Jl+1⨁Jl+2⨁⋯⨁Jκ.(i)
The block diagonal matrixJo=J1⨁J2⨁⋯⨁Jl, where
(41)Ji=[λiλi⋱λi]∈ℳτi,
is also a diagonal matrix with diagonal elements the eigenvalue λi, for i=1,…,l. Consequently, the dimension of Jo is s×s,s≜∑i=1lτi.(ii)
Also, the block matrixJj=Jj,1⨁Jj,2⨁⋯⨁Jj,dj∈ℳτj, where(42)Jj,zj=[λj1λj1λj⋱⋱1λj]∈ℳzjforj=l+1,l+2,…,k,zj=1,2,…,dj.According to the classical theory of ordinary differential equations, the solution of system (32) is given by the following lemma.Lemma 6.
The solution of subsystem (32) is given by
(43)z̲(t)=C∫-∞teJ(t-τ)b̲uo(τ)dτ,
with initial condition z̲(0-)=0̲rpt.Proof.
Consider the transformation(44)z̲(t)=Cζ̲(t),
where nonsingular C∈ℳrp, and ζ̲(0-)=C-1z̲(0-)=0̲rpt.
Substituting (44) into (43), we obtain
(45)Cζ̲′(t)=RCζ̲(t)+L̲uo(t)⇔ζ̲′(t)=C-1RCζ̲(t)+C-1L̲uo(t).
Furthermore, we define b̲≜C-1L̲, such as the last equation is transformed into
(46)ζ̲′(t)=Jζ̲(t)+b̲uo(t).
Now, according to the relevant theory of first-order differential systems of (46) form, see for instant [16], using also (44), the solution is expressed  by (43).Definition 7.
The exponential matrixeJ(t-τ) is defined as
(47)eJ(t-τ)=eJo(t-τ)⨁eJl+1(t-τ)⨁eJl+2(t-τ)⨁⋯⨁eJκ(t-τ)∈ℳrp,
where
(48)eJo(t-τ)=eJ1(t-τ)⨁eJ2(t-τ)⨁⋯⨁eJl(t-τ)∈ℳs,eJi(t-τ)=eλi(t-τ)⨁eλi(t-τ)⨁⋯⨁eλi(t-τ)∈ℳτi,
for i=1,…,l.
Furthermore,
(49)eJj(t-τ)=eλj1(t-τ)⨁eλj2(t-τ)⨁⋯⨁eλjdj(t-τ)∈ℳτj,
where
(50)eJj,zj(t-τ)=eλj(t-τ)[1t-τ⋯(t-τ)μzj-1(μzj-1)!01⋯(t-τ)μzj-2(μzj-2)!⋮⋮⋱⋮00⋯1]∈ℳμzj,
where μzj are the Weyr characteristics via Ferrer diagrams, for j=l+1,l+2,…,k and zj=1,2,…,dj. Note that ρj=maxzi=1,2,…,djμzj is the index of annihilation for the eigenvalue λj.
Now, taking into consideration (21) the solution (22) is transposed into
(51)z̲(t)=limσ→0{CeJt∫-∞te-Jτb̲u(τ)dτ},
or equivalently
(52)z̲(t)=C[limσ→0{eJot∫-∞te-Joτb̲su(τ)dτ}limσ→0{eJl+1t∫-∞te-Jl+1τb̲τl+1u(τ)dτ}⋮limσ→0{eJkt∫-∞te-Jκτb̲τκu(τ)dτ}],
where b̲=[b̲stb̲τl+1t⋯b̲τκt]t.Remark 1.
[4, 5] Given the large number of terms involved, in order to make our calculations affordable, we consider the fact that Φ(t/σ) and its derivatives tend to zero very strongly with t→0 (note also that σ→0). Thus, by letting t=K·σ, where K is chosen large enough (i.e., K→∞) the assumption as stated above is valid, that is,
(53)Φ(tσ)=t=KσΦ(K)→0,
and its derivatives Φ(i)(t/σ)=t=KσΦ(i)(K)→K→∞0, for i=0,1,2,…,k-1.Now, using the Remark1, we obtain that limσ→0eJit=t=KσI, for i=1,2,…,l,l+1,l+2,…,k, and Φ-1(K+λiσ)→Φ-1(K), as well. Moreover, for the analytic determination of vector (3), we need to calculate the integral ∫-∞te-Jτb̲u(τ)dτ.In this part of the paper, two subsystems are derived; see the following lemmas, and AppendicesA and B.Lemma 8.
For the diagonal matrixJo, the following integral is given by (54):
(54)limσ→0∫-∞te-J0tb̲su(τ)dτ=Φ-1(K)[b̲τ1∑k=0n-1αkλ1k⋮b̲τi∑k=0n-1αkλik⋮b̲τl∑k=0n-1αkλlk]∈ℳs1.Proof.
See AppendixA.Lemma 9.
For the diagonal matrixJj=Jj1⨁Jj2⨁⋯⨁Jjdj∈ℳτj, the following integral is given by (55):
(55)limσ→0∫-∞te-Jjtb̲τju(τ)dτ=Φ-1(K)[Vj1Vj2⋮Vjzj⋮Vjdj]·a̲,
where
(56)Vjzj≜[∑c=1μzjbjzjc1(c-1)!·dc-1λj0dλjc-1∑c=1μzjbjzjc1(c-1)!·dc-1λjdλjc-1⋯∑c=1μzjbjzjc1(c-1)!·dc-1λjn-1dλjc-1∑c=2μzjbjzjc1(c-2)!·dc-2λj0dλjc-2∑c=2μzjbjzjc1(c-2)!·dc-2λjdλjc-2⋯∑c=2μzjbjzjc1(c-2)!·dc-2λjn-1dλjc-2⋮⋮⋱⋮∑c=μzjμzjbjzjc1(c-μzj)!·dc-μzjλj0dλjc-μzj∑c=μzjμzjbjzjc1(c-μzj)!·dc-μzjλjdλjc-μzj⋯∑c=μzjμzjbjzjc1(c-μzj)!·dc-μzjλjn-1dλjc-μzj],a̲=[α0α1α2⋮αn-1].Proof.
See AppendixB.Now, we revisit (54), thus
(57)Φ-1(K)[b̲τ1∑k=0n-1αkλ1k⋮b̲τi∑k=0n-1αkλik⋮b̲τl∑k=0n-1αkλlk]=Φ-1(K)(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl)[∑k=0n-1αkλ1k∑k=0n-1αkλ2k⋮∑k=0n-1αkλlk]=Φ-1(K)(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl)×[1λ1λ12⋯λ1n-11λ2λ22⋯λ2n-1⋮⋮⋮⋱⋮1λlλl2⋯λln-1]︸Vl[α0α1α2⋮αn-1]=Φ-1(K)(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl)Vla̲.
Note that Vl is the well-known Vandermonde matrix.Combining (52)–(57), we obtain the following system:(58)C-1z̲(0+)=Φ-1(K)×(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl︸l⨁Iτl+1⨁⋯⨁Iτκ)×[VlV(l+1)1⋮V(l+1)dl+1⋮Vj1⋮Vjdj⋮Vκ1⋮Vκdκ]a̲,
where Vl∈ℳsn and Vjzj∈ℳμzjn with j=l+1,l+2,…,κ and zj=1,2,…,dj. Furthermore, we obtain the initial condition(59)C-1z̲(0+)=[z̲lt(0+)z̲(l+1)1t(0+)⋯z̲(l+1)dl+1t(0+)⋯z̲j1t(0+)⋯z̲jdjt(0+)⋯z̲κ1t(0+)⋯z̲κdκt(0+)].The system (58) can now be divided into two types of subsystems:(S1)
(60)1Φ-1(K)z̲l(0+)=(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl)Vla̲,(S2)
(61)1Φ-1(K)z̲jzj(0+)=Vjzja̲,
for j=l+1,l+2,…,κ and zj=1,2,…,dj.Proposition 10.
System (60) is solvable if
(62)zi1bi1=zi2bi2=⋯=ziτibiτi,
for every nonzero element of vectors b̲τ1,b̲τ2,…,b̲τl, where z̲i(0+)=[zi1zi2⋯ziτi]t, for every i=1,2,…,l.
Moreover, if one of the elements of vectorsb̲τ1,b̲τ2,…,b̲τl is zero, then the relative element of the ith-row of the vector z̲τi(0+) should be also zero.Proof.
System (60) contains l-subsystems of the following type:
(63)(0̲τ1⨁⋯⨁b̲τi⨁⋯⨁0̲τl)Vlα̲=1Φ-1(K)(Iτ1⨁⋯⨁Iτi⨁⋯⨁Iτl)z̲l(0+),
or equivalently
(64)[bi1bi1λi⋯bi1λin-1bi2bi2λi⋯bi2λin-1⋮⋮⋱⋮biτibiτiλi⋯biτiλin-1][α0α1⋮αn-1]=1Φ-1(K)[zi1(0+)zi2(0+)⋮ziτi(0+)].(i)
If one of the coefficientsbiγ=0,for γ= 1, 2,…, τi, then the relative element of the ith row of the vector z̲τi(0+) should be also zero (in order to obtain solution).(ii)
If every of the coefficientsbiγ≠0, for γ= 1, 2,…, τi, then we have(65)bi1(α0+α1λi+⋯+αn-1λin-1)=zi11Φ-1(K)bi2(α0+α1λi+⋯+αn-1λin-1)=zi21Φ-1(K)⋮biτi(α0+α1λi+⋯+αn-1λin-1)=ziτi1Φ-1(K)}⇔α0+α1λi+⋯+αn-iλin-1=zi1bi11Φ-1(K)⋮α0+α1λi+⋯+αn-iλin-1=ziτibiτi1Φ-1(K)}⇒zi1bi1=zi2bi2=⋯=ziτibiτi
for every i=1,2,…,l.
Consequently, system (60) is solvable if (62) is satisfied for every i=1,2,…,l.Now, we will work with subsystems (61) which can be written as follows:(66)[∑c=1μzjbjzjc(c-1)!dc-1λj0dλjc-1∑c=1μzjbjzjc(c-1)!dc-1λjdλjc-1∑c=1μzjbjzjc(c-1)!dc-1λj2dλjc-1⋯∑c=1μzjbjzjc(c-1)!dc-1λjn-1dλjc-1⋮⋮⋮⋱⋮bjzjμzj-1bjzjμzj-1λj+bjzjμzjbjzjμzj-1λj2+2λjbjzjμzj⋯bjzjμzj-1λjn-1+(n-1)λjn-2bjzjμzjbjzjμzjbjzjμzjλj,bjzjμzjλj2⋯bjzjμzjλjn-1]α̲=1Φ-1(K)z̲jzj(0+).The coefficient matrixVjzjcan be transposed to the following equivalent matrix; see (67). Note that Vjzj is a μzj×n matrix. Moreover, from the μzj elements bjzjc, for c=1,2,…,μzj, we can assume that bjzjμzj=bjzjμzj-1=⋯=bjzjμzj-μ=0. Thus, we obtain(67)Vjzj*=[000⋯11(μzj-μ)!dμzj-μλjμzj+1-μdλjμzj-μ⋯1(μzj-μ)!dμzj-μλjn-1dλjμzj-μ⋮⋮⋮⋯⋮⋱⋮001⋯⋯12!d2λjn-1dλj2012λj⋯⋯(n-1)λjn-11λjλj2⋯⋯λjn-1000⋯00⋯0⋮⋮⋮⋱⋮⋮⋮⋮000⋯00⋯0].In order to be able to understand (67) better, the following example is helpful.Example 1.
We take the coefficient matrixV(68)[b3b3λ+b4b3λ2+2λb4+b5b3λ3+3λ2b4+3λb5+b6b3λ4+4λ3b4+6λ2b5+2λb6b4b4λ+b5b4λ2+2λb5+b6b4λ3+3λ2b5+3λb6b4λ4+4λ3b5+6λ2b6b5b5λ+b6b5λ2+2λb6b5λ3+3λ2b6b5λ4+4λ3b6b6b6λb6λ2b6λ3b6λ4].
We assume that b6=0 and b5≠0. Since b5≠0, we are not interested in b3, b4.
Thus, under our assumption, we obtain
(69)[b3b3λ+b4b3λ2+2λb4+b5b3λ3+3λ2b4+3λb5b3λ4+4λ3b4+6λ2b5b4b4λ+b5b4λ2+2λb5b4λ3+3λ2b5b4λ4+4λ3b5b5b5λb5λ2b5λ3b5λ400000].
Afterwards, we multiply the 3rd row with -b4/b5 and it is added to the 2nd row. Moreover, we multiply the 3rd row with -b3/b5 and it is added with the 1st row. Then, the following equivalent matrix is derived:
(70)[0b42λb4+b53λ2b4+3λb54λ3b4+6λ2b50b52λb53λ2b54λ3b5b5b5λb5λ2b5λ3b5λ400000].
Now, we multiply the 2nd row with -b4/b5 and it is added with the 1st row. Thus, we obtain
(71)[00b53λb56λ2b50b52λb53λ2b54λ3b5b5b5λb5λ2b5λ3b5λ400000].
Finally, we multiply the matrix with the element 1/b5and we conclude to obtain
(72)[0013λ6λ2012λ3λ24λ31λλ2λ3λ400000]=[(1)′′2!(λ)′′2!(λ2)′′2!(λ3)′′2!(λ4)′′2!(1)′1!(λ)′1!(λ2)′1!(λ3)′1!(λ4)′1!1λλ2λ3λ400000].Remark 2.
Considering the results already presented, it is clear that there exists a nonsingular matrixLjzj, μzj×μzj such that
(73)LjzjVjzj=Vjzj*.
Thus, system (61) can be transformed into
(74)1Φ-1(K)Ljzjz̲jzj(0+)=LjzjVjzja̲⇔Vjzj*a̲=z̲jzj*(0+),
where
(75)z̲jzj*(0+)=1Φ-1(K)Ljzjz̲jzj(0+),∀j=l+1,l+2,…,κ,∀zj=1,2,…,dj.Proposition 11.
Subsystems (74) are solvable when the elements of vectors
(76)z̲jzj*(0+),foreveryzj=1,2,…,dj,j=l+1,l+2,…,κ
are included into the vector with the greatest dimension, that is, ρj× 1 where
(77)ρj=maxμzj,zj=1,2,…,dj.
Equivalently, if one assumes that
(78)z̲j1*(0+)=[γ1γ2⋮γρj],
then each of the following vectors z̲j2*(0+), z̲j3*(0+),…, z̲jdj*(0+) should be vectors of [γ1γ2⋮γξ] type  with ξ≤ρj.Proof.
We take thedj subsystems; as follows:
(79)Vj1*a̲=z̲j1*(0+),⋯Vjdj*a̲=z̲jdj*(0+).
The matrixVj1* has μ1 rows,Vj2* has μ2 rows,⋮Vjdj* has μdjrows.
Without loss of generality, we assume that μ1≤μ2≤⋯≤μdj=ρj.
Looking carefully at the type of matrices
(80)Vj1*,Vj2*,…,Vjdj*,
we can easily verify that the μ1-first rows of them are identically same. Thus, it is necessary the relevant μ1-first rows of z̲j1*(0+),z̲j2*(0+),…,z̲jdj*(0+) to be also identically the same. Analogously, the μ2-first rows of Vj1*, Vj2*,…,Vjdj* should be identically the same with the relevant μ2-first rows of z̲j1*(0+),z̲j2*(0+),…,z̲jdj*(0+). And so on until the μdj=ρj row.(81)Φ-1(K)(b̲τ1⨁b̲τ2⨁⋯⨁b̲τl︸l⨁1Φ-1(K)[Iτl+1⨁⋯⨁Iτκ]b̲τ1⨁b̲τ2⨁⋯⨁b̲τl︸l)×[VlV(l+1)1*⋮V(l+1)dl+1*⋮Vj1*⋮Vjdj*⋮Vκ1*⋮Vκdκ*]a̲=[z̲l(0+)z̲(l+1)1*(0+)⋮z̲(l+1)dl+1*(0+)⋮z̲j1*(0+)⋮z̲jdj*(0+)⋮z̲κ1*(0+)⋮z̲κdκ*(0+)].Consequently, it is time to use the results of Propositions10 and 11. Thus, system (81) is solvable if we obtain the following.(i)
The first nonzero elements of vector coefficientb̲τ1,b̲τ2,…,b̲τl (practically speaking, without loss of generality, we assume that the first elements are nonzero).(ii)
The matrixV(q+1)zq+1* with dimension ρl+1×n, where ρl+1=maxμzl+1,zl+1=1,2,…,dl+1.⋮(iii)
The matrixVjzj* with dimension ρj×n, whereρj=maxμzj,  zj=1,2,…,dj.⋮(iv)
The matrixVκzκ* with dimension ρκ×n, whereρκ=maxμzκ, zκ=1,2,…,dκ.Consequently, system (81) is transposed to the solvable system (83)(82)Φ-1(K)(bτ1⨁bτ2⨁⋯⨁bτl︸l⨁1Φ-1(K)[Iτl+1⨁⋯⨁Iτκ]bτ1⨁bτ2⨁⋯⨁bτl︸l)×[VlV(l+1)zl+1*V(l+2)zl+2*⋮Vκdκ*]a̲=[z̲l(0+)z̲(l+1)zl+1*(0+)z̲(l+2)z2*(0+)⋮z̲κzκ*(0+)]
or equivalently,(83)[VlV(l+1)zl+1*V(l+2)zl+2*⋮Vκdκ*]a̲=[z̲l*(0+)z̲(l+1)zl+1*(0+)z̲(l+2)z2*(0+)⋮z̲κzκ*(0+)],
where z̲l*(0+)=(1/Φ-1(K))(1/bτ1⨁1/bτ2⨁⋯⨁1/bτl︸l)z̲l(0+), since (bτ1⨁bτ2⨁⋯⨁bτl) is  nonsingular.Remark 3.
The matrices given by (67) with some row transformations can be transformed to the following:
(84)[1λjλj2λj3⋯⋯λjn-1012λj3λj2(n-1)λjn-20013λj(n-1)(n-2)2!λjn-3⋮011(μzj-μ)!dμzj-μλjμzj-μ+1dλjμzj-μ⋯1(μzj-μ)!dμzj-μλjn-1dλjμzj-μ000⋮⋮⋮000].
The matrix (84) is denoted by Vjzj**. Thus, there is a nonsingular matrix Λjzjsuch as
(85)ΛjzjVjzj*=Vjzj**.
Then, subsystems (74) are transformed to
(86)ΛjzjVjzj*a̲=Λjzjz̲jzj*(0+)⇔Vjzj**a̲=z̲jzj**(0+),
where Λjzjz̲jzj*(0+)=z̲jzj**(0+) for every zj=1,2,…,dj, and j=l+1,l+2,…,κ. Consequently, system (83) is transformed to
(87)[VlV(l+1)zl+1**V(l+2)zl+2**⋮Vκdκ**]a̲=[z̲l*(0+)z̲(l+1)zl+1**(0+)z̲(l+2)z2**(0+)⋮z̲κzκ**(0+)].Comment 1.
Without loss of generality, we can assume that the matricesVjdj** do not contain zero rows. On contrary, if there exist some zero rows, the solvability of system (87) is not affected; only the number of nonzero rows which are included in (87) is  changed.Remark 4.
System (87) can be further transposed to a more convenient system. Analytically, if we multiply the 1st row of Vandermonde matrix Vl, that is, [1λ1λ12⋯λ1n-1] with the number (−1) and it is added to the 1st row of each of V(l+1)zl+1**,V(l+2)zl+2**,…,Vκzκ**, then Vjzj** is given by
(88)[0λj-λ1λj2-λ12⋯λjn-1-λ1n-1012λj⋯(n-1)λjn-2⋮1⋮⋱00⋯01,1(ρj-1)!dρj-1λjρj+1dλjρj-1,…,1(ρj-1)!dρj-1λjn-1dλjρj-1].
(Note that we have assumed that the matrices Vjzj**, for j=l+1,l+2,…,κ, do not cotain zero rows; see also Comment)We can easily see that the 1st row of matrix (88) can be rewritten as what follows.The  elementλjκ+λjκ-1λ1+⋯+λjλ1κ-1+λ1κ=∑m1,m2=0,∑i=12mi=κκλjm1λ1m2.Thus, the first row is presented as(89)(λj-λ1)[01∑m1,m2=0∑i=12mi=11λjm1λ1m2∑m1,m2=0∑i=12mi=22λjm1λ1m2⋯∑m1,m2=0∑i=12mi=n-2n-2λjm1λ1m2].Since, the elementλj-λ1≠0, we can multiply by left the equation (88) with a properly chosen transformation matrix, so as to obtain(90)[01∑m1m2=0∑i=12mi=11λjm1λ1m2⋯∑m1m2=0∑i=12mi=n-2n-2λjm1λ1m2012λj⋯(n-1)λjn-2O0⋯11(ρj-1)!dρj-1λjρj+1dλjρj-1⋯1(ρj-1)!dρj-1λjn-1dλjρj-1].Finally, this subsection is concluded by considering system (91), where the matrices Ajzjfor j=l+1,l+2,…,κ are derived by taking into account a properly chosen transformation left-matrix Z as follows:(91)Z[VlV(l+1)zl+1**V(l+2)zl+2**⋮Vκdκ**]a̲=Z[z̲l*(0+)z̲(l+1)zl+1**(0+)z̲(l+2)z2**(0+)⋮z̲κzκ**(0+)]⇔[VlAl+1Al+2⋮Aκ]a̲=[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)],
where(92)[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)]=Z[z̲l*(0+)z̲(l+1)zl+1**(0+)z̲(l+2)z2**(0+)⋮z̲κzκ**(0+)],
with
(93)Vl∈ℳln,Vj∈ℳjnforj=l+1,l+2,…,κ,z̲l*(0+)∈ℳl1,d̲l+2(0+)∈ℳj1,forj=l+1,l+2,…,κ.ρj=maxzi=1,2,…,djμzj is the index of annihilation for the eigenvalue λj.Consequently, the system (91) contains the following subsystems(94)Vla̲=z̲l*(0+),Al+1a̲=d̲l+1(0+),⋯Aκa̲=d̲κ(0+)},
where Vl, Aj for j=l+1,l+2,…,κ are nonsquare matrices.In order to solve system (91), we should use the relevant theory of generalized inverses. Some basic elements are briefly presented in the next subsection.


## 3.3. Solving Subsystem (91)

This subsection is beginning with the brief presentation of the most basic results of{1}-generalized inverses.Definition 12.
For a matrixA∈ℳpq, the X∈ℳqp is called {1}-inverses of matrix A, if
(95)AXA=A.The following proposition provides the means for finding{1}-inverses; see [19, 20] for proofs and details.Proposition 13.
LetA∈ℳpq, with rankA=r<min{p,q} and P∈ℳp, Q∈ℳq, invertible such that
(96)PAQ=Ir⨁𝕆p-r,q-r.
Then, every X, {1}-inverse of matrixA, is written as follows:
(97)X=Q(Ir⨁𝕆p-r,q-r)P,
for arbitrary L∈ℳ(q-r)(p-r).Definition 14.
We define any{1}-inverse of matrixA byA(1).The following result is due to [6].Theorem 15.
LetA∈ℳpq, B∈ℳdν, D∈ℳpν. Then the matrix equation
(98)AXB=D,
is consistent if and only if for some A(1), B(1)(99)AA(1)DB(1)B=D
and the general solution is given by
(100)X=A(1)DB(1)+Y-A(1)AYBB(1)
for arbitrary Y∈ℳpd.The following characterization of the set ofA(1) is due to [20].Corollary 16.
LetA∈ℳpq and A(1)∈ℳqp, then any other {1}-inverse Ã(1)∈ℳqp of matrixA is given by
(101)Ã(1)=A(1)+Z-A(1)AZAA(1)
for some Z∈ℳqp.A special case of{1}-inverses is the left/right inverses of a full column/row rank matrix, respectively. The proofs and further details are presented extensively by [19, 20].Definition 17.
Consider the following matrices.(a)
LetPi(α) be a m×m-matrix(102)Pi(α)=[1𝕆1⋱α1⋱𝕆1].
(The nonzero element a is in the ith row and the ith column.)
Thus, whenever a matrixA is multiplied from the left by Pi(α), then its ith row is multiplied by the nonzero number a.
(b)
LetPi(j,a) be an m×m-matrix(103)Pi(j,α)=[101⋱1⋯α⋱⋮1⋱01].
(The nonzero element a is in the ith row and the jth column.)
Thus, whenever a matrixA is multiplied from the left by Pi(j,a), then its jth row is multiplied by the nonzero number a and it is added to thejth row of A.
(c)
LetQi(a) be an n×n-matrix(104)Qi(α)=[11⋱1α1⋱1].
(The nonzero element a is in the ith-row and the ith-column).
Thus, whenever a matrixA is multiplied from the right by Qi(α), then its ith column is multiplied by the nonzero number a.
(d)
LetQi(j,a) be an n×n-matrix(105)Qi(j,α)=[11⋱1⋮1⋱α⋯1⋱1].
(The nonzero element a is in the jth row and the ith column.)
Thus, whenever a matrixA is multiplied from the right by Qi(j,a), then its ith column is multiplied by the nonzero number a and it is added to thejth column of A.Now, in order to calculate the generalized inverses of Vandermonde and the relative to it matrices, we use the recent results of paper [6]. In this context, the following propositions are relevant.Proposition 18.
The{1}-inverse of the Vandermonde matrix is given by
(106)Vl(1)=Q[Il𝕆n-l,l]P,
where
(107)P=∏s=l-11∏j=ls+1Pj(1λj-λs)Pj(s,-1)∈ℳl,
(the multiplication counts backwards, starting from l→l-1→⋯ etc.)
(108)Q=∏s=1l∏r=s+1nQr(s,-∑m1,…,ms=0m1+⋯+ms=r-sr-s∏ζ=1sλζmζ)∈ℳn,
and the general solution of system Vla̲=z̲l*(0+) is given by
(109)a̲=Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲,
for an arbitrarily chosen vector ψ̲∈ℳn1.Proof.
The proof is a straightforward application of Theorem15. Analytically,
(110)PVlQ=[Il𝕆l,n-l]⇔Vl=P-1[Il𝕆l,n-l]Q-1.
In order for the matrix Q[Il𝕆n-l,l]P to be the {1}-inverse of Vl, we should prove the following equality
(111)VlQ[Il𝕆n-l,l]PVl=Vl.
Thus,
(112)VlQ[Il𝕆n-l,l]PVl=P-1[Il𝕆l,n-l]Q-1Q[Il𝕆n-l,l]PP-1[Il𝕆l,n-l]Q-1=P-1[Il𝕆l,n-l][Il𝕆n-l,l][Il𝕆l,n-l]Q-1=P-1(Il·Il+𝕆l,n-l·𝕆n-l,l)[Il𝕆l,n-l]Q-1=P-1[Il𝕆l,n-l]Q-1=Vl.
Hence, the solution of system Vla̲=z̲l*(0+) is given by (113).Proposition 19.
The{1}-inverse of the matrix Al for j=l+1,l+2,…,κ is given by
(113)Aj(1)=Qj[0̲pjtIpj𝕆n-pj-1,pj]Pj,
where
(114)Pj=∏s=2pjPpj-s+2(1λj-λ1)Ppj-s+2(pj-s+1,-1),Qj=∏s=2pj+1∏m=s+1nQm(s,-1(s-2)!ds-2dλjs-2(∑m1,m2=0m1+m2=m-2m-2λ1m1λjm2)).
for j=l+1,l+2,…,κ, and the general solution of systems Aja̲=d̲j(0+) is given by
(115)a̲=Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲j
for an arbitrarily chosen vector ψ̲j∈ℳn1 for j=l+1,l+2,…,κ.Proof.
The proof is also a straightforward application of Theorem15. Analytically,
(116)PjAjQj=[0̲pjIpj𝕆pj,n-pj-1]⇔Aj=Pj-1[0̲pjIpj𝕆pj,n-pj-1]Qj-1.
In order for the matrix
(117)Qj[0̲pjtIpj𝕆n-pj-1,pj]Pj
to be the {1}-inverse of Aj, we should prove the following equality:
(118)AjQj[0̲pjtIpj𝕆n-pj-1,pj]PjAj=Aj.
Thus,
(119)AjQj[0̲pjtIpj𝕆n-pj-1,pj]PjAj=Pj-1[0̲pjIpj𝕆pj,n-pj-1]Qj-1Qj[0̲pjtIpj𝕆n-pj-1,pj]PjPj-1×[0̲pjIpj𝕆pj,n-pj-1]Qj-1=Pj-1[0̲pjIpj𝕆pj,n-pj-1][0̲pjtIpj𝕆n-pj-1,pj]×[0̲pjIpj𝕆pj,n-pj-1]Qj-1=Pj-1(0̲pj·0̲pjt+Ipj·Ipj+𝕆pj,n-pj-1·𝕆n-pj-1,pj)×[0̲pjIpj𝕆pj,n-pj-1]Qj-1=Pj-1[0̲pjIpj𝕆pj,n-pj-1]Qj-1=Aj.
Hence, the solution of system Aja̲=d̲j(0+) is given by (115).


## 3.4. The Solution of System (91)

This subsection concludes the whole discussion. The following theorem holds.Theorem 20.
System (91) is solvable, that is,
(120)Aa̲=b̲,
where
(121)A≜[VlAl+1Al+2⋮Aκ],b̲≜[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)],
if the following conditions are satisfied:
(122)Vl[Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲]=d̲l,(123)Aj[Vl(1)z̲l(0+)+(In-Vl(1)Vl)ψ̲]=d̲j,(124)Ai[Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲]=d̲i
for i≠j
and i,j=l+1,l+2,…,κ.Proof.
Following Propositions18 and 19, we have      (i)
for systemVla̲=z̲l*(0+), the solution is given by a̲=Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲,(ii)
for systemsAja̲=d̲j(0+), the solution is given by a̲=Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲ for arbitrarily chosen vector ψ̲∈ℳn1 for j=l+1,l+2,…,κ.
There are infinitely many solutions, since the vectorψ̲∈ℳn1 is arbitrary. Moreover, each of the above subsystems also contains the solution of system Aa̲=b̲. Thus, we assume that the desired solution is given by a̲, then it should also satisfy the first system, that is, Vla̲=z̲l*(0+). Thus, we take
(125)a̲=Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲,Aa̲=b̲⇔A(Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲)=b̲⇔AVl(1)z̲l*(0+)+A(In-Vl(1)Vl)ψ̲=b̲⇔AVl(1)z̲l*(0+)+(A-AVl(1)Vl)ψ̲=b̲⇔[VlAl+1Al+2⋮Aκ]Vl(1)z̲l*(0+)+([VlAl+1Al+2⋮Aκ]-[VlAl+1Al+2⋮Aκ]Vl(1)Vl)ψ̲≜[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)]⇔[z̲l*(0+)Al+1Vl(1)z̲l*(0+)Al+2Vl(1)z̲l*(0+)⋮AκVl(1)z̲l*(0+)]+[𝕆lAl+1(In-Vl(1)Vl)Al+2(In-Vl(1)Vl)⋮Aκ(In-Vl(1)Vl)]ψ̲=[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)]⇔AjVl(1)z̲l*(0+)+Aj(In-Vl(1)Vl)ψ̲=d̲j(0+)⇔Aj[Vl(1)z̲l*(0+)+(In-Vl(1)Vl)ψ̲]=d̲j(0+)
for every j=l+1,l+2…,κ. So (123) has been derived. Analogously, the system Aa̲=b̲ should also satisfy every subsystem Aja̲=d̲j(0+) forj=l+1,l+2,…,κ.
Consequently, we have that
(126)a̲=Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲,foreveryj=l+1,l+2,…,κ,Aa̲=b̲⇔A(Aj(1)d̲j(0+)+(In-Aj(1)Aj)ψ̲)=b̲⇔AAj(1)d̲j(0+)+A(In-Aj(1)Aj)ψ̲=b̲⇔[VlAl+1Al+2⋮Aκ]Aj(1)d̲j(0+)+([VlAl+1Al+2⋮Aκ]-[VlAl+1Al+2⋮Aκ]Aj(1)Aj)ψ̲≜[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)]⇔[VlAj(1)d̲j(0+)Al+1Aj(1)d̲j(0+)Al+2Aj(1)d̲j(0+)⋮AκAj(1)d̲j(0+)]+[Vl(In-Aj(1)Aj)Al+1(In-Aj(1)Aj)Al+2(In-Aj(1)Aj)⋮Aκ(In-Aj(1)Aj)]ψ̲=[z̲l*(0+)d̲l+1(0+)d̲l+2(0+)⋮d̲κ(0+)].
From the above equalities, the expressions (122)–(124) are obtained.With this important theorem, the discussion of system (32) has finished. In the next section, we present the solution of system (24).


## 4. The Solution of Subsystem (24)

The differential system of (15) is given by(127)Hqy̲p(r)(t)=y̲q(t)+b̲̃quo(t).
We start this subsection by observing that—as is well known, see Section 2—there exists a q*∈ℕ such that Hqq*=𝕆, that is, q* is the annihilation index of Hq. We obtain(128)Hqy̲p(r)(t)=y̲q(t)+b̲̃quo(t)⇒Hqq*y̲p(r)(t)=Hqq*-1y̲q(t)+Hqq*-1b̲̃quo(t)
or equivalently(129)Hqq*-1y̲q(t)+Hqq*-1b̲̃quo(t)=0̲q.Afterwards, we present a condition, in order to accept the already known solutiona̲ (see Section 3).We know that the nilpotent block matrixHqis given by(130)Hq1⨁Hq2⨁⋯⨁Hqμ.
Now, taking into consideration that Hqq*=𝕆, it follows that for each(131)Hq*q*-1=[0⋯010⋯00⋮⋯⋮⋮0⋯00],q*=max{qj:j=1,2,…,μ}.Lemma 21 (see [5]).
All the elements of the block diagonal matrixHqq*-1 are zero, that is, Hijq*-1=0 apart from the ξ-elements at positions (1,q*), (q*+1,2q*),…,((ξ-1)q*+1,ξq*) which are one. Note that ξ is the number of maximum block matrices with the same dimension, that is, q*×q*, where q*=max{qj:j=1,2,…,r}.Theorem 22.
The condition which makes the system (24) solvable and provides an acceptable choice of vector a̲ is given by
(132)Hqq*-1y̲q(0+)=0̲q.Proof.
First, we obtain system (129), that is, Hqq*-1y̲q(t)+Hqq*-1b̲̃quo(t)=0̲q. Then, for the solution of system (129), we take into account that
(133)u0(t)=limσ→0(α01σΦ(tσ)+α11σ2Φ′(tσ)+⋯+αn-11σnΦ(n-1)(tσ)).
Following Remark 1, we can assume that t=Kσ with K big enough so that
(134)Φ(k)(K)→0,foreveryk=1,2,…,n.
Moreover, we have Kσ→0+ and t→0+. Then, we take
(135)limt→0+(Hqq*-1y̲q(t)+Hqq*-1b̲̃quo(t))=0̲q⇔Hqq*-1limt→0+y̲q(t)+Hqq*-1b̲̃qlimσ→0u(t).
However,
(136)limσ→0u(t)=limσ→0(α01σΦ(tσ)+⋯+αn-11σnΦ(n-1)(tσ))=limσ→0(α01σΦ(K)+⋯+1σnΦ(n-1)(K))=0,limt→0+y̲q(t)=y̲q(0+).
Thus, (132) is derived.An equivalent proof to [5] for Theorem 23 is provided.Theorem 23.
A necessary condition for obtaining the solution of system (24) is given by
(137)x̲(0+)=(In-G(1)G)X,
where
(138)G=[q̃p+q*,1q̃p+q*,1⋯q̃p+q*,nq̃p+2q*,1q̃p+2q*,2⋯q̃p+2q*,n⋮⋮⋯⋮q̃p+ξq*,1q̃p+ξq*,2⋯q̃p+ξq*,n]∈ℳξn;(q̃ij)i=p+q*,…,p+rq*j=1,2,…,n
are elements of the matrix Q̃, the G(1) is {1}-inverse of matrix G, and for an arbitrarily chosen X∈ℳn1.Proof.
Considering the transformation (25), we obtain
(139)y̲(0+)=Q-1x̲(0+)
or equivalently by defining Q̃≜Q-1(140)[y̲p(0+)y̲q(0+)]=(q̃ij)i,j=1,2,…,n[x̲p(0+)x̲q(0+)]⇔[y̲p(0+)y̲q(0+)]=(∑j=1nq̃ijxj(0+))i=1,2,…,n
and finally
(141)Hqq*-1y̲q(0+)=0̲q⇒{∑j=1nq̃p+q*,jxj(0+)=0,∑j=1nq̃p+2q*,jxj(0+)=0,⋮∑j=1nq̃p+ξq*,jxj(0+)=0.
System (141) can be expressed as follows:
(142)Gx̲(0+)=0̲,
where
(143)G=[q̃p+q*,1q̃p+q*,1⋯q̃p+q*,nq̃p+2q*,1q̃p+2q*,2⋯q̃p+2q*,n⋮⋮⋯⋮q̃p+ξq*,1q̃p+ξq*,2⋯q̃p+ξq*,n]∈ℳξn.
The proof is concluded by using the results of Theorem 22.


## 5. An Illustrative Numerical Application

In this section, a numerical example of the above method is presented. We consider the second-order system(144)Fx̲′′(t)=Gx̲(t)+b̲u(t),
where the matrix coefficients are given by(145)F=[01-101101-63-21-1-21-11-10-1-11-32-10-1-100000000010-1-100-31-1100],G=[101-10-1-1-2-2-21134-1201-2-11-10-110120010-1-10-11-11011-2-1-21123],
and vector coefficient b̲=[1001101]t.Let us suppose that the system at time0- is(146)x̲(0-)=x̲′(0-)=[0000000]t,
and at time 0+it achieves(147)x̲(0+)=x̲′(0+)=[-3-223121]t.
From the regularity of sF-G, there exist nonsingular matrices in ℝ7×7(148)P=[000100-110000100-101001100100000-1010000000100000100],Q=[100100-1010000001001111010110011010000-100100001000],
such that(149)PFQ=Fw=[I2𝕆5,2𝕆5,2H5],PGQ=Gw=[J2(1)𝕆5,2𝕆5,2I5],
where I2,J2(1),H5 are given by (17).Consequently, system (144) is divided into the following subsystems:(150)y̲2′′(t)=[2102]y̲2(t)+b̲2uo(t),[0100010000100]y̲5′′(t)=y̲5(t)+b̲5uo(t).
Moreover, we obtain(151)uo(t)=aoδ(t)+a1δ(1)(t)+⋯+a5δ(5)(t)+a6δ(6)(t).
Before we go further with the analytic determination of the unknown coefficients ai, we should verify the important condition (132). First, we calculate the inverse matrix of Q,(152)Q-1=(q̃ij)i,j=1,2,…,n=[1010-1-1-10100000-11-1100100000011-21-110-1-11-110110010-1-10],y̲(0+)=Q-1x̲(0+)=[y̲2(0+)y̲5(0+)]=[-5-2-3105-1]t.
Consequently, it is not difficult to verify that condition (132) holds, that is, H52y̲5(0+)=0̲5.Furthermore, theb̲̃≜Pb̲=[0122101]T should be calculated. Thus(153)b̲̃2=[01],b̲̃5=[22101].
Now, the approximate expression for (151) is given by(154)u(t)=aoσΦ(tσ)+a1σ2Φ(1)(tσ)+⋯+a5σ5Φ(5)(tσ)+a6σ6Φ(6)(tσ).
System (150) is a standard form of nonhomogeneous  second-order linear differential equations of Apostol-Kolodner type. Thus, using the results of Section 3.1, we obtain(155)z̲′(t)=Rz̲(t)+L̲uo(t),
where z̲(t)=[z̲1t(t)z̲2t(t)]t and the coefficient matrices R and L̲ are given by(156)R=[𝕆2I2J2(1)𝕆2]=[0010000111000100],L̲=[0̲2b̲̃2]=[0001].
Considering (25), at time 0+ we obtain(157)y̲(0+)=Q-1x̲(0+)=y̲′(0+)=Q-1x̲′(0+).
Moreover,
(158)y̲2(0+)=[-5-2],y̲2′(0+)=[-5-2].
Furthermore,
(159)z̲(0+)=[y̲2t(0+)y̲2′t(0+)]t=[-5-2-5-2].
There exists an invertible matrix(160)C=[1/41/41/4-1/40-1/201/2-1/401/4001/201/2]
such that J=C-1RC is Jordan canonical form of matrix R. Analytically,(161)J=[-110-1]⨁[1101]=J2(-1)⨁J2(1).
Now, taking into consideration (52) and Remark 1 the solution is given by(162)C-1z̲(t)=[limσ→0{∫-∞te-J2(-1)τb̲2u(τ)dτ}limσ→0{∫-∞te-J2(1)τb̲2u(τ)dτ}],
where
(163)b̲≜C-1L̲=[b̲2b̲2]=[0101].
Using (55), we obtain(164)C-1z̲(t)=[-∫-∞tτeτu(τ)dτ∫-∞teτu(τ)dτ-∫-∞te-ττu(τ)dτ∫-∞te-τu(τ)dτ]=Φ-1(K)[∑k=06αkdλ1kdλ1∑k=06αkλ1k∑k=06αkdλ2kdλ2∑k=06αkλ2k]=Φ-1(K)[01-23-45-61-11-11-1101234561111111]a̲,A=[Aτ1=2Aτ2=2]=[1-11-11-1101-23-45-611111110123456]
or equivalently(165)Aτ1=2=[1-11-11-1101-23-45-6],Aτ2=2=[11111110123456].
Thus,(166)P1(2,1)Aτ1=2=[10-12-34-501-23-45-6]=Ãτ1=2,Ãτ1=2Q1(3,1)Q2(3,2)Q1(4,-2)Q2(4,-3)Q1(5,3)×Q2(5,4)Q1(6,-4)Q2(6,-5)Q1(7,5)Q2(7,6)=[10000000100000],
where(167)P1=[1101],Q1=[101-23-45012-34-5600100000001000000010000000100000001].
Moreover,(168)P1(2,-1)Aτ2=2=[10-1-2-3-4-50123456]=Ãτ2=2,Ãτ1=2Q1(3,1)Q2(3,2)Q1(4,2)Q2(4,-3)Q1(5,3)×Q2(5,-4)Q1(6,4)Q2(6,-5)Q1(7,5)Q2(7,-6)=[10000000100000],
where(169)P2=[1-101],Q2=[101234501-2-3-4-5-600100000001000000010000000100000001].
Thus, considering the result of Section 3, the {1}-inverse of matrices Aτ1=2 and Aτ2=2 are given by,(170)Q1[I2𝕆5,2]P1,Q2[I2𝕆5,2]P2,respectively,
Finally, the general solution is given by the expression(171)a̲=1Φ-1(K)Aτ1=2(1)d̲τ1=2(0+)+(I7-Aτ1=2(1)Aτ1=2)ψ̲,
where the following conditions should also be satisfied:(172)Aτ1=2[1Φ-1(K)Aτ2=2(1)d̲τ2=2(0+)+(I7-Aτ2=2(1)Aτ2=2)ψ̲]=d̲τ1=2,Aτ2=2[1Φ-1(K)Aτ1=2(1)d̲τ1=2(0+)+(I7-Aτ1=2(1)Aτ1=2)ψ̲]=d̲τ2=2.
Analytically,(173)d̲(0+)=C-1z̲(0+)=[d̲τ1=2(0+)d̲τ2=2(0+)]=1Φ-1(K)[-20-22-4],a̲=1Φ-1(K)[-2000000]+[001-23-45002-34-5611111][ψ1ψ2ψ3ψ4ψ5ψ6ψ7],
where
(174)[1-11-11-1101-23-45-6]×[1Φ-1(K)[-2000000]+[001-23-45002-34-5611111][ψ1ψ2ψ3ψ4ψ5ψ6ψ7]]=1Φ-1(K)[-20]⇔1Φ-1(K)[-20]+[-ψ3+ψ4-ψ5+ψ6-ψ72ψ3-3ψ4+4ψ5-5ψ6+6ψ7]=1Φ-1(K)[-20],[11111110123456]×[1Φ-1(K)[-18-400000]+[001234500-2-3-4-5-611111][ψ1ψ2ψ3ψ4ψ5ψ6ψ7]]=1Φ-1(K)[-22-4]⇔1Φ-1(K)[-22-4]-[ψ3+ψ4+ψ5+ψ6+ψ72ψ3+3ψ4+4ψ5+5ψ6+6ψ7]=1Φ-1(K)[-22-4].
Following the above conditions, we can see that ψ4=ψ6=0, ψ3,ψ5, and ψ7 are arbitrary. Thus, we conclude this numerical example with the determination of the unknown coefficients ai, for i=0,1,…,6(175)a̲=[a0a1a2a3a4a5a6]=[-2Φ-1(K)+ψ3+3ψ5+5ψ72ψ3+4ψ5+6ψ7ψ30ψ50ψ7]
for arbitrary ψi∈ℝ, for i=3,5,7 and Φ-1(K)=(1/2π)∫-∞Ke-x2/2dx; K is large enough (i.e., K→∞) that the assumption limx→K(1/2π)e-x2/2=0 is valid.


## 6. Conclusions

In this paper, an analytical method and several sufficient conditions were proposed and discussed for the changing of the state of a higher-order linear descriptor (regular) differential system in (almost) zero time by using the impulse function and its derivatives. The input vector has to be made up a linear combination of theδ-function of Dirac and its derivatives approximated using the normal (Gaussian) probability density function. Using the tools of linear algebra and generalized inverses, exact calculations of the relevant matrices are obtained.To date, no other approximating function has been used in the analysis of the problem studied in this paper. It will be of interest to use some other approximations and obtain comparative results. In [14, 18, 21], there are only some hints about the minimum time. However, there is no known formula which can calculate the minimum time considering some other significant problem parameters, such as the volatility σ, Kand so  forth.Since we have applied an approximation for the impulse input, we transfer the initial stateε-close to a desired state. Consequently, a natural question is to ask how close we can get, and how we can calculate that distance, and if it is possible to minimize it.



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*Source: 562484-2009-10-12.xml*