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

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 Dirac 𝛿-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.

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 Appendices A 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,

𝐹𝑥^(𝑟)(𝑡)=𝐺𝑥(𝑡)+𝑏𝑢(𝑡) (1) with known initial conditions

𝑥(𝑡_𝑜),𝑥^′(𝑡_𝑜),…,𝑥^(𝑟−1)(𝑡_𝑜), (2) where 𝐹,𝐺∈ℳ(𝑛×𝑛;𝔽), and 𝑏∈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)), 𝑥(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 ℳ_𝑛=ℳ(𝑛×𝑛;𝔽) and ℳ_𝑛𝑚=ℳ(𝑛×𝑚;𝔽).

In order to solve this problem, the appropriate input vector has to be made up as a linear combination of the 𝛿-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. Section 2 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 an 𝑛th-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:

𝑢_𝑜(𝑡)=𝑎_𝑜𝛿(𝑡)+𝑎₁𝛿⁽¹⁾(𝑡)+𝑎₂𝛿⁽²⁾(𝑡)+⋯+𝑎_𝑛−1𝛿^(𝑛−1)=(𝑡)_{𝑛−1𝑘=1}𝑎_𝑘𝛿^(𝑘)(𝑡), (3) where 𝛿^(𝑘)(𝑡) or 𝑑^𝑘𝛿(𝑡)/𝑑𝑡^𝑘 is the 𝑘th-derivative of the Dirac 𝛿-function, and 𝑎_𝑖 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

𝑥(0⁻)=𝑥^′(0⁻)=⋯𝑥^(𝑟−1)(0⁻)=[00⋯0]^𝑡, (4) and at time 0⁺, it achieves

𝑥(0⁺)=𝑥⁰₁𝑥⁰₂⋯𝑥⁰_𝑛^𝑡,𝑥^′(0⁺)=𝑥¹₁𝑥¹₂⋯𝑥¹_𝑛^𝑡,…,𝑥^(𝑟−1)(0⁺)=𝑥₁^𝑟−1𝑥₂^𝑟−1⋯𝑥_𝑛^𝑟−1𝑡. (5)

With the following definitions, a brief presentation of the most important elements of matrix pencil theory is given.

Definition 1. Given 𝐹,𝐺∈ℳ_𝑛𝑚 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 pencil 𝑠𝐹−𝐺 is said to be strictly equivalent to the pencil 𝑠𝐹−𝐺 if and only if there exist nonsingular 𝑃∈ℳ_𝑚 and 𝑄∈ℳ_𝑛 such as 𝑃(𝑠𝐹−𝐺)𝑄=𝑠𝐹−𝐺. (6)

In this paper, we consider the case that pencil is regular. Thus, the strict equivalence relation can be defined rigorously on the set of regular pencils as follows.

This is the set of elementary divisors (e.d.s) obtained by factorizing the invariant polynomials 𝑓_𝑖(𝑠,𝑠)̂ into powers of homogeneous polynomials irreducible over field 𝔽.

In the case where 𝑠𝐹−𝐺 is a regular, we have e.d. of the following types:

(i)e.d. of the type 𝑠^𝑝 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 type 𝑠^𝑞̂ are called infinite elementary divisors (i.e.d.).

Let 𝐵₁,𝐵₂,…,𝐵_𝑛 be elements of ℳ_𝑛. The direct sum of them denoted by 𝐵₁⨁𝐵₂⨁⋯⨁𝐵_𝑛 is the 𝑏𝑙𝑜𝑐𝑘diag{𝐵₁,𝐵₂,…,𝐵_𝑛}.

Then, the complex Weierstrass form 𝑠𝐹_𝑤−𝑄_𝑤 of the regular pencil 𝑠𝐹−𝐺 is defined by

𝑠𝐹_𝑤−𝑄_𝑤≜𝑠𝐼_𝑝−𝐽_𝑝𝑠𝐻_𝑞−𝐼_𝑞. (7) Now, the Jordan type element, that is, 𝐽_𝑝, is uniquely defined by the set of f.e.d.

(𝑠−𝑎₁)^𝑝1,…,(𝑠−𝑎_𝜈)^𝑝𝜈,^𝜈_𝑗=1𝑝_𝑗=𝑝, (8) of 𝑠𝐹−𝐺 and has the form

𝑠𝐼_𝑝−𝐽_𝑝≜𝑠𝐼_𝑝1−𝐽_𝑝1(𝑎₁)⋯𝑠𝐼_𝑝𝜈−𝐽_𝑝𝜈(𝑎_𝜈), (9) where

𝐼_𝑝≜𝐼_𝑝1𝐼_𝑝2⋯𝐼_𝑝𝜈,𝐽_𝑝≜𝐽_𝑝1(𝑎₁)𝐽_𝑝2(𝑎₂)⋯𝐽_𝑝𝜈(𝑎_𝜈). (10) The 𝑞 blocks of the second, that is, 𝑠𝐻_𝑞−𝐼_𝑞, see (7), are uniquely defined by the set of i.e.d.

𝑠̂^𝑞1,…,𝑠̂^𝑞𝜎,^𝜎_𝑗=1𝑞_𝑗=𝑞 (11) of 𝑠𝐹−𝐺 and has the form

𝑠𝐻_𝑞−𝐼_𝑞≜𝑠𝐻_𝑞1−𝐼_𝑞1⋯𝑠𝐻_𝑞𝜎−𝐼_𝑞𝜎, (12) where

𝐼_𝑞≜𝐼_𝑞1𝐼_𝑞2⋯𝐼_𝑞𝜎,𝐻_𝑞≜𝐻_𝑞1𝐻_𝑞2⋯𝐻_𝑞𝜎. (13) Furthermore, 𝐻_𝑞 is a nilpotent element of ℳ_𝑛 with index 𝑞=max{𝑞_𝑗∶𝑗=1,2,…,𝜎}, where 𝐻𝑞_𝑞=𝕆, (14) and 𝐼_𝑝𝑗,𝐽_𝑝𝑗(𝑎_𝑗),𝐻_𝑞𝑗 are defined as

𝐼_𝑝𝑗=10⋯001⋯0⋮⋮⋱⋮00⋯1∈ℳ_𝑝𝑗,𝐽_𝑝𝑗(𝑎_𝑗𝑎)=^𝑗10⋯00𝑎_𝑗1⋯0⋮⋮⋱⋮⋮000𝑎_𝑗10000𝑎_𝑗∈ℳ_𝑝𝑗,𝐻_𝑞𝑗=010⋯0001⋯0⋮⋮⋱⋮⋮0000100000∈ℳ_𝑞𝑗. (15) Moreover, for the matrices 𝐹 and 𝐺, we have the parameterization

𝑃𝐹𝑄=𝐹_𝑤=𝐼_𝑝𝐻_𝑞,𝑃𝐺𝑄=𝐺_𝑤=𝐽_𝑝𝐼_𝑞. (16)(17) Since the state 𝑥(0⁺) which we wish to reach is specified, we need to determine the 𝑛 unknown coefficients 𝑎_𝑖, 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 Dirac 𝛿-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

𝛿(𝑡)=lim_𝑎→0𝛿_𝑎(𝑡), (18) where𝛿_𝑎(𝑡) is called a nascent delta function. This limit is in the sense that

lim_𝑎→0^+∞_−∞𝛿_𝑎(𝑡)𝑓(𝑡)𝑑𝑡=𝑓(0). (19)

Some wellknown and very useful in applications nascent 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

𝛿(𝑡)=lim_𝜎→01𝜎√𝑒2𝜋^{−𝑡2/2𝜎2}=lim_𝜎→01𝜎𝑡Φ(𝜎), (20) where Φ(𝑥)=(1/√2𝜋)𝑒^−𝑥2/2.

So, the approximate expression for the controller (2) is given by

1𝑢(𝑡)=𝜎𝑡Φ(𝜎)𝑎_𝑜+1𝜎²Φ⁽¹⁾(𝑡𝜎)𝑎₁1+⋯+𝜎^𝑛Φ^(𝑛−1)(𝑡𝜎)𝑎_𝑛−1=^𝑛−1_𝑘=01𝜎^𝑘+1Φ^(𝑘)(𝑡𝜎)𝑎_𝑘. (21) Then, we take the limit

𝑢_𝑜(𝑡)=lim_𝜎→0𝑢(𝑡). (22) 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: 𝑦_𝑝^(𝑟)(𝑡)=𝐽_𝑝𝑦_𝑝(𝑡)+𝑏̃_𝑝𝑢_𝑜𝐻(𝑡),_𝑞𝑦_𝑞^(𝑟)(𝑡)=𝑦_𝑞(𝑡)+𝑏̃_𝑞𝑢_𝑜(𝑡). (23)(24)

Proof. Consider the transformation 𝑥(𝑡)=𝑄𝑦(𝑡). (25) Substituting the previous expression into (1) and considering also (3), we obtain 𝐹𝑄𝑦^(𝑟)(𝑡)=𝐺𝑄𝑦(𝑡)+𝑏𝑢_𝑜(𝑡). (26) Multiplying by 𝑃, we arrive at 𝐹_𝑤𝑦^(𝑟)(𝑡)=𝐺_𝑤𝑦(𝑡)+𝑃𝑏𝑢_𝑜(𝑡). (27) Now, we denote 𝑦𝑦(𝑡)=^𝑝𝑦(𝑡)_𝑞,(𝑡)𝑃𝑏≜𝑏̃=𝑏̃_𝑝𝑏̃_𝑞, (28) where 𝑝+𝑞=𝑛.
Taking into account the following expressions, that is,
𝐼_𝑝𝕆_𝑝,𝑞𝕆_𝑞,𝑝𝐻_𝑞𝑦_𝑝^(𝑟)𝑦(𝑡)_𝑞(𝑟)=𝐽(𝑡)_𝑝𝕆_𝑝,𝑞𝕆_𝑞,𝑝𝐼_𝑞𝑦_𝑝𝑦(𝑡)_𝑞+𝑏̃(𝑡)𝑏^𝑝̃_𝑞𝑢_𝑜(𝑡), (29) 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

𝑧₁(𝑡)=𝑦_𝑝𝑧(𝑡),₂(𝑡)=𝑦^′_𝑝⋮𝑧(𝑡),_𝑟(𝑡)=𝑦_𝑝^(𝑟−1)(𝑡). (30) Then, we have the following system of ordinary differential equations:

𝑧^′₁(𝑡)=𝑧₂𝑧(𝑡),_′2(𝑡)=𝑧₃⋮𝑧(𝑡),_′𝑟(𝑡)=𝐽_𝑝𝑧₁(𝑡)+𝑏̃_𝑝𝑢_𝑜(𝑡). (31) Finally, (31) can be expressed by using vector-matrix equations

𝐳^′(𝑡)=𝐑𝐳(𝑡)+𝐋𝑢_𝑜(𝑡), (32) where 𝐳(𝑡)=[^𝑧𝑡₁^{(𝑡)𝑧𝑡}₂^{(𝑡)⋯𝑧𝑡}_𝑟^(𝑡)]^𝑡 (()^𝑡 is the transpose tensor) and the coefficient matrices 𝐑 and 𝐋 are given by

𝕆𝐑=^𝑝𝐼_𝑝𝕆_𝑝⋯𝕆_𝑝𝕆_𝑝𝕆_𝑝𝐼_𝑝⋯𝕆_𝑝𝕆⋮⋮⋮⋱⋮_𝑝𝕆_𝑝𝕆_𝑝⋯𝐼_𝑝𝐽_𝑝𝕆_𝑝𝕆_𝑝⋯𝕆_𝑝∈ℳ_𝑟𝑝,𝐋=0_𝑝0_𝑝⋮𝑏̃_𝑝∈ℳ_(𝑟𝑝)1, (33) with corresponding dimension of 𝐳(𝑡), 𝑟𝑝×1.

The state of system (1) at time 0⁻ is 𝑥(0⁻)=[^00⋯0]^𝑡 and at time 0⁺ it achieves 𝑥(0⁺)=[^{𝑥1𝑥2⋯𝑥𝑛}]^𝑡.

Now, considering (25), at time 0⁻ we have

𝑦(0⁻)=𝑄⁻¹𝑥(0⁻)=𝑦^′(0⁻)=𝑄⁻¹𝑥^′(0⁻)=⋯=𝑦^(𝑟−1)(0⁻)=𝑄⁻¹𝑥^(𝑟−1)(0⁻)=0^𝑡, (34) and at time 0⁺ we obtain

𝑦(0⁺)=𝑄⁻¹𝑥(0⁺)=𝜒⁰₁𝜒⁰₂⋯𝜒⁰_𝑛^𝑡,𝑦^′(0⁺)=𝑄⁻¹𝑥^′(0⁺)=𝜒¹₁𝜒¹₂⋯𝜒¹_𝑛^𝑡𝑦,…,_(𝑟−1)(0⁺)=𝑄⁻¹𝑥^(𝑟−1)(0⁺)=𝜒₁^𝑟−1𝜒₂^𝑟−1⋯𝜒_𝑛^𝑟−1𝑡. (35) Moreover, 𝑦_𝑝^(𝑖)(0⁻)=0^𝑡_𝑝∈ℳ_𝑝,𝑦_𝑝^(𝑖)(0⁺)=𝜒^𝑖₁𝜒^𝑖₂⋯𝜒^𝑖_𝑝^𝑡∈ℳ_𝑝,𝑦_𝑞^(𝑖)(0⁻)=0^𝑡_𝑞∈ℳ_𝑞,𝑦_𝑞^(𝑖)(0⁺)=𝜒^𝑖_𝑝+1𝜒^𝑖_𝑝+2⋯𝜒^𝑖_𝑛^𝑡∈ℳ_𝑞, (36) where 𝜒^𝑖_𝑗∈𝔽 for 𝑖=0,1,2,…,𝑟−1, and 𝑗=1,2,…,𝑛.

Furthermore,

𝐳(0⁻)=𝑧^𝑡₁(0⁻)𝑧^𝑡₂(0⁻)⋯𝑧^𝑡_𝑟(0⁻)^𝑡=𝑦^𝑡_𝑝(0⁻)𝑦_𝑝^′𝑡(0⁻)⋯𝑦_𝑝^{(𝑟−1)𝑡}(0⁻)^𝑡=0^𝑡_𝑟𝑝,𝐳(0⁺)=𝑧^𝑡₁(0⁺)𝑧^𝑡₂(0⁺)⋯𝑧^𝑡_𝑟(0⁺)^𝑡=𝑦^𝑡_𝑝(0⁺)𝑦_𝑝^′𝑡(0⁺)⋯𝑦_𝑝^{(𝑟−1)𝑡}(0⁺)^𝑡=[𝜒⁰₁𝜒⁰₂⋯𝜒⁰_𝑝𝜒¹₁𝜒¹₂⋯𝜒¹_𝑝⋯𝜒₁^𝑟−1𝜒₂^𝑟−1⋯𝜒_𝑝^𝑟−1]^𝑡. (37)

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 matrix 𝐑 is given by 𝜑(𝜆)=^𝜅_𝑖=1(𝜆−𝜆₁)^𝜏𝑖, (38) with 𝜆_𝑖≠𝜆_𝑗 for 𝑖≠𝑗 and ∑^𝜅_𝑖=1𝜏_𝑖=𝑟𝑝. Without loss of generality, we assume that 𝑑₁=𝜏₁,𝑑₂=𝜏₂𝑑,…,_𝑙=𝜏_𝑙,for1<𝑙<𝜅,𝑑_𝑙+1<𝜏_𝑙+1,…,𝑑_𝑘<𝜏_𝜅, (39) where 𝑑_𝑖,𝜏_𝑖,𝑖=1,2,…,𝜅 is the geometric and algebraic multiplicity of the given eigenvalues 𝜆_𝑖, respectively.

Generally speaking, the matrix 𝐑 is not diagonalizable. However, we can generate 𝑟𝑝 linearly independent vectors 𝑣₁,𝑣₂,…,𝑣_𝑟𝑝 and 𝑟𝑝×𝑟𝑝 similarity transformation 𝐶=[𝑣₁,𝑣₂,…,𝑣_𝑟𝑝] that takes 𝐑 into the Jordan canonical form, as the following definition clarifies.

Definition 5. There exists an invertible matrix 𝐶∈ℳ_𝑟𝑝 such as 𝐉=𝐶⁻¹𝐑𝐶, 𝐉∈ℳ_𝑟𝑝; is it Jordan canonical form of matrix 𝐑. Analytically, 𝐉=𝐽_𝑜𝐽_𝑙+1𝐽_𝑙+2⋯𝐽_𝜅. (40)(i)The block diagonal matrix 𝐽_𝑜=𝐽₁⨁𝐽₂⨁⋯⨁𝐽_𝑙, where 𝐽_𝑖=𝜆_𝑖𝜆_𝑖⋱𝜆_𝑖∈ℳ_𝜏𝑖, (41) is also a diagonal matrix with diagonal elements the eigenvalue 𝜆_𝑖, for 𝑖=1,…,𝑙. Consequently, the dimension of 𝐽_𝑜 is 𝑠×𝑠,𝑠≜∑^𝑙_𝑖=1𝜏_𝑖.(ii) Also, the block matrix 𝐽_𝑗=𝐽_𝑗,1⨁𝐽_𝑗,2⨁⋯⨁𝐽_{𝑗,𝑑𝑗}∈ℳ_𝜏𝑗, where 𝐽_{𝑗,𝑧𝑗}=𝜆_𝑗1𝜆_𝑗1𝜆_𝑗⋱𝜆⋱1_𝑗∈ℳ_𝑧𝑗for𝑗=𝑙+1,𝑙+2,…,𝑘,𝑧_𝑗=1,2,…,𝑑_𝑗. (42)

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 𝐳(𝑡)=𝐶^𝑡_−∞𝑒^{𝐉(𝑡−𝜏)}𝐛𝑢_𝑜(𝜏)𝑑𝜏, (43) with initial condition 𝐳(0⁻)=0^𝑡_𝑟𝑝.

Proof. Consider the transformation 𝐳(𝑡)=𝐶𝜁(𝑡), (44) where nonsingular 𝐶∈ℳ_𝑟𝑝, and 𝜁(0⁻)=𝐶⁻¹𝐳(0⁻)=0^𝑡_𝑟𝑝.
Substituting (44) into (43), we obtain
𝐶𝜁^′(𝑡)=𝐑𝐶𝜁(𝑡)+𝐋𝑢_𝑜(𝑡)⇔𝜁^′(𝑡)=𝐶⁻¹𝐑𝐶𝜁(𝑡)+𝐶⁻¹𝐋𝑢_𝑜(𝑡). (45) Furthermore, we define 𝐛≜𝐶⁻¹𝐋, such as the last equation is transformed into 𝜁^′(𝑡)=𝐉𝜁(𝑡)+𝐛𝑢_𝑜(𝑡). (46) 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 matrix 𝑒^{𝐽(𝑡−𝜏)} is defined as 𝑒^{𝐉(𝑡−𝜏)}=𝑒^{𝐽𝑜(𝑡−𝜏)}𝑒^{𝐽𝑙+1(𝑡−𝜏)}𝑒^{𝐽𝑙+2(𝑡−𝜏)}⋯𝑒^{𝐽𝜅(𝑡−𝜏)}∈ℳ_𝑟𝑝, (47) where 𝑒^{𝐽𝑜(𝑡−𝜏)}=𝑒^{𝐽1(𝑡−𝜏)}𝑒^{𝐽2(𝑡−𝜏)}⋯𝑒^{𝐽𝑙(𝑡−𝜏)}∈ℳ_𝑠,𝑒^{𝐽𝑖(𝑡−𝜏)}=𝑒^{𝜆𝑖(𝑡−𝜏)}𝑒^{𝜆𝑖(𝑡−𝜏)}⋯𝑒^{𝜆𝑖(𝑡−𝜏)}∈ℳ_𝜏𝑖, (48) for 𝑖=1,…,𝑙.
Furthermore,
𝑒^{𝐽𝑗(𝑡−𝜏)}=𝑒^{𝜆𝑗1(𝑡−𝜏)}𝑒^{𝜆𝑗2(𝑡−𝜏)}⋯𝑒^{𝜆𝑗𝑑𝑗(𝑡−𝜏)}∈ℳ_𝜏𝑗, (49) where 𝑒^{𝐽𝑗𝑗,𝑧(𝑡−𝜏)}=𝑒^{𝜆𝑗(𝑡−𝜏)}1𝑡−𝜏⋯(𝑡−𝜏)^{𝜇𝑧𝑗−1}(𝜇_𝑧𝑗−1)!01⋯(𝑡−𝜏)^{𝜇𝑧𝑗−2}(𝜇_𝑧𝑗−2)!⋮⋮⋱⋮00⋯1∈ℳ_𝜇𝑧𝑗, (50) where 𝜇_𝑧𝑗 are the Weyr characteristics via Ferrer diagrams, for 𝑗=𝑙+1,𝑙+2,…,𝑘 and 𝑧_𝑗=1,2,…,𝑑_𝑗. Note that 𝜌_𝑗=max_{𝑧𝑖=1,2,…,𝑑𝑗}𝜇_𝑧𝑗 is the index of annihilation for the eigenvalue 𝜆_𝑗.
Now, taking into consideration (21) the solution (22) is transposed into
𝐳(𝑡)=lim_𝜎→0{𝐶𝑒^𝐉𝑡𝑡_−∞𝑒^−𝐉𝜏𝐛𝑢(𝜏)𝑑𝜏}, (51) or equivalently 𝐳(𝑡)=𝐶lim_𝜎→0{𝑒^{𝐽𝑜𝑡𝑡}_−∞𝑒^{−𝐽𝑜𝜏}𝐛_𝑠𝑢(𝜏)𝑑𝜏}lim_𝜎→0{𝑒^{𝐽𝑙+1𝑡𝑡}_−∞𝑒^{−𝐽𝑙+1𝜏}𝐛_𝜏𝑙+1⋮𝑢(𝜏)𝑑𝜏}lim_𝜎→0{𝑒^{𝐽𝑘𝑡𝑡}_−∞𝑒^{−𝐽𝜅𝜏}𝐛_𝜏𝜅𝑢(𝜏)𝑑𝜏}, (52) where 𝐛=[^𝐛𝑡_𝑠^𝐛𝑡_𝜏𝑙+1^⋯𝐛𝑡_𝜏𝜅]^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, 𝑡Φ(𝜎)=^{𝑡=𝐾𝜎}Φ(𝐾)→0, (53) and its derivatives Φ^(𝑖)(𝑡/𝜎)=^{𝑡=𝐾𝜎}Φ^(𝑖)(𝐾)−−−−−−^𝐾→∞→0, for 𝑖=0,1,2,…,𝑘−1.

Now, using the Remark 1, we obtain that lim_𝜎→0𝑒=^{𝐽𝑖𝑡𝑡=𝐾𝜎}𝐼, for 𝑖=1,2,…,𝑙,𝑙+1,𝑙+2,…,𝑘, and Φ⁻¹(𝐾+𝜆_𝑖𝜎)→Φ⁻¹(𝐾), as well. Moreover, for the analytic determination of vector (3), we need to calculate the integral ∫^𝑡_−∞𝑒^−𝐽𝜏𝐛𝑢(𝜏)𝑑𝜏.

In this part of the paper, two subsystems are derived; see the following lemmas, and Appendices A and B.

Lemma 8. For the diagonal matrix 𝐽_𝑜, the following integral is given by (54): lim_𝜎→0^𝑡_−∞𝑒^−𝐽0𝑡𝐛_𝑠𝑢(𝜏)𝑑𝜏=Φ⁻¹𝐛(𝐾)^{𝜏1𝑛−1𝑘=0}𝛼_𝑘𝜆^𝑘₁⋮𝐛_𝜏𝑖^𝑛−1_𝑘=0𝛼_𝑘𝜆^𝑘_𝑖⋮𝐛_𝜏𝑙^𝑛−1_𝑘=0𝛼_𝑘𝜆^𝑘_𝑙∈ℳ_𝑠1. (54)

Proof. See Appendix A.

Lemma 9. For the diagonal matrix 𝐽_𝑗=𝐽_𝑗1⨁𝐽_𝑗2⨁⋯⨁𝐽_𝑗𝑑𝑗∈ℳ_𝜏𝑗, the following integral is given by (55): lim_𝜎→0^𝑡_−∞𝑒^{−𝐽𝑗𝑡}𝐛_𝜏𝑗𝑢(𝜏)𝑑𝜏=Φ⁻¹𝐕(𝐾)^𝑗1𝐕_𝑗2⋮𝐕_𝑗𝑧𝑗⋮𝐕_𝑗𝑑𝑗⋅𝐚, (55) where 𝐕_𝑗𝑧𝑗≜^{𝜇𝑧𝑗𝑐=1}𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−1)!^𝑐−1𝜆⁰_𝑗𝑑𝜆_𝑗^{𝜇𝑐−1𝑧𝑗}_𝑐=1𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−1)!^𝑐−1𝜆_𝑗𝑑^𝑐−1_𝜆𝑗⋯^𝜇𝑧𝑗_𝑐=1𝐛^𝑐_j𝑧𝑗1⋅𝑑(𝑐−1)!^𝑐−1𝜆_𝑗^𝑛−1_𝜇^𝑐−1_𝜆𝑗𝑑_{𝑧𝑗𝑐=2}𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−2)!^𝑐−2𝜆⁰_𝑗𝑑𝜆_𝑗^{𝜇𝑐−2𝑧𝑗}_𝑐=2𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−2)!^𝑐−2𝜆_𝑗𝑑^𝑐−2_𝜆𝑗⋯^𝜇𝑧𝑗_𝑐=2𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−2)!^𝑐−2𝜆_𝑗^𝑛−1𝑑^𝑐−2_𝜆𝑗⋮⋮⋱⋮_{𝜇𝑧𝑗𝑐=𝜇𝑧𝑗}𝐛^𝑐_𝑗𝑧𝑗1(𝑐−𝜇_𝑧𝑗⋅𝑑)!^{𝑐−𝜇𝑧𝑗}𝜆⁰_𝑗𝑑𝜆^{𝑐−𝜇𝑧𝑗}_𝑗^{𝜇𝑧𝑗𝑐=𝜇𝑧}_𝑗𝐛^𝑐_𝑗𝑧𝑗1(𝑐−𝜇_𝑧𝑗⋅𝑑)!^{𝑐−𝜇𝑧𝑗}𝜆_𝑗𝑑^{𝑐−𝜇𝑧𝑗}_𝜆𝑗⋯^𝜇𝑧𝑗_{𝑐=𝜇𝑧𝑗}𝐛^𝑐_𝑗𝑧𝑗1(𝑐−𝜇_𝑧𝑗⋅𝑑)!^{𝑐−𝜇𝑧𝑗}𝜆_𝑗^𝑛−1𝑑^{𝑐−𝜇𝑧𝑗}_𝜆𝑗,𝐚=𝛼₀𝛼₁𝛼₂⋮𝛼_𝑛−1. (56)

Proof. See Appendix B.

Now, we revisit (54), thus Φ⁻¹𝐛(𝐾)^{𝜏1𝑛−1𝑘=0}𝛼_𝑘𝜆^𝑘₁⋮𝐛_𝜏𝑖^𝑛−1_𝑘=0𝛼_𝑘𝜆^𝑘_𝑖⋮𝐛_𝜏𝑙^𝑛−1_𝑘=0𝛼_𝑘𝜆^𝑘_𝑙=Φ⁻¹(𝐾)(𝐛_𝜏1𝐛_𝜏2⋯𝐛_𝜏𝑙)^{𝑛−1𝑘=0}𝛼_𝑘𝜆^𝑘_{1𝑛−1𝑘=0}𝛼_𝑘𝜆^𝑘_{2𝑛−1𝑘=0}⋮𝛼_𝑘𝜆^𝑘_𝑙=Φ⁻¹(𝐾)(𝐛_𝜏1𝐛_𝜏2⋯𝐛_𝜏𝑙)×1𝜆₁𝜆²₁⋯𝜆₁^𝑛−11𝜆₂𝜆²₂⋯𝜆₂^𝑛−1⋮⋮⋮⋱⋮1𝜆_𝑙𝜆²_𝑙⋯𝜆_𝑙^𝑛−1_𝐕𝑙𝛼₀𝛼₁𝛼₂⋮𝛼_𝑛−1=Φ⁻¹(𝐾)(𝐛_𝜏1𝐛_𝜏2⋯𝐛_𝜏𝑙)𝐕_𝑙𝐚. (57) Note that 𝐕_𝑙 is the well-known Vandermonde matrix.

Combining (52)–(57), we obtain the following system:

𝐶⁻¹𝐳(0⁺)=Φ⁻¹(𝐾)×(𝐛_𝜏1𝐛_𝜏2⋯𝐛_𝜏𝑙𝑙𝐼_𝜏𝑙+1⋯𝐼_𝜏𝜅)×𝐕_𝑙𝐕_(𝑙+1)1⋮𝐕_{(𝑙+1)𝑑𝑙+1}⋮𝐕_𝑗1⋮𝐕_𝑗𝑑𝑗⋮𝐕_𝜅1⋮𝐕_𝜅𝑑𝜅𝐚, (58) where 𝐕_𝑙∈ℳ_𝑠𝑛 and 𝐕_𝑗𝑧𝑗∈ℳ_{𝜇𝑧𝑗𝑛} with 𝑗=𝑙+1,𝑙+2,…,𝜅 and 𝑧_𝑗=1,2,…,𝑑_𝑗. Furthermore, we obtain the initial condition

𝐶⁻¹𝐳(0⁺)=[𝐳^𝑡_𝑙(0⁺)𝐳^𝑡_(𝑙+1)1(0⁺)⋯𝐳^𝑡_{(𝑙+1)𝑑𝑙+1}(0⁺)⋯𝐳^𝑡_𝑗1(0⁺)⋯𝐳^𝑡_𝑗𝑑𝑗(0⁺)⋯𝐳^𝑡_𝜅1(0⁺)⋯𝐳^𝑡_𝜅𝑑𝜅(0⁺)]. (59)

The system (58) can now be divided into two types of subsystems:

(S1) 1Φ⁻¹𝐳(𝐾)^𝑙(0⁺)=(𝐛_𝜏1𝐛_𝜏2⋯𝐛_𝜏𝑙)𝐕_𝑙𝐚, (60)

(S2) 1Φ⁻¹𝐳(𝐾)^𝑗𝑧𝑗(0⁺)=𝐕_𝑗𝑧𝑗𝐚, (61) for 𝑗=𝑙+1,𝑙+2,…,𝜅 and 𝑧_𝑗=1,2,…,𝑑_𝑗.

Proposition 10. System (60) is solvable if 𝑧¹_𝑖𝑏¹_𝑖=𝑧²_𝑖𝑏²_𝑖𝑧=⋯=^𝜏𝑖𝑖𝑏^𝜏𝑖_𝑖, (62) for every nonzero element of vectors 𝐛_𝜏1,𝐛_𝜏2,…,𝐛_𝜏𝑙, where 𝑧_𝑖(0⁺)=[^𝑧1_𝑖^𝑧2_𝑖^{⋯𝑧𝜏𝑖}_𝑖]^𝑡, for every 𝑖=1,2,…,𝑙.
Moreover, if one of the elements of vectors 𝐛_𝜏1,𝐛_𝜏2,…,𝐛_𝜏𝑙 is zero, then the relative element of the 𝑖th-row of the vector 𝐳_𝜏𝑖(0⁺) should be also zero.

Proof. System (60) contains 𝑙-subsystems of the following type: (0_𝜏1⋯𝐛_𝜏𝑖⋯0_𝜏𝑙)𝐕_𝑙𝜶=1Φ⁻¹(𝐾)(𝐼_𝜏1⋯𝐼_𝜏𝑖⋯𝐼_𝜏𝑙)𝐳_𝑙(0⁺), (63) or equivalently 𝑏¹_𝑖𝑏¹_𝑖𝜆_𝑖⋯𝑏¹_𝑖𝜆_𝑖^𝑛−1𝑏²_𝑖𝑏²_𝑖𝜆_𝑖⋯𝑏²_𝑖𝜆_𝑖^𝑛−1𝑏⋮⋮⋱⋮_𝜏𝑖𝑖𝑏^𝜏𝑖_𝑖𝜆_𝑖⋯𝑏^𝜏𝑖_𝑖𝜆_𝑖^𝑛−1𝛼₀𝛼₁⋮𝛼_𝑛−1=1Φ⁻¹𝑧(𝐾)^1𝑖(0⁺)𝑧²_𝑖(0⁺)⋮𝑧^𝜏𝑖_𝑖(0⁺). (64)(i)If one of the coefficients 𝑏^𝛾_𝑖=0,for 𝛾= 1, 2,…, 𝜏_𝑖, then the relative element of the 𝑖th row of the vector 𝐳_𝜏𝑖(0⁺) should be also zero (in order to obtain solution).(ii) If every of the coefficients 𝑏^𝛾_𝑖≠0, for 𝛾= 1, 2,…, 𝜏_𝑖, then we have𝑏¹_𝑖(𝛼₀+𝛼₁𝜆_𝑖+⋯+𝛼_𝑛−1𝜆_𝑖^𝑛−1)=𝑧¹_𝑖1Φ⁻¹𝑏(𝐾)_2𝑖(𝛼₀+𝛼₁𝜆_𝑖+⋯+𝛼_𝑛−1𝜆_𝑖^𝑛−1)=𝑧²_𝑖1Φ⁻¹⋮𝑏(𝐾)_𝜏𝑖𝑖(𝛼₀+𝛼₁𝜆_𝑖+⋯+𝛼_𝑛−1𝜆_𝑖^𝑛−1)=𝑧^𝜏𝑖_𝑖1Φ⁻¹𝛼(𝐾)}⇔₀+𝛼₁𝜆_𝑖+⋯+𝛼_𝑛−𝑖𝜆_𝑖^𝑛−1=𝑧¹_𝑖𝑏¹_𝑖1Φ⁻¹⋮𝛼(𝐾)₀+𝛼₁𝜆_𝑖+⋯+𝛼_𝑛−𝑖𝜆_𝑖^𝑛−1=𝑧^𝜏𝑖_𝑖𝑏^𝜏𝑖_𝑖1Φ⁻¹}⇒𝑧(𝐾)_1𝑖𝑏¹_𝑖=𝑧²_𝑖𝑏²_𝑖𝑧=⋯=^𝜏𝑖𝑖𝑏^𝜏𝑖_𝑖 (65) 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:

^𝜇𝑧𝑗_𝑐=1𝑏^𝑐_𝑗𝑧𝑗𝑑(𝑐−1)!^𝑐−1𝜆⁰_𝑗𝑑𝜆_𝑗^{𝜇𝑐−1𝑧𝑗}_𝑐=1𝑏^𝑐_𝑗𝑧𝑗𝑑(𝑐−1)!^𝑐−1𝜆_𝑗𝑑𝜆_𝑗^{𝜇𝑐−1𝑧𝑗}_𝑐=1𝑏^𝑐_𝑗𝑧𝑗𝑑(𝑐−1)!^𝑐−1𝜆²_𝑗𝑑𝜆_𝑗^𝑐−1⋯^𝜇𝑧𝑗_𝑐=1𝑏^𝑐_𝑗𝑧𝑗𝑑(𝑐−1)!^𝑐−1𝜆_𝑗^𝑛−1𝑑𝜆_𝑗^𝑐−1𝑏⋮⋮⋮⋱⋮_{𝜇𝑧𝑗𝑗−1𝑧𝑗}𝑏^𝜇𝑧𝑗_𝑗⁻¹_𝑧𝑗𝜆_𝑗+𝑏^𝜇𝑧𝑗_𝑗𝑧𝑗𝑏^𝜇𝑧𝑗_𝑗⁻¹_𝑧𝑗𝜆²_𝑗+2𝜆_𝑗𝑏^𝜇𝑧𝑗_𝑗𝑧𝑗⋯𝑏^𝜇𝑧𝑗_𝑗⁻¹_𝑧𝑗𝜆_𝑗^𝑛−1+(𝑛−1)𝜆_𝑗^𝑛−2𝑏^𝜇𝑧𝑗_𝑗𝑧𝑗𝑏^𝜇𝑧𝑗_𝑗𝑧𝑗𝑏^𝜇𝑧𝑗_𝑗𝑧𝑗𝜆_𝑗,𝑏^𝜇𝑧𝑗_𝑗𝑧𝑗𝜆²_𝑗⋯𝑏^𝜇𝑧𝑗_𝑗𝑧𝑗𝜆_𝑗^𝑛−1𝛼=1Φ⁻¹𝐳(𝐾)^𝑗𝑧𝑗. (0⁺)(66)

The coefficient matrix 𝐕_𝑗𝑧𝑗can be transposed to the following equivalent matrix; see (67). Note that 𝐕_𝑗𝑧𝑗 is a 𝜇_𝑧𝑗×𝑛 matrix. Moreover, from the 𝜇_𝑧𝑗 elements 𝑏^𝑐_𝑗𝑧𝑗, for 𝑐=1,2,…,𝜇_𝑧𝑗, we can assume that 𝑏^𝜇𝑧𝑗_𝑗𝑧𝑗=𝑏^𝜇𝑧𝑗_𝑗⁻¹_𝑧𝑗=⋯=𝑏^𝜇𝑧𝑗_𝑗^−𝜇_𝑧𝑗=0. Thus, we obtain

𝐕^∗_𝑗𝑧𝑗=1000⋯1(𝜇_𝑧𝑗𝑑−𝜇)!^{𝜇𝑧𝑗−𝜇}𝜆^𝜇𝑧𝑗_𝑗^+1−𝜇𝑑𝜆^𝜇𝑧𝑗_𝑗^−𝜇⋯1(𝜇_𝑧𝑗𝑑−𝜇)!^{𝜇𝑧𝑗−𝜇}𝜆_𝑗^𝑛−1𝑑𝜆^𝜇𝑧𝑗_𝑗^−𝜇1⋮⋮⋮⋯⋮⋱⋮001⋯⋯𝑑2!²𝜆_𝑗^𝑛−1𝑑𝜆²_𝑗012𝜆_𝑗⋯⋯(𝑛−1)𝜆_𝑗^𝑛−11𝜆_𝑗𝜆²_𝑗⋯⋯𝜆_𝑗^𝑛−1000⋯00⋯0⋮⋮⋮⋱⋮⋮⋮⋮000⋯00⋯0. (67)

In order to be able to understand (67) better, the following example is helpful.

Example 1. We take the coefficient matrix 𝐕𝑏₃𝑏₃𝜆+𝑏₄𝑏₃𝜆²+2𝜆𝑏₄+𝑏₅𝑏₃𝜆³+3𝜆²𝑏₄+3𝜆𝑏₅+𝑏₆𝑏₃𝜆⁴+4𝜆³𝑏₄+6𝜆²𝑏₅+2𝜆𝑏₆𝑏₄𝑏₄𝜆+𝑏₅𝑏₄𝜆²+2𝜆𝑏₅+𝑏₆𝑏₄𝜆³+3𝜆²𝑏₅+3𝜆𝑏₆𝑏₄𝜆⁴+4𝜆³𝑏₅+6𝜆²𝑏₆𝑏₅𝑏₅𝜆+𝑏₆𝑏₅𝜆²+2𝜆𝑏₆𝑏₅𝜆³+3𝜆²𝑏₆𝑏₅𝜆⁴+4𝜆³𝑏₆𝑏₆𝑏₆𝜆𝑏₆𝜆²𝑏₆𝜆³𝑏₆𝜆⁴. (68) We assume that 𝑏₆=0 and 𝑏₅≠0. Since 𝑏₅≠0, we are not interested in 𝑏₃, 𝑏₄.
Thus, under our assumption, we obtain
𝑏₃𝑏₃𝜆+𝑏₄𝑏₃𝜆²+2𝜆𝑏₄+𝑏₅𝑏₃𝜆³+3𝜆²𝑏₄+3𝜆𝑏₅𝑏₃𝜆⁴+4𝜆³𝑏₄+6𝜆²𝑏₅𝑏₄𝑏₄𝜆+𝑏₅𝑏₄𝜆²+2𝜆𝑏₅𝑏₄𝜆³+3𝜆²𝑏₅𝑏₄𝜆⁴+4𝜆³𝑏₅𝑏₅𝑏₅𝜆𝑏₅𝜆²𝑏₅𝜆³𝑏₅𝜆⁴00000. (69) 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: 0𝑏₄2𝜆𝑏₄+𝑏₅3𝜆²𝑏₄+3𝜆𝑏₅4𝜆³𝑏₄+6𝜆²𝑏₅0𝑏₅2𝜆𝑏₅3𝜆²𝑏₅4𝜆³𝑏₅𝑏₅𝑏₅𝜆𝑏₅𝜆²𝑏₅𝜆³𝑏₅𝜆⁴00000. (70) Now, we multiply the 2nd row with −𝑏₄/𝑏₅ and it is added with the 1st row. Thus, we obtain 00𝑏₅3𝜆𝑏₅6𝜆²𝑏₅0𝑏₅2𝜆𝑏₅3𝜆²𝑏₅4𝜆³𝑏₅𝑏₅𝑏₅𝜆𝑏₅𝜆²𝑏₅𝜆³𝑏₅𝜆⁴00000. (71) Finally, we multiply the matrix with the element 1/𝑏₅and we conclude to obtain 0013𝜆6𝜆²012𝜆3𝜆²4𝜆³1𝜆𝜆²𝜆³𝜆⁴=00000(1)^′′2!(𝜆)^′′2!(𝜆²)^′′2!(𝜆³)^′′2!(𝜆⁴)^′′2!(1)^′1!(𝜆)^′1!(𝜆²)^′1!(𝜆³)^′1!(𝜆⁴)^′1!1𝜆𝜆²𝜆³𝜆⁴.00000 (72)

Remark 2. Considering the results already presented, it is clear that there exists a nonsingular matrix 𝐿_𝑗𝑧𝑗, 𝜇_𝑧𝑗×𝜇_𝑧𝑗 such that 𝐿_𝑗𝑧𝑗𝐕_𝑗𝑧𝑗=𝐕^∗_𝑗𝑧𝑗. (73) Thus, system (61) can be transformed into 1Φ⁻¹𝐿(𝐾)^𝑗𝑧𝑗𝐳_𝑗𝑧𝑗(0⁺)=𝐿_𝑗𝑧𝑗𝐕_𝑗𝑧𝑗𝐚⇔𝐕^∗_𝑗𝑧𝑗𝐚=𝐳^∗_𝑗𝑧𝑗(0⁺), (74) where 𝐳^∗_𝑗𝑧𝑗(0⁺1)=Φ⁻¹𝐿(𝐾)^𝑗𝑧𝑗𝐳_𝑗𝑧𝑗(0⁺),∀𝑗=𝑙+1,𝑙+2,…,𝜅,∀𝑧_𝑗=1,2,…,𝑑_𝑗. (75)

Proposition 11. Subsystems (74) are solvable when the elements of vectors 𝐳^∗_𝑗𝑧𝑗(0⁺),forevery𝑧_𝑗=1,2,…,𝑑_𝑗,𝑗=𝑙+1,𝑙+2,…,𝜅 (76) are included into the vector with the greatest dimension, that is, 𝜌_𝑗× 1 where 𝜌_𝑗=max𝜇_𝑧𝑗,𝑧_𝑗=1,2,…,𝑑_𝑗. (77) Equivalently, if one assumes that 𝐳^∗_𝑗1(0⁺𝛾)=¹𝛾₂⋮𝛾_𝜌𝑗, (78) then each of the following vectors 𝐳^∗_𝑗2(0⁺), 𝐳^∗_𝑗3(0⁺),…, 𝐳^∗_𝑗𝑑𝑗(0⁺) should be vectors of [^𝛾1𝛾2_⋮𝛾𝜉] type with 𝜉≤𝜌_𝑗.

Proof. We take the 𝑑_𝑗 subsystems; as follows: 𝐕^∗_𝑗1𝐚=𝐳^∗_𝑗1(0⁺⋯𝐕),_{∗𝑗𝑑𝑗}𝐚=𝐳^∗_𝑗𝑑𝑗(0⁺). (79) The matrix𝐕^∗_𝑗1 has 𝜇₁ rows,𝐕^∗_𝑗2 has 𝜇₂ rows, ⋮𝐕^∗_𝑗𝑑𝑗 has 𝜇_𝑑𝑗rows. Without loss of generality, we assume that 𝜇₁≤𝜇₂≤⋯≤𝜇_𝑑𝑗=𝜌_𝑗.
Looking carefully at the type of matrices
𝐕^∗_𝑗1,𝐕^∗_𝑗2,…,𝐕^∗_𝑗𝑑𝑗, (80) we can easily verify that the 𝜇₁-first rows of them are identically same. Thus, it is necessary the relevant 𝜇₁-first rows of 𝐳^∗_𝑗1(0⁺),𝐳^∗_𝑗2(0⁺),…,𝐳^∗_𝑗𝑑𝑗(0⁺) to be also identically the same. Analogously, the 𝜇₂-first rows of 𝐕^∗_𝑗1, 𝐕^∗_𝑗2,…,𝐕^∗_𝑗𝑑𝑗 should be identically the same with the relevant 𝜇₂-first rows of 𝐳^∗_𝑗1(0⁺),𝐳^∗_𝑗2(0⁺),…,𝐳^∗_𝑗𝑑𝑗(0⁺). And so on until the 𝜇_𝑑𝑗=𝜌_𝑗 row.

Φ⁻¹(𝐾)(𝐛_𝜏1𝐛_𝜏2⋯𝐛_𝜏𝑙𝑙1Φ⁻¹(𝐾)[𝐼_𝜏𝑙+1⋯𝐼_𝜏𝜅×𝐕])_𝑙𝐕^∗_(𝑙+1)1⋮𝐕^∗_{(𝑙+1)𝑑𝑙+1}⋮𝐕^∗_𝑗1⋮𝐕^∗_𝑗𝑑𝑗⋮𝐕^∗_𝜅1⋮𝐕^∗_𝜅𝑑𝜅𝐚=𝐳_𝑙(0⁺)𝐳^∗_(𝑙+1)1(0⁺)⋮𝐳^∗_{(𝑙+1)𝑑𝑙+1}(0⁺)⋮𝐳^∗_𝑗1(0⁺)⋮𝐳^∗_𝑗𝑑𝑗(0⁺)⋮𝐳^∗_𝜅1(0⁺)⋮𝐳^∗_𝜅𝑑𝜅(0⁺). (81)

Consequently, it is time to use the results of Propositions 10 and 11. Thus, system (81) is solvable if we obtain the following.

(i) The first nonzero elements of vector coefficient 𝐛_𝜏1,𝐛_𝜏2,…,𝐛_𝜏𝑙 (practically speaking, without loss of generality, we assume that the first elements are nonzero).(ii) The matrix 𝐕^∗_{(𝑞+1)𝑧𝑞+1} with dimension 𝜌_𝑙+1×𝑛, where 𝜌_𝑙+1=max𝜇_𝑧𝑙+1,𝑧_𝑙+1=1,2,…,𝑑_𝑙+1.

⋮

(iii) The matrix 𝐕^∗_𝑗𝑧𝑗 with dimension 𝜌_𝑗×𝑛, where𝜌_𝑗=max𝜇_𝑧𝑗, 𝑧_𝑗=1,2,…,𝑑_𝑗.

⋮

(iv) The matrix 𝐕^∗_𝜅𝑧𝜅 with dimension 𝜌_𝜅×𝑛, where𝜌_𝜅=max𝜇_𝑧𝜅, 𝑧_𝜅=1,2,…,𝑑_𝜅.

Consequently, system (81) is transposed to the solvable system (83)

Φ⁻¹(𝐾)(𝑏_𝜏1𝑏_𝜏2⋯𝑏_𝜏𝑙𝑙1Φ⁻¹(𝐾)[𝐼_𝜏𝑙+1⋯𝐼_𝜏𝜅×𝐕])_𝑙𝐕^∗_{(𝑙+1)𝑧𝑙+1}𝐕^∗_{(𝑙+2)𝑧𝑙+2}⋮𝐕^∗_𝜅𝑑𝜅𝐚=𝐳_𝑙(0⁺)𝐳^∗_{(𝑙+1)𝑧𝑙+1}(0⁺)𝐳^∗_{(𝑙+2)𝑧2}(0⁺)⋮𝐳^∗_𝜅𝑧𝜅(0⁺) (82) or equivalently,

𝐕_𝑙𝐕^∗_{(𝑙+1)𝑧𝑙+1}𝐕^∗_{(𝑙+2)𝑧𝑙+2}⋮𝐕^∗_𝜅𝑑𝜅𝐚=𝐳^∗_𝑙(0⁺)𝐳^∗_{(𝑙+1)𝑧𝑙+1}(0⁺)𝐳^∗_{(𝑙+2)𝑧2}(0⁺)⋮𝐳^∗_𝜅𝑧𝜅(0⁺), (83) where 𝐳^∗_𝑙(0⁺)=(1/Φ⁻¹(𝐾))(1/𝑏_𝜏11/𝑏_𝜏2⋯1/𝑏_𝜏𝑙𝑙)𝐳_𝑙(0⁺), since (𝑏_𝜏1⨁𝑏_𝜏2⨁⋯⨁𝑏_𝜏𝑙) is nonsingular.

Remark 3. The matrices given by (67) with some row transformations can be transformed to the following: 1𝜆_𝑗𝜆²_𝑗𝜆³_𝑗⋯⋯𝜆_𝑗^𝑛−1012𝜆_𝑗3𝜆²_𝑗(𝑛−1)𝜆_𝑗^𝑛−20013𝜆_𝑗(𝑛−1)(𝑛−2)𝜆2!^𝑗𝑛−3⋮101(𝜇_𝑧𝑗𝑑−𝜇)!^{𝜇𝑧𝑗−𝜇}𝜆^𝜇𝑧𝑗_𝑗^−𝜇+1𝑑𝜆^𝜇𝑧𝑗_𝑗^−𝜇⋯1(𝜇_𝑧𝑗𝑑−𝜇)!^{𝜇𝑧𝑗−𝜇}𝜆_𝑗^𝑛−1𝑑𝜆^𝜇𝑧𝑗_𝑗^−𝜇000⋮⋮⋮000. (84) The matrix (84) is denoted by 𝐕_𝑗^∗∗_𝑧𝑗. Thus, there is a nonsingular matrix Λ_𝑗𝑧𝑗such as Λ_𝑗𝑧𝑗𝐕^∗_𝑗𝑧𝑗=𝐕_𝑗^∗∗_𝑧𝑗. (85) Then, subsystems (74) are transformed to Λ_𝑗𝑧𝑗𝐕^∗_𝑗𝑧𝑗𝐚=Λ_𝑗𝑧𝑗𝐳^∗_𝑗𝑧𝑗(0⁺)⇔𝐕_𝑗^∗∗_𝑧𝑗𝐚=𝐳_𝑗^∗∗_𝑧𝑗(0⁺), (86) where Λ_𝑗𝑧𝑗𝐳^∗_𝑗𝑧𝑗(0⁺)=𝐳_𝑗^∗∗_𝑧𝑗(0⁺) for every 𝑧_𝑗=1,2,…,𝑑_𝑗, and 𝑗=𝑙+1,𝑙+2,…,𝜅. Consequently, system (83) is transformed to 𝐕_𝑙𝐕^∗∗_{(𝑙+1)𝑧𝑙+1}𝐕^∗∗_{(𝑙+2)𝑧𝑙+2}⋮𝐕_𝜅^∗∗_𝑑𝜅𝐚=𝐳^∗_𝑙(0⁺)𝐳^∗∗_{(𝑙+1)𝑧𝑙+1}(0⁺)𝐳^∗∗_{(𝑙+2)𝑧2}(0⁺)⋮𝐳_𝜅^∗∗_𝑧𝜅(0⁺). (87)

Comment 1. Without loss of generality, we can assume that the matrices 𝐕_𝑗^∗∗_𝑑𝑗 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 𝐕_𝑙, that is, [^1𝜆₁^𝜆2₁^⋯𝜆₁^𝑛−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},…,𝐕_𝜅^∗∗_𝑧𝜅, then 𝐕_𝑗^∗∗_𝑧𝑗 is given by 0𝜆_𝑗−𝜆₁𝜆²_𝑗−𝜆²₁⋯𝜆_𝑗^𝑛−1−𝜆₁^𝑛−1012𝜆_𝑗⋯(𝑛−1)𝜆_𝑗^𝑛−21⋮1⋮⋱00⋯01,(𝜌_𝑗𝑑−1)!^𝜌𝑗−1𝜆^𝜌𝑗+1_𝑗𝑑𝜆^𝜌𝑗_𝑗⁻¹1,…,(𝜌_𝑗𝑑−1)!^𝜌𝑗−1𝜆_𝑗^𝑛−1𝑑𝜆^𝜌𝑗_𝑗⁻¹. (88) (Note that we have assumed that the matrices 𝐕_𝑗^∗∗_𝑧𝑗, 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 element 𝜆^𝜅_𝑗+𝜆_𝑗^𝜅−1𝜆₁+⋯+𝜆_𝑗𝜆₁^𝜅−1+𝜆^𝜅₁=∑^𝜅_{𝑚1,𝑚2=0,∑2𝑖=1𝑚𝑖=𝜅}𝜆^𝑚1_𝑗𝜆^𝑚2₁.

Thus, the first row is presented as

(𝜆_𝑗−𝜆₁)[01¹_{𝑚1,𝑚2∑=02𝑖=1𝑚𝑖=1}𝜆^𝑚1_𝑗𝜆^𝑚2₁²_{𝑚1,𝑚2∑=02𝑖=1𝑚𝑖=2}𝜆^𝑚1_𝑗𝜆^𝑚2₁⋯^𝑛−2_{𝑚1,𝑚2∑=02𝑖=1𝑚𝑖=𝑛−2}𝜆^𝑚1_𝑗𝜆^𝑚2₁]. (89)

Since, the element 𝜆_𝑗−𝜆₁≠0, we can multiply by left the equation (88) with a properly chosen transformation matrix, so as to obtain

01¹_{𝑚1𝑚2∑=02𝑖=1𝑚𝑖=1}𝜆^𝑚1_𝑗𝜆^𝑚2₁⋯^𝑛−2_{𝑚1𝑚2∑=02𝑖=1𝑚𝑖=𝑛−2}𝜆^𝑚1_𝑗𝜆^𝑚2₁012𝜆_𝑗⋯(𝑛−1)𝜆_𝑗^𝑛−2𝑂10⋯1(𝜌_𝑗𝑑−1)!^𝜌𝑗−1𝜆^𝜌𝑗_𝑗⁺¹𝑑𝜆^𝜌𝑗_𝑗⁻¹⋯1(𝜌_𝑗𝑑−1)!^𝜌𝑗−1𝜆_𝑗^𝑛−1𝑑𝜆^𝜌𝑗_𝑗⁻¹. (90)

Finally, this subsection is concluded by considering system (91), where the matrices 𝐀_𝑗𝑧𝑗for 𝑗=𝑙+1,𝑙+2,…,𝜅 are derived by taking into account a properly chosen transformation left-matrix Z as follows:

𝑍𝐕_𝑙𝐕^∗∗_{(𝑙+1)𝑧𝑙+1}𝐕^∗∗_{(𝑙+2)𝑧𝑙+2}⋮𝐕_𝜅^∗∗_𝑑𝜅𝐚𝐳=𝑍^∗𝑙(0⁺)𝐳^∗∗_{(𝑙+1)𝑧𝑙+1}(0⁺)𝐳^∗∗_{(𝑙+2)𝑧2}(0⁺)⋮𝐳_𝜅^∗∗_𝑧𝜅(0⁺)⇔𝐕_𝑙𝐀_𝑙+1𝐀_𝑙+2⋮𝐀_𝜅𝐚=𝐳^∗_𝑙(0⁺)𝐝_𝑙+1(0⁺)𝐝_𝑙+2(0⁺)⋮𝐝_𝜅(0⁺), (91) where

𝐳^∗_𝑙(0⁺)𝐝_𝑙+1(0⁺)𝐝_𝑙+2(0⁺)⋮𝐝_𝜅(0⁺)𝐳=𝑍^∗𝑙(0⁺)𝐳^∗∗_{(𝑙+1)𝑧𝑙+1}(0⁺)𝐳^∗∗_{(𝑙+2)𝑧2}(0⁺)⋮𝐳_𝜅^∗∗_𝑧𝜅(0⁺), (92) with 𝐕_𝑙∈ℳ_ln,𝐕_𝑗∈ℳ_𝑗𝑛𝐳for𝑗=𝑙+1,𝑙+2,…,𝜅,_∗𝑙(0⁺)∈ℳ_𝑙1,𝐝_𝑙+2(0⁺)∈ℳ_𝑗1,for𝑗=𝑙+1,𝑙+2,…,𝜅. (93)𝜌_𝑗=max_{𝑧𝑖=1,2,…,𝑑𝑗}𝜇_𝑧𝑗 is the index of annihilation for the eigenvalue 𝜆_𝑗.

Consequently, the system (91) contains the following subsystems

𝐕_𝑙𝐚=𝐳^∗_𝑙(0⁺𝐀),_𝑙+1𝐚=𝐝_𝑙+1(0⁺⋯𝐀),_𝜅𝐚=𝐝_𝜅(0⁺)}, (94) where 𝐕_𝑙, 𝐀_𝑗 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 matrix 𝐴∈ℳ_𝑝𝑞, the 𝑋∈ℳ_𝑞𝑝 is called {1}-inverses of matrix 𝐴, if 𝐴𝑋𝐴=𝐴. (95)

The following proposition provides the means for finding {1}-inverses; see [19, 20] for proofs and details.

Proposition 13. Let 𝐴∈ℳ_pq, with rank𝐴=𝑟<min{𝑝,𝑞} and 𝑃∈ℳ_𝑝, 𝑄∈ℳ_𝑞, invertible such that 𝑃𝐴𝑄=𝐼_𝑟𝕆_{𝑝−𝑟,𝑞−𝑟}. (96) Then, every 𝑋, {1}-inverse of matrix𝐴, is written as follows: 𝑋=𝑄(𝐼_𝑟𝕆_{𝑝−𝑟,𝑞−𝑟})𝑃, (97) 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. Let 𝐴∈ℳ_𝑝𝑞, 𝐵∈ℳ_𝑑𝜈, 𝐷∈ℳ_𝑝𝜈. Then the matrix equation 𝐴𝑋𝐵=𝐷, (98) is consistent if and only if for some 𝐴⁽¹⁾, 𝐵⁽¹⁾𝐴𝐴⁽¹⁾𝐷𝐵⁽¹⁾𝐵=𝐷 (99) and the general solution is given by 𝑋=𝐴⁽¹⁾𝐷𝐵⁽¹⁾+𝑌−𝐴⁽¹⁾𝐴𝑌𝐵𝐵⁽¹⁾ (100) for arbitrary 𝑌∈ℳ_𝑝𝑑.

The following characterization of the set of 𝐴⁽¹⁾ is due to [20].

Corollary 16. Let 𝐴∈ℳ_𝑝𝑞 and 𝐴⁽¹⁾∈ℳ_𝑞𝑝, then any other {1}-inverse 𝐴⁽¹⁾∈ℳ_𝑞p of matrix𝐴 is given by 𝐴⁽¹⁾=𝐴⁽¹⁾+𝑍−𝐴⁽¹⁾𝐴𝑍𝐴𝐴⁽¹⁾ (101) for some 𝑍∈ℳ_𝑞𝑝.

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)Let 𝑃_𝑖(𝛼) be a 𝑚×𝑚-matrix𝑃_𝑖1⋱𝛼1⋱(𝛼)=1𝕆𝕆1. (102) (The nonzero element 𝑎 is in the 𝑖th row and the 𝑖th column.)
Thus, whenever a matrix 𝐴 is multiplied from the left by 𝑃_𝑖(𝛼), then its 𝑖th row is multiplied by the nonzero number 𝑎.
(b)Let 𝑃_𝑖(𝑗,𝑎) be an 𝑚×𝑚-matrix𝑃_𝑖1⋱1⋱(𝑗,𝛼)=101⋯𝛼⋱⋮01. (103) (The nonzero element 𝑎 is in the 𝑖th row and the 𝑗th column.)
Thus, whenever a matrix 𝐴 is multiplied from the left by 𝑃_𝑖(𝑗,𝑎), then its 𝑗th row is multiplied by the nonzero number 𝑎 and it is added to the𝑗th row of 𝐴.
(c)Let 𝑄_𝑖(𝑎) be an 𝑛×𝑛-matrix𝑄_𝑖11⋱1𝛼1⋱1(𝛼)=. (104) (The nonzero element 𝑎 is in the 𝑖th-row and the 𝑖th-column).
Thus, whenever a matrix 𝐴 is multiplied from the right by 𝑄_𝑖(𝛼), then its 𝑖th column is multiplied by the nonzero number 𝑎.
(d)Let 𝑄_𝑖(𝑗,𝑎) be an 𝑛×𝑛-matrix𝑄_𝑖11⋱1⋱⋱1(𝑗,𝛼)=⋮1𝛼⋯1. (105) (The nonzero element 𝑎 is in the 𝑗th row and the 𝑖th column.)
Thus, whenever a matrix 𝐴 is multiplied from the right by 𝑄_𝑖(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 𝐕_𝑙⁽¹⁾𝐼=𝐐^𝑙𝕆_{𝑛−𝑙,𝑙}𝐏, (106) where 𝐏=¹_{𝑠=𝑙−1}^𝑠+1_𝑗=𝑙𝑃_𝑗(1𝜆_𝑗−𝜆_𝑠)𝑃_𝑗(𝑠,−1)∈ℳ_𝑙, (107) (the multiplication counts backwards, starting from 𝑙→𝑙−1→⋯ etc.) 𝐐=^𝑙𝑛_{𝑠=1𝑟=𝑠+1}𝑄_𝑟(𝑠,−^𝑟−𝑠_{𝑚1,…,𝑚𝑠𝑚=01+⋯+𝑚𝑠}^𝑠_{=𝑟−𝑠𝜁=1}𝜆^𝑚𝜁_𝜁)∈ℳ_𝑛, (108) and the general solution of system 𝐕_𝑙𝐚=𝐳^∗_𝑙(0⁺) is given by 𝐚=𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)+(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓, (109) for an arbitrarily chosen vector 𝜓∈ℳ_𝑛1.

Proof. The proof is a straightforward application of Theorem 15. Analytically, 𝐏𝐕_𝑙𝐐=𝐼_𝑙𝕆_{𝑙,𝑛−𝑙}⇔𝐕_𝑙=𝐏⁻¹𝐼_𝑙𝕆_{𝑙,𝑛−𝑙}𝐐⁻¹. (110) In order for the matrix 𝐐[^𝐼𝑙_{𝕆𝑛−𝑙,𝑙}]𝐏 to be the {1}-inverse of 𝐕_𝑙, we should prove the following equality 𝐕_𝑙𝐐𝐼_𝑙𝕆_{𝑛−𝑙,𝑙}𝐏𝐕_𝑙=𝐕_𝑙. (111) Thus, 𝐕_𝑙𝐐𝐼_𝑙𝕆_{𝑛−𝑙,𝑙}𝐏𝐕_𝑙=𝐏⁻¹𝐼_𝑙𝕆_{𝑙,𝑛−𝑙}𝐐⁻¹𝐐𝐼_𝑙𝕆_{𝑛−𝑙,𝑙}𝐏𝐏⁻¹𝐼_𝑙𝕆_{𝑙,𝑛−𝑙}𝐐⁻¹=𝐏⁻¹𝐼_𝑙𝕆_{𝑙,𝑛−𝑙}𝐼_𝑙𝕆_{𝑛−𝑙,𝑙}𝐼_𝑙𝕆_{𝑙,𝑛−𝑙}𝐐⁻¹=𝐏⁻¹(𝐼_𝑙⋅𝐼_𝑙+𝕆_{𝑙,𝑛−𝑙}⋅𝕆_{𝑛−𝑙,𝑙})𝐼_𝑙𝕆_{𝑙,𝑛−𝑙}𝐐⁻¹=𝐏⁻¹𝐼_𝑙𝕆_{𝑙,𝑛−𝑙}𝐐⁻¹=𝐕_𝑙. (112) Hence, the solution of system 𝐕_𝑙𝐚=𝐳^∗_𝑙(0⁺) is given by (113).

Proposition 19. The {1}-inverse of the matrix 𝐀_𝑙 for 𝑗=𝑙+1,𝑙+2,…,𝜅 is given by 𝐀_𝑗⁽¹⁾=𝐐_𝑗0^𝑡_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑛−𝑝𝑗−1,𝑝𝑗}𝐏_𝑗, (113) where 𝐏_𝑗=^𝑝𝑗_𝑠=2𝑃_{𝑝𝑗−𝑠+2}(1𝜆_𝑗−𝜆₁)𝑃_{𝑝𝑗−𝑠+2}(𝑝_𝑗𝐐−𝑠+1,−1),_𝑗=^{𝑝𝑗+1𝑛}_{𝑠=2𝑚=𝑠+1}𝑄_𝑚1(𝑠,−𝑑(𝑠−2)!𝑑𝜆^𝑠−2_𝑗^𝑠−2(^𝑚−2_{𝑚1,𝑚2𝑚=01+𝑚2=𝑚−2}𝜆^𝑚1₁𝜆^𝑚2_𝑗)). (114) for 𝑗=𝑙+1,𝑙+2,…,𝜅, and the general solution of systems 𝐀_𝑗𝐚=𝐝_𝑗(0⁺) is given by 𝐚=𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)+(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝜓_𝑗 (115) for an arbitrarily chosen vector 𝜓_𝑗∈ℳ_𝑛1 for 𝑗=𝑙+1,𝑙+2,…,𝜅.

Proof. The proof is also a straightforward application of Theorem 15. Analytically, 𝐏_𝑗𝐀_𝑗𝐐_𝑗=0_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑝𝑗,𝑛−𝑝𝑗−1}⇔𝐀_𝑗=𝐏_𝑗⁻¹0_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑝𝑗,𝑛−𝑝𝑗−1}𝐐_𝑗⁻¹. (116) In order for the matrix 𝐐_𝑗0^𝑡_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑛−𝑝𝑗−1,𝑝𝑗}𝐏_𝑗 (117) to be the {1}-inverse of 𝐀_𝑗, we should prove the following equality: 𝐀_𝑗𝐐_𝑗0^𝑡_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑛−𝑝𝑗−1,𝑝𝑗}𝐏_𝑗𝐀_𝑗=𝐀_𝑗. (118) Thus, 𝐀_𝑗𝐐_𝑗0^𝑡_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑛−𝑝𝑗−1,𝑝𝑗}𝐏_𝑗𝐀_𝑗=𝐏_𝑗⁻¹0_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑝𝑗,𝑛−𝑝𝑗−1}𝐐_𝑗⁻¹𝐐_𝑗0^𝑡_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑛−𝑝𝑗−1,𝑝𝑗}𝐏_𝑗𝐏_𝑗⁻¹×0_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑝𝑗,𝑛−𝑝𝑗−1}𝐐_𝑗⁻¹=𝐏_𝑗⁻¹0_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑝𝑗,𝑛−𝑝𝑗−1}0^𝑡_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑛−𝑝𝑗−1,𝑝𝑗}×0_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑝𝑗,𝑛−𝑝𝑗−1}𝐐_𝑗⁻¹=𝐏_𝑗⁻¹(0_𝑝𝑗⋅0^𝑡_𝑝𝑗+𝐼_𝑝𝑗⋅𝐼_𝑝𝑗+𝕆_{𝑝𝑗,𝑛−𝑝𝑗−1}⋅𝕆_{𝑛−𝑝𝑗−1,𝑝𝑗})×0_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑝𝑗,𝑛−𝑝𝑗−1}𝐐_𝑗⁻¹=𝐏_𝑗⁻¹0_𝑝𝑗𝐼_𝑝𝑗𝕆_{𝑝𝑗,𝑛−𝑝𝑗−1}𝐐_𝑗⁻¹=𝐀_𝑗. (119) Hence, the solution of system 𝐀_𝑗𝐚=𝐝_𝑗(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, 𝐀𝐚=𝐛, (120) where 𝐕𝐀≜^𝑙𝐀_𝑙+1𝐀_𝑙+2⋮𝐀_𝜅,𝐛≜𝐳^∗_𝑙(0⁺)𝐝_𝑙+1(0⁺)𝐝_𝑙+2(0⁺)⋮𝐝_𝜅(0⁺), (121) if the following conditions are satisfied: 𝐕_𝑙[𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)+(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝜓]=𝐝_𝑙𝐀_𝑗,[𝐕_𝑙⁽¹⁾𝐳_𝑙(0⁺)+(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓]=𝐝_𝑗𝐀_𝑖,[𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)+(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝜓]=𝐝_𝑖 (122)(123)(124) for 𝑖≠𝑗 and 𝑖,𝑗=𝑙+1,𝑙+2,…,𝜅.

Proof. Following Propositions 18 and 19, we have (i)for system𝐕_𝑙𝐚=𝐳^∗_𝑙(0⁺), the solution is given by 𝐚=𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)+(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓,(ii)for systems 𝐀_𝑗𝐚=𝐝_𝑗(0⁺), the solution is given by 𝐚=𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)+(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝜓 for arbitrarily chosen vector 𝜓∈ℳ_𝑛1 for 𝑗=𝑙+1,𝑙+2,…,𝜅.
There are infinitely many solutions, since the vector 𝜓∈ℳ_𝑛1 is arbitrary. Moreover, each of the above subsystems also contains the solution of system 𝐀𝐚=𝐛. Thus, we assume that the desired solution is given by 𝐚, then it should also satisfy the first system, that is, 𝐕_𝑙𝐚=𝐳^∗_𝑙(0⁺). Thus, we take
𝐚=𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)+(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓,𝐀𝐚=𝐛⇔𝐀(𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)+(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓)=𝐛⇔𝐀𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)+𝐀(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓=𝐛⇔𝐀𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)+(𝐀−𝐀𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓=𝐛⇔𝐕_𝑙𝐀_𝑙+1𝐀_𝑙+2⋮𝐀_𝜅𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺𝐕)+(^𝑙𝐀_𝑙+1𝐀_𝑙+2⋮𝐀_𝜅−𝐕_𝑙𝐀_𝑙+1𝐀_𝑙+2⋮𝐀_𝜅𝑉_𝑙⁽¹⁾𝑉_𝑙)𝜓≜𝐳^∗_𝑙(0⁺)𝐝_𝑙+1(0⁺)𝐝_𝑙+2(0⁺)⋮𝐝_𝜅(0⁺)⇔𝐳^∗_𝑙(0⁺)𝐀_𝑙+1𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)𝐀_𝑙+2𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)⋮𝐀_𝜅𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)+𝕆_𝑙𝐀_𝑙+1(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝐀_𝑙+2(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)⋮𝐀_𝜅(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓=𝐳^∗_𝑙(0⁺)𝐝_𝑙+1(0⁺)𝐝_𝑙+2(0⁺)⋮𝐝_𝜅(0⁺)⇔𝐀_𝑗𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)+𝐀_𝑗(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓=𝐝_𝑗(0⁺)⇔𝐀_𝑗[𝐕_𝑙⁽¹⁾𝐳^∗_𝑙(0⁺)+(𝐼_𝑛−𝐕_𝑙⁽¹⁾𝐕_𝑙)𝜓]=𝐝_𝑗(0⁺) (125) for every 𝑗=𝑙+1,𝑙+2…,𝜅. So (123) has been derived. Analogously, the system 𝐀𝐚=𝐛 should also satisfy every subsystem 𝐀_𝑗𝐚=𝐝_𝑗(0⁺) for𝑗=𝑙+1,𝑙+2,…,𝜅.
Consequently, we have that
𝐚=𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)+(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝜓,forevery𝑗=𝑙+1,𝑙+2,…,𝜅,𝐀𝐚=𝐛⇔𝐀(𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)+(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝜓)=𝐛⇔𝐀𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)+𝐀(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝜓=𝐛⇔𝐕_𝑙𝐀_𝑙+1𝐀_𝑙+2⋮𝐀_𝜅𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺𝐕)+(^𝑙𝐀_𝑙+1𝐀_𝑙+2⋮𝐀_𝜅−𝐕_𝑙𝐀_𝑙+1𝐀_𝑙+2⋮𝐀_𝜅𝐀_𝑗⁽¹⁾𝐀_𝑗)𝜓≜𝐳^∗_𝑙(0⁺)𝐝_𝑙+1(0⁺)𝐝_𝑙+2(0⁺)⋮𝐝_𝜅(0⁺)⇔𝐕_𝑙𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)𝐀_𝑙+1𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)𝐀_𝑙+2𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)⋮𝐀_𝜅𝐀_𝑗⁽¹⁾𝐝_𝑗(0⁺)+𝐕_𝑙(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝐀_𝑙+1(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝐀_𝑙+2(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)⋮𝐀_𝜅(𝐼_𝑛−𝐀_𝑗⁽¹⁾𝐀_𝑗)𝜓=𝐳^∗_𝑙(0⁺)𝐝_𝑙+1(0⁺)𝐝_𝑙+2(0⁺)⋮𝐝_𝜅(0⁺). (126) 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

𝐻_𝑞𝑦_𝑝^(𝑟)(𝑡)=𝑦_𝑞(𝑡)+𝑏̃_𝑞𝑢_𝑜(𝑡). (127) We start this subsection by observing that—as is well known, see Section 2—there exists a 𝑞^∗∈ℕ such that 𝐻^𝑞∗_𝑞=𝕆, that is, 𝑞^∗ is the annihilation index of 𝐻_𝑞. We obtain

𝐻_𝑞𝑦_𝑝^(𝑟)(𝑡)=𝑦_𝑞(𝑡)+𝑏̃_𝑞𝑢_𝑜(𝑡)⇒𝐻^𝑞∗_𝑞𝑦_𝑝^(𝑟)(𝑡)=𝐻^𝑞∗_𝑞⁻¹𝑦_𝑞(𝑡)+𝐻^𝑞∗_𝑞⁻¹𝑏̃_𝑞𝑢_𝑜(𝑡) (128) or equivalently

𝐻^𝑞∗_𝑞⁻¹𝑦_𝑞(𝑡)+𝐻^𝑞∗_𝑞⁻¹𝑏̃_𝑞𝑢_𝑜(𝑡)=0_𝑞. (129)

Afterwards, we present a condition, in order to accept the already known solution 𝐚 (see Section 3).

We know that the nilpotent block matrix 𝐻_𝑞is given by

𝐻_𝑞1𝐻_𝑞2⋯𝐻_𝑞𝜇. (130) Now, taking into consideration that 𝐻^𝑞∗_𝑞=𝕆, it follows that for each

𝐻^𝑞∗_𝑞⁻¹_∗=,𝑞0⋯010⋯00⋮⋯⋮⋮0⋯00_∗=max{𝑞_𝑗∶𝑗=1,2,…,𝜇}. (131)

Lemma 21 (see [5]). All the elements of the block diagonal matrix 𝐻^𝑞∗_𝑞⁻¹ are zero, that is, 𝐻^𝑞∗−1_𝑖𝑗=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{𝑞_𝑗∶𝑗=1,2,…,𝑟}.

Theorem 22. The condition which makes the system (24) solvable and provides an acceptable choice of vector 𝐚 is given by 𝐻^𝑞∗_𝑞⁻¹𝑦_q(0⁺)=0_𝑞. (132)

Proof. First, we obtain system (129), that is, 𝐻^𝑞∗_𝑞⁻¹𝑦_𝑞(𝑡)+𝐻^𝑞∗_𝑞⁻¹𝑏̃_𝑞𝑢_𝑜(𝑡)=0_𝑞. Then, for the solution of system (129), we take into account that 𝑢₀(𝑡)=lim_𝜎→0(𝛼₀1𝜎𝑡Φ(𝜎)+𝛼₁1𝜎²Φ^′(𝑡𝜎)+⋯+𝛼_𝑛−11𝜎^𝑛Φ^(𝑛−1)(𝑡𝜎)). (133) Following Remark 1, we can assume that 𝑡=𝐾𝜎 with 𝐾 big enough so that Φ^(𝑘)(𝐾)→0,forevery𝑘=1,2,…,𝑛. (134) Moreover, we have 𝐾𝜎→0⁺ and 𝑡→0⁺. Then, we take lim_𝑡→0+(𝐻^𝑞∗_𝑞⁻¹𝑦_𝑞(𝑡)+𝐻^𝑞∗_𝑞⁻¹𝑏̃_𝑞𝑢_𝑜(𝑡))=0_𝑞⇔𝐻^𝑞∗_𝑞⁻¹lim_𝑡→0+𝑦_𝑞(𝑡)+𝐻^𝑞∗_𝑞⁻¹𝑏̃_𝑞lim_𝜎→0𝑢(t). (135) However, lim_𝜎→0𝑢(𝑡)=lim_𝜎→0(𝛼₀1𝜎𝑡Φ(𝜎)+⋯+𝛼_𝑛−11𝜎^𝑛Φ^(𝑛−1)(𝑡𝜎))=lim_𝜎→0(𝛼₀1𝜎1Φ(𝐾)+⋯+𝜎^𝑛Φ^(𝑛−1)(𝐾))=0,lim_𝑡→0+𝑦_𝑞(𝑡)=𝑦_𝑞(0⁺). (136) 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 𝑥(0⁺)=(𝐼_𝑛−𝐺⁽¹⁾𝐺)𝑋, (137) where 𝑞𝐺=^{𝑝+𝑞∗,1}𝑞_{𝑝+𝑞∗,1}⋯𝑞_{𝑝+𝑞∗,𝑛}𝑞_{𝑝+2𝑞∗,1}𝑞_{𝑝+2𝑞∗,2}⋯𝑞_{𝑝+2𝑞∗,𝑛}𝑞⋮⋮⋯⋮_{𝑝+𝜉𝑞∗,1}𝑞_{𝑝+𝜉𝑞∗,2}⋯𝑞_{𝑝+𝜉𝑞∗,𝑛}∈ℳ_𝜉𝑛;(𝑞_𝑖𝑗)_{𝑖=𝑝+𝑞}^∗_{,…,𝑝+𝑟𝑞}^∗_{𝑗=1,2,…,𝑛} (138) are elements of the matrix 𝑄, the 𝐺⁽¹⁾ is {1}-inverse of matrix 𝐺, and for an arbitrarily chosen 𝑋∈ℳ_𝑛1.

Proof. Considering the transformation (25), we obtain 𝑦(0⁺)=𝑄⁻¹𝑥(0⁺) (139) or equivalently by defining 𝑄≜𝑄⁻¹𝑦_𝑝(0⁺)𝑦_𝑞(0⁺)=(𝑞_𝑖𝑗)_{𝑖,𝑗=1,2,…,𝑛}𝑥_𝑝(0⁺)𝑥_𝑞(0⁺)⇔𝑦_𝑝(0⁺)𝑦_𝑞(0⁺)=(^𝑛_𝑗=1𝑞_𝑖𝑗𝑥_𝑗(0⁺))_{𝑖=1,2,…,𝑛} (140) and finally 𝐻^𝑞∗_𝑞⁻¹𝑦_𝑞(0⁺)=0_𝑞⇒^𝑛𝑗=1𝑞_{𝑝+𝑞∗,𝑗}𝑥_𝑗(0⁺)=0,_𝑛𝑗=1𝑞_{𝑝+2𝑞∗,𝑗}𝑥_𝑗(0⁺⋮)=0,_𝑛𝑗=1𝑞_{𝑝+𝜉𝑞∗,𝑗}𝑥_𝑗(0⁺)=0. (141) System (141) can be expressed as follows: 𝐺𝑥(0⁺)=0, (142) where 𝑞𝐺=^{𝑝+𝑞∗,1}𝑞_{𝑝+𝑞∗,1}⋯𝑞_{𝑝+𝑞∗,𝑛}𝑞_{𝑝+2𝑞∗,1}𝑞_{𝑝+2𝑞∗,2}⋯𝑞_{𝑝+2𝑞∗,𝑛}𝑞⋮⋮⋯⋮_{𝑝+𝜉𝑞∗,1}𝑞_{𝑝+𝜉𝑞∗,2}⋯𝑞_{𝑝+𝜉𝑞∗,𝑛}∈ℳ_𝜉𝑛. (143) 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

𝐹𝑥^′′(𝑡)=𝐺𝑥(𝑡)+𝑏𝑢(𝑡), (144) where the matrix coefficients are given by

,,𝐹=01−101101−63−21−1−21−11−10−1−11−32−10−1−100000000010−1−100−31−1100𝐺=101−10−1−1−2−2−21134−1201−2−11−10−110120010−1−10−11−11011−2−1−21123 (145) and vector coefficient 𝑏=[^1001101]^𝑡.

Let us suppose that the system at time 0⁻ is

𝑥(0⁻)=𝑥^′(0⁻)=0000000^𝑡, (146) and at time 0⁺it achieves

𝑥(0⁺)=𝑥^′(0⁺)=−3−223121^𝑡. (147) From the regularity of 𝑠𝐹−𝐺, there exist nonsingular matrices in ℝ^7×7

,,𝑃=000100−110000100−101001100100000−1010000000100000100𝑄=100100−1010000001001111010110011010000−100100001000 (148) such that

𝑃𝐹𝑄=𝐹_𝑤=𝐼₂𝕆_5,2𝕆_5,2𝐻₅,𝑃𝐺𝑄=𝐺_𝑤=𝐽₂(1)𝕆_5,2𝕆_5,2𝐼₅, (149) where 𝐼₂,𝐽₂(1),𝐻₅ are given by (17).

Consequently, system (144) is divided into the following subsystems:

𝑦₂^′′(𝑡)=𝑦2102²(𝑡)+𝑏₂𝑢_𝑜𝑦(𝑡),0100010000100^5′′(𝑡)=𝑦₅(𝑡)+𝑏₅𝑢_𝑜(𝑡). (150) Moreover, we obtain

𝑢_𝑜(𝑡)=𝑎_𝑜𝛿(𝑡)+𝑎₁𝛿⁽¹⁾(𝑡)+⋯+𝑎₅𝛿⁽⁵⁾(𝑡)+𝑎₆𝛿⁽⁶⁾(𝑡). (151) Before we go further with the analytic determination of the unknown coefficients 𝑎_𝑖, we should verify the important condition (132). First, we calculate the inverse matrix of 𝑄,

𝑄⁻¹=(𝑞_𝑖𝑗)_{𝑖,𝑗=1,2,…,𝑛}=,𝑦1010−1−1−10100000−11−1100100000011−21−110−1−11−110110010−1−10(0⁺)=𝑄⁻¹𝑥(0⁺𝑦)=²(0⁺)𝑦₅(0⁺)=−5−2−3105−1^𝑡. (152) Consequently, it is not difficult to verify that condition (132) holds, that is, 𝐻²₅𝑦₅(0⁺)=0₅.

Furthermore, the 𝑏̃≜𝑃𝑏=[^0122101]^𝑇 should be calculated. Thus

𝑏̃₂=01,𝑏̃₅=22101. (153) Now, the approximate expression for (151) is given by

𝑎𝑢(𝑡)=^𝑜𝜎𝑡Φ(𝜎𝑎)+¹𝜎²Φ⁽¹⁾(𝑡𝜎)𝑎+⋯+⁵𝜎⁵Φ⁽⁵⁾(𝑡𝜎𝑎)+⁶𝜎⁶Φ⁽⁶⁾(𝑡𝜎). (154) 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

𝐳^′(𝑡)=𝐑𝐳(𝑡)+𝐋𝑢_𝑜(𝑡), (155) where 𝐳(𝑡)=[^𝑧𝑡₁^{(𝑡)𝑧𝑡}₂^(𝑡)]^𝑡 and the coefficient matrices 𝐑 and 𝐋 are given by

𝕆𝐑=²𝐼₂𝐽₂(1)𝕆₂=,𝐋0010000111000100=0₂𝑏̃₂=0001. (156) Considering (25), at time 0⁺ we obtain

𝑦(0⁺)=𝑄⁻¹𝑥(0⁺)=𝑦^′(0⁺)=𝑄⁻¹𝑥^′(0⁺). (157) Moreover, 𝑦₂(0⁺)=−5−2,𝑦^′₂(0⁺)=−5−2. (158) Furthermore, 𝐳(0⁺)=𝑦^𝑡₂(0⁺)𝑦₂^′𝑡(0⁺)^𝑡=−5−2−5−2. (159) There exists an invertible matrix

𝐶=1/41/41/4−1/40−1/201/2−1/401/4001/201/2 (160) such that 𝐉=𝐶⁻¹𝐑𝐶 is Jordan canonical form of matrix 𝐑. Analytically,

𝐉=−110−11101=𝐽₂(−1)𝐽₂(1). (161) Now, taking into consideration (52) and Remark 1 the solution is given by

𝐶⁻¹𝐳(𝑡)=lim_𝜎→0{^𝑡_−∞𝑒^{−𝐽2(−1)𝜏}𝐛₂𝑢(𝜏)𝑑𝜏}lim_𝜎→0{^𝑡_−∞𝑒^{−𝐽2(1)𝜏}𝐛₂𝑢(𝜏)𝑑𝜏}, (162) where 𝐛≜𝐶⁻¹𝐋=𝐛₂𝐛₂=0101. (163) Using (55), we obtain

𝐶⁻¹𝐳(𝑡)=−^𝑡_−∞𝜏𝑒^𝜏𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^𝜏−𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^−𝜏𝜏𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^−𝜏𝑢(𝜏)𝑑𝜏=Φ⁻¹(𝐾)^6𝑘=0𝛼_𝑘𝑑𝜆^𝑘₁𝑑𝜆_16𝑘=0𝛼_𝑘𝜆^𝑘_16𝑘=0𝛼_𝑘𝑑𝜆^𝑘₂𝑑𝜆_26𝑘=0𝛼_𝑘𝜆^𝑘₂=Φ⁻¹(𝐾)𝐚01−23−45−61−11−11−1101234561111111,𝐀𝐀=^𝜏1=2𝐀_𝜏2=2=1−11−11−1101−23−45−611111110123456 (164) or equivalently

𝐀_𝜏1=2=,𝐀1−11−11−1101−23−45−6_𝜏2=2=.11111110123456 (165) Thus,

𝑃₁(2,1)𝐀_𝜏1=2==𝐀10−12−34−501−23−45−6^𝜏1=2,𝐀_𝜏1=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 (166) where

𝐏₁=1101,𝐐₁=101−23−45012−34−5600100000001000000010000000100000001. (167) Moreover,

𝑃₁(2,−1)𝐀_𝜏2=2==𝐀10−1−2−3−4−50123456^𝜏2=2,𝐀_𝜏1=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 (168) where

𝐏₂=,𝐐1−101₂=.101234501−2−3−4−5−600100000001000000010000000100000001 (169) Thus, considering the result of Section 3, the {1}-inverse of matrices 𝐀_𝜏1=2 and 𝐀_𝜏2=2 are given by,

𝑄₁𝐼₂𝕆_5,2𝐏₁,𝐐₂𝐼₂𝕆_5,2𝐏₂,respectively, (170) Finally, the general solution is given by the expression

𝐚=1Φ⁻¹𝐀(𝐾)^𝜏(1)1=2𝐝_𝜏1=2(0⁺)+(𝐼₇−𝐀_𝜏⁽¹⁾₁₌₂𝐀_𝜏1=2)𝜓, (171) where the following conditions should also be satisfied:

𝐀_𝜏1=2[1Φ⁻¹𝐀(𝐾)^𝜏(1)2=2𝐝_𝜏2=2(0⁺)+(𝐼₇−𝐀_𝜏⁽¹⁾₂₌₂𝐀_𝜏2=2)𝜓]=𝐝_𝜏1=2,𝐀_𝜏2=2[1Φ⁻¹𝐀(𝐾)^𝜏(1)1=2𝐝_𝜏1=2(0⁺)+(𝐼₇−𝐀_𝜏⁽¹⁾₁₌₂𝐀_𝜏1=2)𝜓]=𝐝_𝜏2=2. (172) Analytically,

𝐝(0⁺)=𝐶⁻¹𝐳(0⁺𝐝)=^𝜏1=2(0⁺)𝐝_𝜏2=2(0⁺)=1Φ⁻¹0,𝐚(𝐾)−2−22−4=1Φ⁻¹000000+11111𝜓(𝐾)−2001−23−45002−34−56𝜓¹₂𝜓₃𝜓₄𝜓₅𝜓₆𝜓₇, (173) where 11−11−11−1101−23−45−6×[Φ⁻¹000000+11111𝜓(𝐾)−2001−23−45002−34−56𝜓¹₂𝜓₃𝜓₄𝜓₅𝜓₆𝜓₇=1]Φ⁻¹0⇔1(𝐾)−2Φ⁻¹0+(𝐾)−2−𝜓₃+𝜓₄−𝜓₅+𝜓₆−𝜓₇2𝜓₃−3𝜓₄+4𝜓₅−5𝜓₆+6𝜓₇=1Φ⁻¹01(𝐾)−2,11111110123456×[Φ⁻¹00000+11111𝜓(𝐾)−18−4001234500−2−3−4−5−6𝜓¹₂𝜓₃𝜓₄𝜓₅𝜓₆𝜓₇=1]Φ⁻¹⇔1(𝐾)−22−4Φ⁻¹−𝜓(𝐾)−22−4³+𝜓₄+𝜓₅+𝜓₆+𝜓₇2𝜓₃+3𝜓₄+4𝜓₅+5𝜓₆+6𝜓₇=1Φ⁻¹(𝐾)−22−4. (174)(1)(2)(3)(4)(5)(6)(7)(8)(9) 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 𝑎_𝑖, for 𝑖=0,1,…,6

𝐚=𝑎₀𝑎₁𝑎₂𝑎₃𝑎₄𝑎₅𝑎₆=−2Φ⁻¹(𝐾)+𝜓₃+3𝜓₅+5𝜓₇2𝜓₃+4𝜓₅+6𝜓₇𝜓₃0𝜓₅0𝜓₇ (175) for arbitrary 𝜓_𝑖∈ℝ, for 𝑖=3,5,7 and Φ⁻¹(K)=(1/√2𝜋)∫^𝐾_−∞𝑒^−𝑥2/2𝑑𝑥; 𝐾 is large enough (i.e., 𝐾→∞) that the assumption lim_𝑥→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 the 𝛿-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 state 𝜀-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.

Appendices

A. The Analytic Calculation of Integral lim_𝜎→0∫^𝑡_−∞𝑒^−𝐽0𝑡𝑏_𝑠𝑢(𝜏)𝑑𝜏

Consider that

lim_𝜎→0^𝑡_−∞𝑒^−𝐽0𝑡𝐛_𝑠=𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^−𝐽1𝜏⋱⋱𝑒^{−𝐽𝑙𝜏}𝐛_𝜏1𝐛_𝜏2⋮𝐛_𝜏𝑙𝑢(𝜏)𝑑𝜏, (A.1) and finally, we obtain

lim_𝜎→0^𝑡_−∞𝑒^−𝐽0𝑡𝐛_𝑠𝑢(𝜏)𝑑𝜏=^𝑡−∞𝑒^−𝐽1𝜏𝐛_𝜏1⋮𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝐽𝑖𝜏}𝐛_𝜏𝑖⋮𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝐽𝑙𝜏}𝐛_𝜏𝑙𝑢(𝜏)𝑑𝜏∈ℳ_𝑠1 (A.2) with 𝑖=1,2,…,𝑙and 𝐛_𝜏𝑖∈ℳ_(𝜏𝑖)1. Moreover, we take for every 𝑖=1,2,…,𝑙,

^𝑡_−∞𝑒^{−𝐽𝑖𝜏}𝐛_𝜏𝑖𝑢(𝜏)𝑑𝜏=^𝑡_−∞𝑒^{−𝜆𝑖𝜏}𝐛_𝜏𝑖𝑢(𝜏)𝑑𝜏=𝐛_𝜏𝑖^𝑡_−∞𝑒^{−𝜆𝑖𝜏}𝑢(𝜏)𝑑𝜏. (A.3) Afterwards, we calculate the integral ∫^𝑡_−∞𝑒^{−𝜆𝑖𝜏}𝑢(𝜏)𝑑𝜏.

Analytically, considering also (21), we obtain

^𝑡_−∞𝑒^{−𝜆𝑖𝜏}𝑢(𝜏)𝑑𝜏=^𝑡_−∞𝑒^{−𝜆𝑖𝜏𝑛−1}_𝑘=01𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑎_𝑘𝑑𝜏, (A.4) for every 𝑖=1,2,…,𝑞. Thus,

^𝑡_−∞𝑒^{−𝜆𝑖𝜏}𝑢(𝜏)𝑑𝜏=^𝑛−1_𝑘=0𝑎_𝑘^𝑡_−∞𝑒^{−𝜆𝑖𝜏}1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏, (A.5) where (1/𝜎)Φ⁰(𝜏/𝜎)=(1/𝜎)Φ(𝜏/𝜎)andΦ^(𝑚)(𝜏/𝜎)=𝑑^𝑚Φ(𝜏/𝜎)/𝑑(𝜏/𝜎)^𝑚.

Remark 5. It is well-known that ∫^𝑡_−∞(1/√2𝜋)𝑒^−𝜔2/2𝑑𝜔=Φ⁻¹(𝑡). Consequently, ^𝑡_−∞Φ(𝜔)𝑑𝜔=Φ⁻¹(𝑡), (A.6) where Φ⁻¹(𝑡)is the inverse function of normal probability distribution which is also a continuous one.

Lemma 24. One has ^𝑡_−∞𝑒^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏=𝑒^−𝜆𝑡(^𝑘−1_ℓ=0𝜆^{𝑘−ℓ−1}Φ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘𝑒^𝜆2𝜎2/2Φ⁻¹(𝑡𝜎+𝜆𝜎). (A.7)

Proof. First, we calculate the integral ^𝑡_−∞𝑒^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏=𝑒^{(𝜆2𝜎2)/2}⋅1𝜎^𝑡_−∞1√𝑒2𝜋^{−(𝜏/𝜎+𝜆𝜎)2/2}𝑑𝜏. (A.8) Afterwards, we denote (𝜏/𝜎)+𝜆𝜎=𝜔. Then, d 𝜏=𝜎⋅ d 𝜔.
Consequently, we obtain
𝑒^𝜆2𝜎2/21𝜎^{𝑡/𝜎+𝜆𝜎}_−∞1√𝑒2𝜋^−𝜔2/2𝜎𝑑𝜔=𝑒^{𝜆2𝜎2/2𝑡/𝜎+𝜆𝜎}_−∞1√𝑒2𝜋^−𝜔2/2𝑑𝜔. (A.9) Then, ^𝑡_−∞𝑒^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏=𝑒^𝜆2𝜎2/2Φ⁻¹(𝑡𝜎+𝜆𝜎). (A.10) Continuing our calculations, we take ^𝑡_−∞𝑒^−𝜆𝜏Φ⁽¹⁾(𝜏/𝜎)𝜎²=1𝑑𝜏𝜎[𝑒^−𝜆𝜏𝜏Φ(𝜎)]^𝑡_−∞1+𝜆𝜎^𝑡_−∞𝑒^−𝜆𝜏𝜏Φ(𝜎)𝑑𝜏(Butlim_{𝜏→−∞}𝑒^−𝜆𝜏𝜏Φ(𝜎)=0). (A.11) Moreover, considering also (A.10), we obtain ^𝑡_−∞𝑒^−𝜆𝜏Φ⁽¹⁾(𝜏/𝜎)𝜎²=1𝑑𝜏𝜎𝑒^−𝜆𝑡𝑡Φ(𝜎)+𝜆𝑒^𝜆2𝜎2/2Φ⁻¹(𝑡𝜎+𝜆𝜎). (A.12) Continuing the above procedure, we have to calculate the integral ∫^𝑡_−∞𝑒^−𝜆𝜏(Φ⁽²⁾(𝜏/𝜎)/𝜎³)𝑑𝜏. Thus, ^𝑡_−∞𝑒^−𝜆𝜏Φ⁽²⁾(𝜏/𝜎)𝜎³𝑑𝜏= (A.13) (we denote 𝜏/𝜎=𝑧⇔𝜏=𝜎𝑧, then d 𝜏=𝜎𝑑𝑧and −∞/𝜎<𝑧<𝑡/𝜎. Note that −∞/𝜎=−∞ because 𝜎>0—by definition of the normal probability distribution) =^𝑡/𝜎_−∞𝑒^{−𝜆𝜎𝑧}Φ⁽²⁾(𝑧)𝜎³=1⋅𝜎𝑑𝑧𝜎²𝑒^−𝜆𝑡Φ⁽¹⁾(𝑡𝜎1)+𝜎²𝜆𝜎^𝑡/𝜎_−∞𝑒^{−𝜆𝜎𝑧}Φ⁽¹⁾(𝑧)𝑑𝑧= (A.14) (we reuse the transformation 𝑧=𝜏/𝜎) =1𝜎²𝑒^−𝜆𝑡Φ⁽¹⁾(𝑡𝜎)1+𝜆[𝜎𝑒^−𝜆𝑡𝑡Φ(𝜎)+𝜆𝑒^{(𝜆2𝜎2)/2}Φ⁻¹(𝑡𝜎+𝜆𝜎)]. (A.15) Consequently, we obtain ^𝑡_−∞𝑒^−𝜆𝜏Φ⁽²⁾(𝜏/𝜎)𝜎³1𝑑𝜏=𝜎²𝑒^−𝜆𝑡Φ⁽¹⁾(𝑡𝜎1)+𝜆𝜎𝑒^−𝜆𝑡𝑡Φ(𝜎)+𝜆²𝑒^𝜆2𝜎2/2Φ⁻¹(𝑡𝜎+𝜆𝜎). (A.16) Following the previous expressions, we assume that ^𝑡_−∞𝑒^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝑡𝜎)𝑑𝜏=𝑒^−𝜆𝑡(^𝑘−1_ℓ=0𝜆^{𝑘−ℓ−1}Φ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘𝑒^𝜆2𝜎2/2Φ⁻¹(𝑡𝜎+𝜆𝜎). (A.17) Now, in order to conclude the proof, the recursive formula should be proved, that is, ^𝑡_−∞𝑒^−𝜆𝜏1𝜎^𝑘+2Φ^(𝑘+1)(𝜏𝜎)𝑑𝜏=𝑒^−𝜆𝑡(^𝑘_ℓ=0𝜆^𝑘−ℓΦ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘+1𝑒^𝜆2𝜎2/2Φ⁻¹(𝑡𝜎+𝜆𝜎). (A.18) Thus, we have ^𝑡_−∞𝑒^−𝜆𝜏1𝜎^𝑘+2Φ^(𝑘+1)(𝜏𝜎𝜏)𝑑𝜏(denote𝜎=1=𝑧)𝜎^𝑘+1𝑒^−𝜆𝑡Φ^(𝑘)(𝑡𝜎1)+𝜆𝜎^{𝑘+1𝑡/𝜎}_−∞𝑒^{−𝜆𝜎𝑧}Φ^(𝑘)(𝑧)𝜎𝑑𝑧 (A.19) (reusing the transformation 𝑧=𝜏/𝜎) ^(A=^.16)1𝜎^𝑘+1𝑒^−𝜆𝑡Φ^(𝑘)(𝑡𝜎)+𝑒^{−𝜆𝑡𝑘−1}_ℓ=0𝜆^𝑘−ℓΦ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1+𝜆^𝑘+1𝑒^𝜆2𝜎2/2Φ⁻¹(𝑡𝜎+𝜆𝜎). (A.20) However, the 1𝜎^𝑘+1𝑒^−𝜆𝑡Φ^(𝑘)(𝑡𝜎)+𝑒^{−𝜆𝑡𝑘−1}_ℓ=0𝜆^𝑘−ℓΦ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1=𝑒^{𝑘−𝜆𝑡}_ℓ=0𝜆^𝑘−ℓΦ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1. (A.21) Consequently, we have proved that ^𝑡_−∞𝜎^1/(𝑘+2)𝑒^−𝜆𝜏Φ^(𝑘+1)(𝜏𝜎)𝑑𝜏=𝑒^−𝜆𝑡(^𝑘_ℓ=0𝜆^𝑘−ℓΦ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘+1𝑒^𝜆2𝜎2/2Φ⁻¹(𝑡𝜎+𝜆𝜎). (A.22) And the proof has been concluded.

Lemma 25. One has ^𝑡_−∞𝑒^{−𝜆𝑖𝜏}𝑢(𝜏)𝑑𝜏=^𝑛−1_𝑘=0𝛼_𝑘[𝑒^{−𝜆𝑖𝑡}(^𝑘−1_ℓ=0𝜆_𝑖^{𝑘−ℓ−1}Φ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘_𝑖𝑒^{𝜆2𝑖𝜎2/2}Φ⁻¹(𝑡𝜎+𝜆_𝑖𝜎)]. (A.23)

Proof. We have seen that ^𝑡_−∞𝑒^{−𝜆𝑖𝜏}𝑢(𝜏)𝑑𝜏_(A.17)=^𝑛−1_𝑘=0𝛼_𝑘[𝑒^{−𝜆𝑖𝑡}(^𝑘−1_ℓ=0𝜆_𝑖^{𝑘−ℓ−1}Φ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘_𝑖𝑒^{𝜆2𝑖𝜎2/2}Φ⁻¹(𝑡𝜎+𝜆_𝑖𝜎)]. (A.24)

Substituting (A.23) into (A.3), we obtain

𝐛_𝜏𝑖^𝑡_−∞𝑒^{−𝜆𝑖𝜏}𝑢(𝜏)𝑑𝜏=𝐛_𝜏𝑖^𝑛−1_𝑘=0𝛼_𝑘[𝑒^{−𝜆𝑖𝑡}(^𝑘−1_ℓ=0𝜆_𝑖^{𝑘−ℓ−1}Φ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘_𝑖𝑒^{𝜆2𝑖𝜎2/2}Φ⁻¹(𝑡𝜎+𝜆_𝑖𝜎)]. (A.25) Thus, combining (A.2) and (A.25), we obtain

lim_𝜎→0^𝑡_−∞𝑒^−𝐽0𝑡𝐛_𝑠𝐛𝑢(𝜏)𝑑𝜏=^{𝜏1𝑛−1𝑘=0}𝛼_𝑘[𝑒^−𝜆1𝑡(^𝑘−1_ℓ=0𝜆₁^{𝑘−ℓ−1}Φ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘₁𝑒^{𝜆21𝜎2/2}Φ⁻¹(𝑡𝜎+𝜆₁⋮𝐛𝜎)]_{𝜏𝑖𝑛−1𝑘=0}𝛼_𝑘[𝑒^{−𝜆𝑖𝑡}(^𝑘−1_ℓ=0𝜆_𝑖^{𝑘−ℓ−1}Φ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘_𝑖𝑒^{𝜆2𝑖𝜎2/2}Φ⁻¹(𝑡𝜎+𝜆_𝑖⋮𝐛𝜎)]_{𝜏𝑙𝑛−1𝑘=0}𝛼_𝑘[𝑒^{−𝜆𝑙𝑡}(^𝑘−1_ℓ=0𝜆_𝑙^{𝑘−ℓ−1}Φ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘_𝑙𝑒^{𝜆2𝑙𝜎2/2}Φ⁻¹(𝑡𝜎+𝜆_𝑙𝜎)]∈ℳ_𝑠1. (A.26)

Lemma 26. One has lim_𝜎→0^𝑛−1_𝑘=0𝛼_𝑘[𝑒^{−𝜆𝑖𝑡}(^𝑘−1_ℓ=0𝜆_𝑖^{𝑘−ℓ−1}Φ^(ℓ)(𝑡/𝜎)𝜎^ℓ+1)+𝜆^𝑘_𝑖𝑒^{𝜆2𝑖𝜎2/2}Φ⁻¹(𝑡𝜎+𝜆_𝑖𝜎)]≈Φ⁻¹(𝐾)^𝑛−1_𝑘=0𝛼_𝑘𝜆^𝑘_𝑖for𝑖=1,2,…,𝑙. (A.27)

Proof. Considering here, Remark  2.1, and denoting t = K 𝜎 with K big enough such as Φ^(𝑘)(𝐾)𝜎^𝑘+1≈0forevery𝑘=0,1,2,…,𝑛−1, (A.28) then we obtain (A.27).

This appendix is finished with (A.29). This equation is a consequence of (A.26) and (A.27)

lim_𝜎→0^𝑡_−∞𝑒^−𝐽0𝑡𝐛_𝑠𝑢(𝜏)𝑑𝜏=Φ⁻¹𝐛(𝐾)^{𝜏1𝑛−1𝑘=0}𝛼_𝑘𝜆^𝑘₁⋮𝐛_𝜏𝑖^𝑛−1_𝑘=0𝛼_𝑘𝜆^𝑘_𝑖⋮𝐛_𝜏𝑙^𝑛−1_𝑘=0𝛼_𝑘𝜆^𝑘_𝑙∈ℳ_𝑠1. (A.29)

B. The Analytic Calculation of Integral lim_𝜎→0∫^𝑡_−∞𝑒^{−𝐽𝑖𝑡}𝑏_𝜏𝑖𝑢(𝜏)𝑑𝜏 for 𝑖=𝑙+1,𝑙+2,…,𝑘

Consider that lim_𝜎→0^𝑡_−∞𝑒^{−𝐽𝑗𝑡}𝐛_𝜏𝑗=𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝐽𝑗1𝜏}𝑒^{−𝐽𝑗2𝜏}⋱𝑒^{−𝐽𝑗𝑗𝑑𝜏}𝐛_𝑗1𝐛_𝑗2⋮𝐛_𝑗𝑑𝑗=𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝐽𝑗1𝜏}𝐓_𝑗1𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝐽𝑗2𝜏}𝐓_𝑗2⋮𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝐽𝑗𝑑𝑗𝜏}𝐓_𝑗𝑑𝑗𝑢(𝜏)𝑑𝜏 (B.1) with 𝑧_𝑗=1,2,…,𝑑_𝑗 and the matrix 𝐛_𝑗𝑧𝑗∈ℳ_{(𝑗𝑧𝑗)1}.

Thus,

^𝑡_−∞𝑒^{−𝐽𝑗𝑧𝑗𝜏}𝐛_𝑗𝑧𝑗𝑢(𝜏)𝑑𝜏=^𝑡_−∞𝑒^{−𝜆𝑗𝜏}(−1)𝜏𝑒^{−𝜆𝑗𝜏}⋯(−1)^{(𝜇𝑧𝑗−1)}𝜏^{(𝜇𝑧𝑗−1)}(𝜇_𝑧𝑗𝑒−1)!^{−𝜆𝑗𝜏}0𝑒^{−𝜆𝑗𝜏}⋯(−1)^{(𝜇𝑧𝑗−2)}𝜏^{(𝜇𝑧𝑗−2)}(𝜇_𝑧𝑗𝑒−2)!^{−𝜆𝑗𝜏}⋮⋮⋱⋮000𝑒^𝜆𝑗𝜏=𝐛¹_𝑗𝑧𝑗𝐛²_𝑗𝑧𝑗⋮𝐛^𝜇𝑗_𝑗𝑧𝑗=𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝜆𝑗𝜏}𝐛_𝑗^1,_𝑧𝑗𝑢(𝜏)𝑑𝜏+^𝑡_−∞(−1)𝜏𝑒^{−𝜆𝑗𝜏}𝐛²_𝑗𝑧𝑗𝑢(𝜏)𝑑𝜏+⋯+^𝑡_−∞(−1)^{(𝜇𝑧𝑗−1)}𝜏^{(𝜇𝑧𝑗−1)}(𝜇_𝑧𝑗𝑒−1)!^{−𝜆𝑗𝜏}𝐛^𝜇𝑗_𝑗𝑧𝑗𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝜆𝑗𝜏}𝐛_𝑗^2,_𝑧𝑗𝑢(𝜏)𝑑𝜏+⋯+^𝑡_−∞(−1)^{(𝜇𝑧𝑗−2)}𝜏^{(𝜇𝑧𝑗−2)}(𝜇_𝑧𝑗𝐛−2)!^{𝜇𝑗𝑗𝑧𝑗}⋮𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝜆𝑗𝜏}𝐛^𝜇𝑗_𝑗𝑧𝑗.𝑢(𝜏)𝑑𝜏 (B.2)

Remark 6. According to (B.2), in order to able be to calculate the integral lim_𝜎→0∫^𝑡_−∞𝑒^{−𝐽𝑖𝑡}𝐛_𝜏𝑖𝑢(𝜏)𝑑𝜏, we should calculate the lim_𝜎→0∫^𝑡_−∞(𝜏^𝜌/𝜌!)𝑒^−𝜆𝜏𝑢(𝜏)𝑑𝜏.

Lemma 27. One has lim_𝜎→0^𝑡_−∞𝜏^𝜌𝑒𝜌!^−𝜆𝜏𝑢(𝜏)𝑑𝜏=(−1)^𝜌(^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝜌𝜆^𝑘𝑑𝜆^𝜌1𝜌!)Φ⁻¹(𝐾). (B.3) However, before we prove Lemma 27, the following lemmas should be considered firstly.

Lemma 28. One has lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏𝑢(𝜏)𝑑𝜏=−^𝑛−1_𝑘=0𝛼_𝑘𝑑𝜆^𝑘Φ𝑑𝜆⁻¹(𝐾). (B.4)

Proof. After some basic calculations, we obtain lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏𝑢(𝜏)𝑑𝜏=^𝑛−1_𝑘=0𝛼_𝑘lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏. (B.5) First, we calculate lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏=lim_𝜎→0(^𝑡_−∞𝜏1𝜎1√𝑒2𝜋^{−((𝜏/𝜎)+𝜆𝜎)2/2}𝑑𝜏⋅𝑒^{(𝜆2𝜎2)/2}). (B.6)Then lim_𝜎→0𝑒^𝜆2𝜎2/2=1 and we obtain lim_𝜎→0∫^𝑡_−∞𝜏(1/𝜎)(1/√2𝜋)𝑒^{−((𝜏/𝜎)+𝜆𝜎)2/2}𝑑𝜏.
Afterwards, we denote (𝜏/𝜎)+𝜆𝜎=𝑧⇔𝜏=𝜎𝑧−𝜆𝜎²and𝑑𝜏=𝜎𝑑𝑧.
And, we take
lim_𝜎→0^{(𝑡/𝜎)+𝜆𝜎}_−∞(𝜎𝑧−𝜆𝜎²)1√12𝜋𝜎𝑒^−𝑧2/2𝜎𝑑𝑧=lim_𝜎→0𝜎[−√𝑒2𝜋^{−((𝑡/𝜎)+𝜆𝜎)2/2}−𝜎²𝜆Φ⁻¹(𝑡𝜎+𝜆𝜎)]. (B.7) In this part, we use also Remark 2.1, with as 𝑡=𝐾𝜎 with 𝜎→0 and 𝐾≫0. Thus lim_𝜎→0𝜎[−√𝑒2𝜋^{−(𝐾+𝜆𝜎)2/2}−𝜎²𝜆Φ⁻¹(𝐾+𝜆𝜎)]=0. (B.8) Moreover, lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏=0. (B.9)
Now, we calculate
lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏Φ⁽¹⁾(𝜏/𝜎)𝜎²𝜏𝑑𝜏(denote𝜎=𝑧.Then𝑑𝜏=𝜎𝑑𝑧)=lim_𝜎→0[[𝑧𝑒^{−𝜆𝜎𝑧}Φ(𝑧)]^𝑡/𝜎_−∞−^𝑡/𝜎_−∞(𝑒^{−𝜆𝜎𝑧}−𝜆𝜎𝑧𝑒^{−𝜆𝜎𝑧})Φ(𝑧)𝑑𝑧]. (B.10) The lim_𝜎→0[𝑧𝑒^{−𝜆𝜎𝑧}Φ(𝑧)]^𝑡/𝜎_−∞=lim_𝜎→0𝑡/𝜎𝑒^−𝜆𝑡Φ(𝑡/𝜎)=0, as we have seen (using Remark  2.1).
Then, we obtain
−lim_𝜎→0^𝑡/𝜎_−∞𝑒^{−𝜆𝜎𝑧}Φ(𝑧)𝑑𝑧+𝜆lim_𝜎→0^𝑡/𝜎_−∞𝜎𝑧𝑒^{−𝜆𝜎𝑧}Φ(𝑧)𝑑𝑧=−lim_𝜎→0^𝑡_−∞𝑒^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏+𝜆lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏= (B.11) (using B.9 and A.10 ) =−Φ⁻¹(K) +𝜆⋅ 0 =−Φ⁻¹(K).
Thus, we conclude
lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏Φ⁽¹⁾(𝑡/𝜎)𝜎²𝑑𝜏=−Φ⁻¹(𝐾). (B.12) Now, we calculate lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏Φ⁽²⁾(𝜏/𝜎)𝜎³𝑑𝜏=lim_𝜎→0(1𝜎[𝑧𝑒^{−𝜆𝜎𝑧}Φ⁽¹⁾(𝑧)]^𝑡/𝜎_−∞−1𝜎^𝑡/𝜎_−∞(𝑒^{−𝜆𝜎𝑧}−𝜆𝜎𝑧𝑒^{−𝜆𝜎𝑧})Φ⁽¹⁾(𝑧)𝑑𝑧) (B.13)  (since 𝜎→0 and 𝑡=𝐾𝜎, then [𝑧𝑒^{−𝜆𝜎𝑧}Φ⁽¹⁾(𝑧)]^𝑡/𝜎_−∞→0). Thus, we obtain lim_𝜎→01(−𝜎^𝑡/𝜎_−∞𝑒^{−𝜆𝜎𝑧}Φ⁽¹⁾1(𝑧)𝑑𝑧+𝜆𝜎^𝑡/𝜎_−∞𝜎𝑧𝑒^{−𝜆𝜎𝑧}Φ⁽¹⁾(𝑧)𝑑𝑧)=lim_𝜎→0(−^𝑡_−∞𝑒^−𝜆𝜏1𝜎²Φ⁽¹⁾(𝜏𝜎)𝑑𝜏+𝜆^𝜏_−∞𝜏𝑒^−𝜆𝜏1𝜎²Φ⁽¹⁾(𝜏𝜎)𝑑𝜏) (B.14) (using B.9 and A.12 ) =−𝜆Φ⁻¹(𝐾)−𝜆Φ⁻¹(𝐾)=−2𝜆Φ⁻¹(𝐾).
Consequently,
lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏Φ⁽²⁾(𝜏/𝜎)𝜎³𝑑𝜏=−2𝜆Φ⁻¹(𝐾). (B.15) Generalized the above expression, we take lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏Φ^(𝑘)(𝜏/𝜎)𝜎^𝑘+1𝑑𝜏=−𝑑𝜆^𝑘Φ𝑑𝜆⁻¹(𝐾). (B.16) Now, it should be proved the recursive (B.17) lim_𝜎→0^𝑡_−∞𝜏𝑒^−𝜆𝜏Φ^(𝑘+1)(𝜏/𝜎)𝜎^𝑘+2𝑑𝜏=−𝑑𝜆^𝑘+1Φ𝑑𝜆⁻¹(𝐾). (B.17) Thus, we obtain lim_𝜎→0^𝑡/𝜎_−∞𝜎𝑧𝑒^{−𝜆𝜎𝑧}Φ^(𝑘+1)(𝑧)𝜎^𝑘+2𝜎𝑑𝑧=lim_𝜎→01(−𝜎^𝑘𝑡_−∞𝑒^−𝜆𝜏Φ^(𝑘)(𝜏/𝜎)𝜎𝑑𝜏+𝜆^𝑡_−∞𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏)𝑑𝜏)= (B.18) (using B.16 and A.16 ) =−(𝑑𝜆^𝑘+1/𝑑𝜆)Φ⁻¹(𝐾). The above expression concludes the recursive proof.

Lemma 29. One has lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏𝑢(𝜏)𝑑𝜏=(−1)²12!(^𝑛−1_𝑘=0𝛼_𝑘𝑑²𝜆^𝑘𝑑𝜆²)Φ⁻¹(𝐾). (B.19)

Proof. Consider that lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏=𝑢(𝜏)𝑑𝜏_{𝑛−1𝑘=0}𝛼_𝑘lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏. (B.20) First, we calculate lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏=lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎1√𝑒2𝜋^{−((𝜏/𝜎)+𝜆𝜎)2/2}𝜏𝑑𝜏=(Denote𝜎+𝜆𝜎=𝑧⇔𝜏=𝜎𝑧−𝜆𝜎²,𝑑𝜏=𝜎𝑑𝑧)=lim_𝜎→0^{𝑡/𝜎+𝜆𝜎}_−∞12!(𝜎𝑧−𝜆𝜎²)²1𝜎1√𝑒2𝜋⁻𝑧²2𝜎𝑑𝑧=lim_𝜎→0(𝜎²12!^{(𝑡/𝜎)+𝜆𝜎}_−∞𝑧²1√𝑒2𝜋^−𝑧2/2𝑑𝑧−2𝜆𝜎³2!^{(𝑡/𝜎)+𝜆𝜎}_−∞𝑧1√𝑒2𝜋^−𝑧2/2𝜆𝑑𝑧+²𝜎⁴2!^{(𝑡/𝜎)+𝜆𝜎}_−∞1√𝑒2𝜋^−𝑧2/2𝑑𝑧). (B.21) Following Remark  2.1, we assume that 𝜎→ 0 and t =𝐾𝜎 with 𝐾≫0. Thus, ^{(𝑡/𝜎)+𝜆𝜎}_−∞𝑧²1√𝑒2𝜋^−𝑧2/2𝑑𝑧≈0,_{(𝑡/𝜎)+𝜆𝜎−∞}𝑧1√𝑒2𝜋^−𝑧2/2𝑑𝑧≈0,_{(𝑡/𝜎)+𝜆𝜎−∞}1√𝑒2𝜋^−𝑧2/2𝑑𝑧≈Φ⁻¹(𝐾). (B.22) Moreover, we take lim_𝜎→0(𝜆²𝜎⁴/2!)Φ⁻¹(𝐾)=0. And lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏=0⋅Φ⁻¹(𝐾)=0. (B.23) Then, we calculate that lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎²Φ⁽¹⁾(𝜏𝜎)𝑑𝜏=lim_𝜎→0^𝑡/𝜎_−∞𝜎²𝑧²𝑒2!^{−𝜆𝜎𝑧}1𝜎²Φ⁽¹⁾(𝑧)𝜎𝑑𝑧=lim_𝜎→0(−^𝑡_−∞𝜏𝑒^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏+𝜆^𝑡_−∞𝜏²𝑒^−𝜆𝜏1𝜎𝜏Φ(𝜎)𝑑𝜏) (B.24) (using (B.9) and (B.23)) = 0.
Thus, we obtain
lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎²Φ⁽¹⁾(𝜏𝜎)𝑑𝜏=0. (B.25) Afterwards, we calculate lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎³Φ⁽²⁾(𝜏𝜎)𝑑𝜏=lim_𝜎→0(−^𝑡_−∞𝜏𝑒^−𝜆𝜏1𝜎²Φ⁽¹⁾(𝜏𝜎+1)𝑑𝜏2𝜆^𝑡_−∞𝜏²𝑒^−𝜆𝜏1𝜎²Φ⁽¹⁾(𝜏)𝑑𝜏)= (B.26) (using (B.12) and (B.25)) =−(−Φ⁻¹(𝐾) + 0) =Φ⁻¹(𝐾).
Hence, we conclude that
^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎³Φ⁽²⁾(𝜏𝜎)𝑑𝜏=Φ⁻¹(𝐾). (B.27) Now, we calculate that lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎⁴Φ⁽³⁾(𝜏𝜎)𝑑𝜏=lim_𝜎→0(−^𝑡_−∞𝜏𝑒^−𝜆𝜏1𝜎³Φ⁽²⁾(𝜏𝜎+1)𝑑𝜏2𝜆^t_−∞𝜏²𝑒^−𝜆𝜏1𝜎³Φ⁽²⁾(𝜏𝜎)𝑑𝜏)= (B.28) (using (B.15) and (B.27))=−(−2𝜆Φ⁻¹(𝐾)) +𝜆Φ⁻¹(𝐾) = 3𝜆Φ⁻¹(𝐾).
Hence, we obtain
lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎⁴Φ⁽³⁾(𝜏𝜎)𝑑𝜏=3𝜆Φ⁻¹(𝐾). (B.29) Combining the equations (B.23)–(B.29), we obtain recursively (B.30) lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎1)𝑑𝜏=𝑑2!²𝜆^𝑘𝑑𝜆²Φ⁻¹(𝐾). (B.30) Thus, in order to conclude, we should prove lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎^𝑘+2Φ^(𝑘+1)(𝜏𝜎1)𝑑𝜏=𝑑2!²𝜆^𝑘+1𝑑𝜆²Φ⁻¹(𝐾). (B.31) Hence, lim_𝜎→0^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎^𝑘+2Φ^(𝑘+1)(𝜏𝜎)𝑑𝜏=lim_𝜎→0112!𝜎^{𝑘−1𝑡/𝜎}_−∞𝑧²𝑒^{−𝜆𝜎𝑧}Φ^(𝑘+1)(𝑧)𝑑𝑧=lim_𝜎→0(−^𝑡_−∞𝜏𝑒^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏+𝜆^𝑡_−∞𝜏²𝑒2!^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏) (B.32) (using (B.16) and (B.30)) =((𝑘+1)𝑘/2!)𝜆^𝑘−1Φ⁻¹(𝐾)=(1/2!)(𝑑²𝜆^𝑘+1/𝑑𝜆²)Φ⁻¹(𝐾).
The proof is concluded by considering.

Lemma 30. One has lim_𝜎→0^𝑡_−∞𝜏^𝜌𝑒𝜌!^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏=(−1)^𝜌1𝑑𝜌!^𝜌𝜆^𝑘𝑑𝜆^𝜌Φ⁻¹(𝐾). (B.33)

Proof. For 𝜌= 1, we have proved (B.16). For 𝜌= 2, we have also proved (B.30). We assume that (B.33) is true.Now, we easily have to prove the following expression for 𝜌+ 1, that is,lim_𝜎→0^𝑡_−∞𝜏^𝜌+1𝑒(𝜌+1)!^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏=(−1)^𝜌+11𝑑(𝜌+1)!^𝜌+1𝜆^𝑘𝑑𝜆^𝜌+1Φ⁻¹(𝐾). (B.34)

Now, we return to the proof of Lemma 27.

Proof. Consider that lim_𝜎→0^𝑡_−∞𝜏^𝜌𝑒𝜌!^−𝜆𝜏=𝑢(𝜏)𝑑𝜏_{𝑛−1𝑘=0}𝛼_𝑘lim_𝜎→0^𝑡_−∞𝜏^𝜌𝑒𝜌!^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏. (B.35) Then, lim_𝜎→0^𝑡_−∞𝜏^𝜌𝑒𝜌!^−𝜆𝜏=𝑢(𝜏)𝑑𝜏_{𝑛−1𝑘=0}𝛼_𝑘⋅lim_𝜎→0^𝑡_−∞𝜏^𝜌𝑒𝜌!^−𝜆𝜏1𝜎^𝑘+1Φ^(𝑘)(𝜏𝜎)𝑑𝜏 (B.36) (considering (B.33)) =^𝑛−1_𝑘=0𝛼_𝑘(−1)^𝜌1𝑑𝜌!^𝜌𝜆^𝑘𝑑𝜆^𝜌Φ⁻¹(𝐾)=(−1)^𝜌1Φ𝜌!⁻¹(𝐾)(^𝑛−1_𝑘=01𝑑𝜌!^𝜌𝜆^𝑘𝑑𝜆^𝜌). (B.37)

Remark 7. In all the proofs of Lemma 27–30, we have used lim_𝜎→0[𝑧^𝜌𝑒^{−𝜆𝜎𝑧}Φ^(𝑘)(𝑧)]^𝑡/𝜎_−∞=0 (B.38) for every 𝜅 and every 𝜌. In this part of Appendix B, we return to expression (B.2) and we calculate lim_𝜎→0^𝑡_−∞𝑒^{−𝐽𝑗𝑧𝑗𝜏}𝐛_𝑗𝑧𝑗𝑢(𝜏)𝑑𝜏= (B.39) (using (A.23) and (B.3), and calculating the limit as 𝜎→ 0, we obtain what follows) lim_𝜎→0^𝑡−∞𝑒^{−𝜆𝑗𝜏𝜇𝑧𝑗}_𝑐=1𝐛^𝑐_𝑗𝑧𝑗(−1)^𝑐−1𝜏^𝑐−1(𝑐−1)!𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝜆𝑗𝜏𝜇𝑧𝑗}_𝑐=2𝐛^𝑐_𝑗𝑧𝑗(−1)^𝑐−2𝜏^𝑐−2⋮(𝑐−2)!𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝜆𝑗𝜏𝜇𝑧𝑗}_{𝑐=𝜇𝑧𝑗}𝐛^𝑐_𝑗𝑧𝑗(−1)^{𝑐−𝜇𝑧𝑗}𝜏^{𝑐−𝜇𝑧𝑗}(𝑐−𝜇_𝑧𝑗=)!𝑢(𝜏)𝑑𝜏_{𝜇𝑧𝑗𝑐=1}𝐛^𝑐_𝑗𝑧𝑗(−1)^𝑐−1⋅lim_𝜎→0^𝑡_−∞𝑒^{−𝜆𝑗𝜏}𝜏^𝑐−1(𝑐−1)!𝑢(𝜏)𝑑𝜏_{𝜇𝑧𝑗𝑐=2}𝐛^𝑐_𝑗𝑧𝑗(−1)^𝑐−2⋅lim_𝜎→0^𝑡_−∞𝑒^{−𝜆𝑗𝜏}𝜏^𝑐−2⋮(𝑐−2)!𝑢(𝜏)𝑑𝜏_{𝜇𝑧𝑗𝑐=𝜇𝑧𝑗}𝐛^𝑐_𝑗𝑧𝑗(−1)^{𝑐−𝜇𝑧𝑗}⋅lim_𝜎→0^𝑡_−∞𝑒^{−𝜆𝑗𝜏}𝜏^{𝑐−𝜇𝑧𝑗}(𝑐−𝜇_𝑧𝑗=)!𝑢(𝜏)𝑑𝜏_{𝜇𝑧𝑗𝑐=1}𝐛^𝑐_𝑗𝑧𝑗(−1)^2(𝑐−1)⋅1((𝑐−1)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−1𝜆^𝑘_𝑗𝑑𝜆_𝑗^𝑐−1)Φ⁻¹(𝐾)_{𝜇𝑧𝑗𝑐=2}𝐛^𝑐_𝑗𝑧𝑗(−1)^2(𝑐−2)⋅1((𝑐−2)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−2𝜆^𝑘_𝑗𝑑𝜆_𝑗^𝑐−2)Φ⁻¹⋮(𝐾)_{𝜇𝑧𝑗𝑐=𝜇𝑧𝑗}𝐛^𝑐_𝑗𝑧𝑗(−1)^{2(𝑐−𝜇𝑧𝑗)}⋅1(𝑐−𝜇_𝑧𝑗()!^𝑛−1_𝑘=0𝛼_𝑘𝑑^{𝑐−𝜇𝑧𝑗}𝜆^𝑘_𝑗𝑑𝜆^{𝑐−𝜇𝑧𝑗}_𝑗)Φ⁻¹.(𝐾) (B.40) Consequently, we obtain lim_𝜎→0^𝑡_−∞𝑒^{−𝐽𝑗𝑧𝑗𝜏}𝐛_𝑗𝑧𝑗𝑢(𝜏)𝑑𝜏=Φ⁻¹(𝐾)^{𝜇𝑧𝑗𝑐=1}𝐛^𝑐_𝑗𝑧𝑗1(𝑐−1)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−1𝜆^𝑘_𝑗𝑑𝜆_𝑗𝜇^𝑐−1_{𝑧𝑗𝑐=2}𝐛_𝑗^𝑐,_𝑧𝑗1(𝑐−2)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−2𝜆^𝑘_𝑗𝑑𝜆_𝑗^𝑐−2_{𝜇𝑧𝑗𝑐=𝜇𝑧𝑗}⋮𝐛^𝑐_𝑗𝑧𝑗1(𝑐−𝜇_𝑧𝑗)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^{𝑐−𝜇𝑧𝑗}𝜆^𝑘_𝑗𝑑𝜆^{𝑐−𝜇𝑧𝑗}_𝑗. (B.41) Combining (B.1) and (B.41), we obtain (B.42) for every𝑗=𝑙+1,𝑙+2,…,𝑘, that is, lim_𝜎→0^𝑡_−∞𝑒^{−𝐽𝑗𝑡}𝐛_𝜏𝑗=𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝐽𝑗1𝜏}𝐛_𝑗1𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝐽𝑗2𝜏}𝐛_𝑗2⋮𝑢(𝜏)𝑑𝜏_𝑡−∞𝑒^{−𝐽𝑗𝑑𝑗𝜏}𝐛_𝑗𝑑𝑗𝑢(𝜏)𝑑𝜏=Φ⁻¹(𝐾)^𝜇1𝑐=1𝐛^𝑐_𝑗11(𝑐−1)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−1𝜆^𝑘_𝑗𝑑𝜆_𝑗𝜇^𝑐−1_1𝑐=2𝐛^𝑐_𝑗11(𝑐−2)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−2𝜆^𝑘_𝑗𝑑𝜆_𝑗^𝑐−2_{𝜇1𝑐=𝜇1}⋮𝐛^𝑐_𝑗11(𝑐−𝜇₁)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−𝜇1𝜆^𝑘_𝑗𝑑𝜆^𝑐−𝜇1_𝑗}𝜇₁×1_𝜇2𝑐=1𝐛^𝑐_𝑗21(𝑐−1)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−1𝜆^𝑘_𝑗𝑑𝜆_𝑗𝜇^𝑐−1_2𝑐=2𝐛^𝑐_𝑗21(𝑐−2)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−2𝜆^𝑘_𝑗𝑑𝜆_𝑗^𝑐−2_{𝜇2𝑐=𝜇2}⋮𝐛^𝑐_𝑗21(𝑐−𝜇₂)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−𝜇2𝜆^𝑘_𝑗𝑑𝜆^𝑐−𝜇2_𝑗}𝜇₂⋮×1_{𝜇𝑑𝑗𝑐=1}𝐛^𝑐_𝑗𝑑𝑗1(𝑐−1)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−1𝜆^𝑘_𝑗𝑑𝜆_𝑗𝜇^𝑐−1_𝑑𝑗c=2𝐛^𝑐_𝑗𝑑𝑗1(𝑐−2)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−2𝜆^𝑘_𝑗𝑑𝜆_𝑗^𝑐−2_{𝜇𝑑𝑗𝑐=𝜇𝑑𝑗}⋮𝐛^𝑐_𝑗𝑑𝑗1(𝑐−𝜇_𝑑𝑗)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^{𝑐−𝜇𝑑𝑗}𝜆^𝑘_𝑗𝑑𝜆^{𝑐−𝜇𝑑𝑗}_𝑗}𝜇_𝑗𝑑𝑗,×1 (B.42) where 𝜇_𝑧𝑗 are the Weyr characteristics via Ferrer diagrams, for 𝑗=𝑙+1,𝑙+2,…,𝜅 and 𝑧_𝑗=1,2,…,𝑑_𝑗. Note that 𝜌_𝑗=max_{𝑧𝑖=1,2,…,𝑑𝑗}𝜇_𝑧𝑗 is the index of annihilation for the eigenvalue 𝜆_𝑗. Note that matrix (B.42) has 𝑑_𝑗-blocks. Each one of the 𝑑_𝑗 blocks can be written as follows: Φ⁻¹(𝐾)^{𝜇𝑧𝑗𝑐=1}𝐛^𝑐_𝑗𝑧𝑗1(𝑐−1)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−1𝜆^𝑘_𝑗𝑑𝜆_𝑗𝜇^𝑐−1_{𝑧𝑗𝑐=2}𝐛^𝑐_𝑗𝑧𝑗1(𝑐−2)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^𝑐−2𝜆^𝑘_𝑗𝑑𝜆_𝑗^𝑐−2_{𝜇𝑧𝑗𝑐=𝜇𝑧𝑗}⋮𝐛^𝑐_𝑗𝑧𝑗1(𝑐−𝜇₂)!^𝑛−1_𝑘=0𝛼_𝑘𝑑^{𝑐−𝜇𝑧𝑗}𝜆^𝑘_𝑗𝑑𝜆^{𝑐−𝜇𝑧𝑗}_𝑗=Φ⁻¹(𝐾)^{𝜇𝑧𝑗𝑐=1}𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−1)!^𝑐−1𝜆⁰_𝑗𝑑𝜆_𝑗^𝑐−1,^𝜇𝑧𝑗_𝑐=1𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−1)!^𝑐−1𝜆_𝑗𝑑^𝑐−1_𝜆𝑗,…,^𝜇𝑧𝑗_𝑐=1𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−1)!^𝑐−1𝜆_𝑗^𝑛−1_𝜇^𝑐−1_𝜆𝑗𝑑_{𝑧𝑗𝑐=2}𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−2)!^𝑐−2𝜆⁰_𝑗𝑑𝜆_𝑗^𝑐−2,^𝜇𝑧𝑗_𝑐=2𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−2)!^𝑐−2𝜆_𝑗𝑑^𝑐−2_𝜆𝑗,…,^𝜇𝑧𝑗_𝑐=2𝐛^𝑐_𝑗𝑧𝑗1⋅𝑑(𝑐−2)!^𝑐−2𝜆_𝑗^𝑛−1𝑑𝜆_𝑗^𝑐−2_{𝜇𝑧𝑗𝑐=𝜇𝑧𝑗}⋮𝐛^𝑐_𝑗𝑧𝑗1(𝑐−𝜇_𝑧𝑗⋅𝑑)!^{𝑐−𝜇𝑧𝑗}𝜆⁰_𝑗𝑑𝜆^{𝑐−𝜇𝑧𝑗}_𝑗,^𝜇𝑧𝑗_{𝑐=𝜇𝑧𝑗}𝐛^𝑐_𝑗𝑧𝑗1(𝑐−𝜇_𝑧𝑗⋅𝑑)!^{𝑐−𝜇𝑧𝑗}𝜆_𝑗𝑑𝜆^{𝑐−𝜇𝑧𝑗}_𝑗,…,^{𝜇𝑧(𝑗)}_{𝑐=𝜇𝑧𝑗}𝐛_𝑗^𝑐,_𝑧𝑗1(𝑐−𝜇_𝑧𝑗⋅𝑑)!^{𝑐−𝜇𝑧𝑗}𝜆_𝑗^𝑛−1𝑑𝜆^{𝑐−𝜇𝑧𝑗}_{𝑗𝐕𝑗𝑗𝑧}⋅𝛼₀𝛼₁𝛼₂⋮𝛼_𝑛−1𝐚=Φ⁻¹(𝐾)𝐕_𝑗𝑧𝑗𝐚, (B.43)forevery𝑗=𝑙+1,𝑙+2,…,𝜅,and𝑧_𝑗=1,2,…,𝑑_𝑗.
Consequently, the matrix (B.42), by using (B.43), can be written as follows
lim_𝜎→0^𝑡_−∞𝑒^{−𝐽𝑗𝑡}𝐛_𝜏𝑗𝑢(𝜏)𝑑𝜏=Φ⁻¹𝐕(𝐾)^𝑗1𝐕_𝑗2⋮𝐕_𝑗𝑧𝑗⋮𝐕_𝑗𝑑𝑗⋅𝐚. (B.44)

Acknowledgments

The authors are very grateful to Editor Professor Amit Bhaya and the anonymous referees for their comments and suggestions, which highly improved the quality of the paper. The second author is also very grateful to Mr. Aristomenis Tsiomos, Mr. Chris Andrianoupolitis, Ms. Maria Kelertzi, Ms. Aggeliki Drakopoulou, and Mr. Michael Filakouris for their important moral support.