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