Solution Properties of Linear Descriptor (Singular) Matrix Differential Systems of Higher Order with (Non-) Consistent Initial Conditions

Abstract

In some interesting applications in control and system theory, linear descriptor (singular) matrix differential equations of higher order with time-invariant coefficients and (non-) consistent initial conditions have been used. In this paper, we provide a study for the solution properties of a more general class of the Apostol-Kolodner-type equations with consistent and nonconsistent initial conditions.

1. Introduction

Linear Time-Invariant (LTI) (i.e., with constant matrix coefficients) descriptor matrix differential systems of type (1.1) with several kinds of inputs

𝐹𝑋^(𝑟)(𝑡)=𝐴𝑋(𝑡)+𝐵𝑈(𝑡),(1.1) where 𝐹,𝐴∈ℳ(𝑛×𝑚;𝔽), 𝐵∈ℳ(𝑛×𝜇;𝔽), and 𝑈∈𝒞^∞(𝔽,ℳ(𝜇×𝑚;𝔽)), often appear in control and system theory. For instance, (1.1) identifies and models effectively many physical, engineering, mechanical, as well as financial phenomena. For instance, we can provide in economy, the well-known, famous input-output Leontief model and its several important extensions, advice [1, 2]. Moreover, in the beginning of this introductive section, we should point out that singular perturbations arise often in systems whose dynamics have sufficiently separate slow and fast parts. Now by considering the classical proportional feedback controller

𝑈(𝑡)=−𝐹𝑋(𝑡),(1.2) we can obtain (1.3), where 𝐺=𝐴−𝐵𝐹.

Our long-term purpose is to study the solution of LTI descriptor matrix differential systems of higher order (1.1) into the mainstream of matrix pencil theory, that is,

𝐹𝑋^(𝑟)(𝑡)=𝐺𝑋(𝑡),(1.3) where, for (1.1), (1.2), and (1.3), 𝑟th is the order of the systems, 𝐹,𝐺∈ℳ(𝑛×𝑚;𝔽) (where matrix 𝐹 is singular), and 𝑋∈𝒞^∞(𝔽,ℳ(𝑚×𝑙;𝔽) (note that 𝔽 can be either ℝ or ℂ). For the sake of simplicity we set in the sequel ℳ_𝑛=ℳ(𝑛×𝑛;𝔽) and ℳ_𝑛,𝑚=ℳ(𝑛×𝑚;𝔽).

Matrix pencil theory has been extensively used for the study of LTI descriptor differential equations of first order; see, for instance, [3–6]. Systems of type (1.3) are more general, including the special case when 𝐹=𝐼_𝑛, where 𝐼_𝑛 is the identity matrix of ℳ_𝑛, since the well-known class of higher order linear matrix differential of Apostol-type equations is derived straightforwardly; see [7–10]. In the same way, system (1.1) might be considered as the more general class of higher order linear descriptor matrix differential equations of Apostol-Kolodner type, since Kolodner has also studied such systems in nondescriptor form; see also [8].

Recently, in [5], the regular case of higher order linear descriptor matrix differential equations of Apostol-Kolodner type has been investigated. The regular case is simpler, since it considers square matrix coefficients and the Weierstrass canonical form has been applied. Actually, the recent work is a nonstraight generalization of [5]. Analytically, in this article, we study the linear descriptor matrix differential equations of higher order whose coefficients are rectangular constant matrices, that is, the singular case is examined. Adopting several different methods for computing the matrix powers and exponential, new formulas representing auxiliary results are obtained. This allows us to prove properties of a large class of linear matrix differential equations of higher order; in particular results of Apostol and Kolodner are recovered; see also [5, 8].

Finally, it should be mentioned that in the classical theory of linear (descriptor) differential systems, see, for instance, [1, 2, 11–13], one of the important features is that not every initial condition 𝑋₀ admits a functional solution. Thus, we shall call 𝑋₀a consistent initial condition for (1.3) at 𝑡_𝑜 if there is a solution for (1.3), which is defined on some interval [𝑡_𝑜,𝑡_𝑜+𝛾], 𝛾>0 such that 𝑋(𝑡_𝑜)=𝑋₀.

On the other hand, it is not rare to appear in some practical significant applications that the assumption of the initial conditions for (1.3) can be nonconsistent, that is, 𝑋(𝑡_𝑜)≠𝑋₀.

2. Mathematical Background and Notations

In this preliminary section, some well-known concepts and definitions for matrix pencils are introduced. This discussion is highly important, in order to understand better the results of Section 3.

Definition 2.1. 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 (where 0 is the zero element of ℳ(1,𝔽)). In any other case, the pencil is called singular.

In this paper, as we are going to see in the next paragraph, we consider the case that the pencil is singular. The next definition is very important, since the notion of strict equivalence between two pencils is presented.

Definition 2.2. The pencil 𝑠𝐹−𝐺 is said to be strictly equivalent to the pencil 𝑠𝐹−𝐺 if and only if there exist nonsingular 𝑃∈ℳ_𝑛 and 𝑄∈ℳ_𝑚 such that 𝑃(𝑠𝐹−𝐺)𝑄=𝑠𝐹−𝐺.(2.1)

The characterization of singular pencils requires the definition of additional sets of invariants known as the minimal indices.

Let us assume that 𝑟=rank_𝔽(𝑠)(𝑠𝐹−𝐺), where 𝔽(𝑠) denotes the field of rational functions in 𝑠 having coefficients in the field 𝔽. The equations

(𝑠𝐹−𝐺)𝑥(𝑠)=0,𝜓^𝑇(𝑠)(𝑠𝐹−𝐺)=0^𝑇(2.2) have nonzero solutions 𝑥(𝑠) and 𝜓(𝑠) which are vectors in the rational vector spaces

𝒩_right(𝑠)≜𝒩_right(𝑠𝐹−𝐺),𝒩_left(𝑠)≜𝒩_left(𝑠𝐹−𝐺),(2.3) respectively, where

𝒩_right(𝑠)≜{𝑥(𝑠)∈𝔽(𝑠)^𝑚∶(𝑠𝐹−𝐺)𝑥(𝑠)=0_𝑛𝒩},_left(𝑠)={𝜓(𝑠)∈𝔽(𝑠)^𝑛∶𝜓^𝑇(𝑠)(𝑠𝐹−𝐺)=0^𝑇_𝑚}.(2.4) The sets of the minimal degrees {𝑣_𝑖,1≤𝑖≤𝑚−𝑟} and {𝑢_𝑗,1≤𝑗≤𝑛−𝑟} are known as column minimal indices (c.m.i.) and row minimal indices (r.m.i.) of 𝑠𝐹−𝐺, respectively. Furthermore, if 𝑟=rank_𝔽(𝑠)(𝑠𝐹−𝐺), it is evident such that

𝑟=^𝑚−𝑟_𝑖=𝑔+1𝑣_𝑖+^𝑛−𝑟_𝑗=ℎ+1𝑢_𝑗+rank_𝔽(𝑠)(𝑠𝐹_𝑤−𝐺_𝑤),(2.5) where 𝑠𝐹_𝑤−𝐺_𝑤 is the complex Weierstrass canonical form; see [3].

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}.

Thus, there exists 𝑃∈ℳ_𝑚 and 𝑄∈ℳ_𝑛 such that the complex Kronecker form 𝑠𝐹_𝑘−𝐺_𝑘 of the singular pencil 𝑠𝐹−𝐺 is defined as follows:

𝑠𝐹_𝑘−𝐺_𝑘≜𝕆_ℎ,𝑔⊕𝑠Λ_𝑣−𝜆_𝑣⊕𝑠Λ^𝑇_𝑢−𝜆^𝑇_𝑢⊕𝑠𝐼_𝑝−𝐽_𝑝⊕𝑠𝐻_𝑞−𝐼_𝑞,(2.6) where 𝑣=∑^𝑚−𝑟_𝑖=𝑔+1𝑣_𝑖, 𝑢=∑^𝑛−𝑟_𝑗=ℎ+1𝑢_𝑗, 𝑝=∑^𝜅_𝑗=1𝑝_𝑗, and 𝑞=∑^𝜎_𝑗=1𝑞_𝑗 (see below). In more details, the following are given.

(S1) Matrix 𝕆_ℎ,𝑔 is uniquely defined by the sets {0,0,…,0}_𝑔 and {0,0,…,0}_ℎ of zero column and row minimal indices, respectively.

(S2) The second normal block 𝑠Λ_𝑣−𝜆_𝑣 is uniquely defined by the set of nonzero column minimal indices (a new arrangement of the indices of 𝑣 must be noted in order to simplify the notation) {𝑣_𝑔+1≤⋯≤𝑣_𝑚−𝑟} of 𝑠𝐹−𝑄 and has the form

𝑠Λ_𝑣−𝜆_𝑣≜𝑠Λ_𝑣𝑔+1−𝜆_𝑣𝑔+1⊕⋯⊕𝑠Λ_𝑣𝑖−𝜆_𝑣𝑖⊕⋯⊕𝑠Λ_{𝑣𝑚−𝑟}−𝜆_{𝑣𝑚−𝑟},(2.7) where Λ_𝑣𝑖=[𝐼_𝑣𝑖⋮0]∈ℳ_{𝑣𝑖,𝑣𝑖+1}, 𝜆_𝑣𝑖=[𝐻_𝑣𝑖⋮𝜀_𝑣𝑖]∈ℳ_{𝑣𝑖,𝑣𝑖+1} for every 𝑖=𝑔+1,𝑔+2,..., 𝑚−𝑟, and 𝐼_𝑣𝑖 and 𝐻_𝑣𝑖 denote the 𝑣_𝑖×𝑣_𝑖 identity and the nilpotent (with index of nilpotency 𝑣_𝑖) matrix, respectively. 0 and 𝜀_𝑣𝑖=[0⋯01]^𝑇 are the zero column and the column with element 1 at the 𝑣_𝑖place, respectively.

(S3) The third normal block 𝑠Λ^𝑇_𝑢−𝜆^𝑇_𝑢 is uniquely determined by the set of nonzero row minimal indices (a new arrangement of the indices of 𝑢 must be noted in order to simplify the notation) {𝑢_ℎ+1≤⋯≤𝑢_𝑛−𝑟} of 𝑠𝐹−𝐺 and has the form

𝑠Λ^𝑇_𝑢−𝜆^𝑇_𝑢≜𝑠Λ^𝑇_𝑢ℎ+1−𝜆^𝑇_𝑢ℎ+1⊕⋯⊕𝑠Λ^𝑇_𝑢𝑗−𝜆^𝑇_𝑢𝑗⊕⋯⊕𝑠Λ^𝑇_{𝑢𝑛−𝑟}−𝜆^𝑇_{𝑢𝑛−𝑟},(2.8) where Λ^𝑇_𝑢𝑗=[^{𝑒𝑇𝑢𝑗⋯}_𝐻𝑢𝑗]∈ℳ_{𝑢𝑗+1,𝑢𝑗}, 𝜆^𝑇_𝑢𝑗=[^0𝑇⋯_𝐼𝑢𝑗]∈ℳ_{𝑢𝑗+1,𝑢𝑗} for every 𝑗=ℎ+1,ℎ+2,…,𝑚−𝑟, and 𝐼_𝑢𝑗 and 𝐻_𝑢𝑗 denote the 𝑢_𝑗×𝑢_𝑗 identity and nilpotent (with index of nilpotency 𝑢_𝑗) matrix, respectively. 0 and 𝑒_𝑢𝑗=[1⋯00]^𝑇 are the zero column and the column with element 1 at the first place, respectively.

(S4-S5) The forth and the fifth normal matrix block is the complex Weierstrass form 𝑠𝐹_𝑤−𝐺_𝑤 of the singular pencil 𝑠𝐹−𝐺 which is defined by

𝑠𝐹_𝑤−𝑄_𝑤≜𝑠𝐼_𝑝−𝐽_𝑝⊕𝑠𝐻_𝑞−𝐼_𝑞,(2.9) where the first normal Jordan-type element is uniquely defined by the set of finite elementary divisors (f.e.d.)

(𝑠−𝑎₁)^𝑝1,…,(𝑠−𝑎_𝜅)^𝑝𝜅,^𝜅_𝑗=1𝑝_𝑗=𝑝(2.10) of 𝑠𝐹−𝐺 and has the form

𝑠𝐼_𝑝−𝐽_𝑝≜𝑠𝐼_𝑝1−𝐽_𝑝1(𝑎₁)⊕⋯⊕𝑠𝐼_𝑝𝜅−𝐽_𝑝𝜅(𝑎_𝜅).(2.11) And also the 𝑞 blocks of the second uniquely defined block 𝑠𝐻_𝑞−𝐼_𝑞 correspond to the infinite elementary divisors (i.e.d.)

𝑠̂^𝑞1,…,𝑠̂^𝑞𝜎,^𝜎_𝑗=1𝑞_𝑗=𝑞(2.12) of 𝑠𝐹−𝐺 and have the form

𝑠𝐻_𝑞−𝐼_𝑞≜𝑠𝐻_𝑞1−𝐼_𝑞1⊕⋯⊕𝑠𝐻_𝑞𝜎−𝐼_𝑞𝜎.(2.13)

Thus 𝐻_𝑞 is a nilpotent element of ℳ_𝑛 with index 𝑞=max{𝑞_𝑗∶𝑗=1,2,…,𝜎}, where

𝐻𝑞_𝑞=𝕆,(2.14) and 𝐼_𝑝𝑗,𝐽_𝑝𝑗(𝑎_𝑗),𝐻_𝑞𝑗 are the matrices

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.(2.15)

In the last part of this introductive section, some elements for the analytic computation of 𝑒^{𝐴(𝑡−𝑡𝑜)}, 𝑡∈[𝑡_𝑜,∞) are provided. To perform this computation, many theoretical and numerical methods have been developed.

Thus, the interested reader might consult papers in [1, 2, 7–10, 14–16], and the references therein. In order to obtain more analytic formulas, the following known results should be mentioned.

Lemma 2.3 (see [15]). 𝑒^{𝐽𝑝𝑗(𝑎𝑗)(𝑡−𝑡𝑜)}=[𝑑_𝑘1𝑘2]_𝑝𝑗,(2.16) where 𝑑_𝑘1𝑘2=𝑒^{𝑎𝑗(𝑡−𝑡𝑜)}(𝑡−𝑡_𝑜)^{𝑘2−𝑘1}(𝑘₂−𝑘₁)!,1≤𝑘₁≤𝑘₂≤𝑝_𝑗,0,otherwise.(2.17)

Another expression for the exponential matrix of Jordan block, see (2.18), is provided by the following lemma.

Lemma 2.4 (see [15]). 𝑒^{𝐽𝑝𝑗(𝑎𝑗)(𝑡−𝑡𝑜)}=^𝑝𝑗−1_𝑖=0𝑓_𝑖(𝑡−𝑡_𝑜)[𝐽_𝑝𝑗(𝑎_𝑗)]^𝑖,(2.18) where the 𝑓_𝑘(𝑡−𝑡_𝑜)^,𝑠 satisfy the following system of 𝑝_𝑗equations: ^𝑝𝑗−1_𝑖=𝑘𝑖𝑘𝑎_𝑗^𝑖−𝑘𝑓_𝑖(𝑡−𝑡_𝑜)=(𝑡−𝑡_𝑜)^𝑘𝑒𝑘!^𝑎𝑗𝑡,𝑘=1,2,...,𝑝_𝑗,[𝐽_𝑝𝑗(𝑎_𝑗)]^𝑖=[𝛽_𝑘^(𝑖)_1𝑘2]_𝑝𝑗,for1≤𝑘₁,𝑘₂≤𝑝_𝑗,(2.19) where 𝛽_𝑘^(𝑖)_1𝑘2=(^𝑖_{𝑘2−𝑘1})𝑎^{𝑖−(𝑘2−𝑘1)}_𝑗.

3. Solution Space for Consistent Initial Conditions

In this section, the main results for consistent initial conditions are analytically presented for the singular case. The whole discussion extends the existing literature; see, for instance [8]. Now, in order to obtain a solution, we deal with consistent initial value problem. More analytically, we consider the system

𝐹𝑋^(𝑟)(𝑡)=𝐺𝑋(𝑡),(3.1) with known 𝑋(𝑡_𝑜),𝑋^′(𝑡_𝑜),…,𝑋^(𝑟−1)(𝑡_𝑜),(3.2) where 𝐹,𝐺∈ℳ_𝑛,𝑚 (where matrix 𝐹 is singular), and 𝑋∈𝒞^∞(𝔽,ℳ_𝑚,𝑙).

From the singularity of 𝑠^𝑟𝐹−𝐺, there exist nonsingular matrices 𝑃∈ℳ_𝑛 and 𝑄∈ℳ_𝑚 such that (see also Section 2)

𝑃𝐹𝑄=𝐹_𝑘=𝕆_ℎ,𝑔⊕Λ_𝑣⊕Λ^𝑇_𝑢⊕𝐼_𝑝⊕𝐻_𝑞,𝑃𝐺𝑄=𝐺_𝑘=𝕆_ℎ,𝑔⊕𝜆_𝑣⊕𝜆^𝑇_𝑢⊕𝐽_𝑝⊕𝐼_𝑞,(3.3) where Λ_𝑣,𝜆_𝑣, Λ^𝑇_𝑢, 𝜆^𝑇_𝑢, 𝐼_𝑝,𝐽_𝑝,𝐻_𝑞, and 𝐼_𝑞 are given by

Λ_𝑣=Λ_𝑣𝑔+1⊕⋯⊕Λ_𝑣𝑖⊕⋯⊕Λ_{𝑣𝑚−𝑟},𝜆_𝑣≜𝜆_𝑣𝑔+1⊕⋯⊕𝜆_𝑣𝑖⊕⋯⊕𝜆_{𝑣𝑚−𝑟},Λ^𝑇_𝑢≜Λ^𝑇_𝑢ℎ+1⊕⋯⊕Λ^𝑇_𝑢𝑗⊕⋯⊕Λ^𝑇_{𝑢𝑛−𝑟},𝜆^𝑇_𝑢≜𝜆^𝑇_𝑢ℎ+1⊕⋯⊕𝜆^𝑇_𝑢𝑗⊕⋯⊕𝜆^𝑇_{𝑢𝑛−𝑟},𝐼_𝑝=𝐼_𝑝1⊕⋯⊕𝐼_𝑝𝜅,𝐽_𝑝=𝐽_𝑝1(𝑎₁)⊕⋯⊕𝐽_𝑝𝜅(𝑎_𝜅𝐻),_𝑞≜𝐻_𝑞1⊕⋯⊕𝐻_𝑞𝜎,𝐼_𝑞=𝐼_𝑞1⊕⋯⊕𝐼_𝑞𝜎.(3.4)

By using the Kronecker canonical form, we might rewrite system (1.3), as the following lemma denotes.

Lemma 3.1. System (1.3) may be divided into five subsystems: 𝕆_ℎ,𝑔𝑌_𝑔^(𝑟)(𝑡)=𝕆_ℎ,𝑔𝑌_𝑔Λ(𝑡),(3.5)_𝑣𝑌_𝑣^(𝑟)(𝑡)=𝜆_𝑣𝑌_𝑣Λ(𝑡),(3.6)_𝑇𝑢𝑌_𝑢^(𝑟)(𝑡)=𝜆^𝑇_𝑢𝑌_𝑢(𝑡),(3.7) the so-called slow subsystem 𝑌_𝑝^(𝑟)(𝑡)=𝐽_𝑝𝑌_𝑝(𝑡),(3.8) and the relative fast subsystem 𝐻_𝑞𝑌_𝑞^(𝑟)(𝑡)=𝑌_𝑞(𝑡).(3.9)

Proof. Consider the transformation 𝑋(𝑡)=𝑄𝑌(𝑡),(3.10) where 𝑄∈ℳ_𝑚 and 𝑌∈𝒞^∞(𝔽,ℳ_𝑚,𝑙). Substituting the previous expression into (1.3), we obtain 𝐹𝑄𝑌^(𝑟)(𝑡)=𝐺𝑄𝑌(𝑡).(3.11) Whereby, multiplying by 𝑃, we arrive at 𝐹_𝑘𝑌^(𝑟)(𝑡)=𝐺_𝑘𝑌(𝑡).(3.12) Moreover, we can write 𝑌(𝑡) as 𝑌(𝑡)=𝑌^𝑇_𝑔(𝑡)𝑌^𝑇_𝑣(𝑡)𝑌^𝑇_𝑢(𝑡)𝑌^𝑇_𝑝(𝑡)𝑌^𝑇_𝑞(𝑡)^𝑇∈ℳ_𝑚,𝑙,(3.13) where 𝑌_𝑔(𝑡)∈𝒞^∞(𝔽,ℳ_𝑔,𝑙), 𝑌_𝑣(𝑡)∈𝒞^∞(𝔽,ℳ_𝑣,𝑙), 𝑌_𝑣(𝑡)∈𝒞^∞(𝔽,ℳ_𝑢,𝑙), 𝑌_𝑝(𝑡)∈ 𝒞^∞(𝔽,ℳ_𝑝,𝑙), and 𝑌_𝑞(𝑡)∈𝒞^∞(𝔽,ℳ_𝑞,𝑙). Note that 𝑔 is the number of zero column entries, 𝑣=∑^𝑚−𝑟_𝑖=𝑔+1𝑣_𝑖, 𝑢=∑^𝑛−𝑟_𝑗=ℎ+1𝑢_𝑗, 𝑝=∑^𝜅_𝑗=1𝑝_𝑗, and 𝑞=∑^𝜎_𝑗=1𝑞_𝑗.
And taking into account the above expressions, we arrive easily at (3.5)–(3.9).

Proposition 3.2. For system (3.5), the elements of the matrix 𝑌_𝑔(𝑡)∈𝒞^∞(𝔽,ℳ(𝑔×𝑙;𝔽)) can be chosen arbitrarily.

Proof. Since 𝕆_ℎ,𝑔, it is profound that any g-column vector can be chosen.

Proposition 3.3. The analytic solution of system Λ_𝑣𝑖𝑌_𝑣^(𝑟)_𝑖(𝑡)=𝜆_𝑣𝑖𝑌_𝑣𝑖(𝑡)(3.14) is given by the expression 𝑌_𝑣𝑖(𝑡)=𝑌₁(𝑡)𝑌₂(𝑡)⋯𝑌_𝑙(𝑡)=[𝑦_𝜆,𝑗(𝑡)]_{𝜆=1,2,…,𝑣𝑖𝑗=1,2,…,𝑙},(3.15) where 𝑦_𝜆,𝑗(𝑡)=⋯𝑦_𝜆+1,𝑗(𝑡)𝑑𝑡⋯𝑑𝑡_𝑟-times+^𝑟_𝜉=1𝑐_{𝜆,𝑟−𝜉+1}𝑡^𝜉−1(𝜉−1)!,(3.16) where 𝑦_𝜆+1,𝑗(𝑡) is an arbitrary function, for every 𝜆=1,2,…,𝑣_𝑖, 𝑖=𝑔+1,…,𝑚−𝑟, and 𝑗=1,2,…,𝑙. (Note that 𝑐_{𝜆,𝑟−𝜉+1} should be uniquely determined via the given initial conditions.)

Proof. System (3.14) is rewritten as [𝐼_𝑣𝑖⋮0]𝑌_𝑣^(𝑟)_𝜄(𝑡)=[𝐻_𝑣𝑖⋮𝜀_𝑣𝑖]𝑌_𝑣𝜄(𝑡),(3.17) for every 𝑖=𝑔+1,𝑔+2,…,𝑚−𝑟. Now, we denote 𝑌_𝑣𝑖Ψ(𝑡)=^𝑣𝑖𝑦(𝑡)₁(𝑡),(3.18) where Ψ_𝑣𝑖(𝑡)∈ℳ_𝑣𝑖𝑙, Ψ_𝑣𝑖(𝑡)=[𝑌₁(𝑡)𝑌₂(𝑡)⋯𝑌_𝑙(𝑡)] with 𝑌_𝑗(𝑡)=[𝑦_1,𝑗(𝑡)𝑦_2,𝑗(𝑡)⋯𝑦_{𝑣𝑖,𝑗}(𝑡)]^𝑇, and 𝑦₁(𝑡)∈ℳ_1𝑙 (vector, 1×𝑙).
Thus,
[𝐼_𝑣𝑖⋮0]Ψ_𝑣^(𝑟)_𝑖𝑦(𝑡)_1(𝑟)(𝑡)=[𝐻_𝑣𝑖⋮𝜀_𝑣𝑖]Ψ_𝑣𝑖𝑦(𝑡)₁(𝑡),(3.19) or, equivalently, we obtain Ψ_𝑣^(𝑟)_𝑖(𝑡)=𝐻_𝑣𝑖Ψ_𝑣𝑖(𝑡)+𝜀_𝑣𝑖𝑦₁(𝑡).(3.20) Note that 𝜀_𝑣𝑖𝑦₁(𝑡) is a matrix with 𝑣_𝑖×𝑣_𝑖-elements as follows 𝜀_𝑣𝑖𝑦₁(𝑡)=𝑌₁(𝑡)𝑌₂(𝑡)⋯𝑌_𝑙≜𝕆(𝑡)^{𝑣𝑖−1,𝑙}𝑦_𝑣𝑖+1,1(𝑡)𝑦_𝑣𝑖+1,2(𝑡)⋯𝑦_{𝑣𝑖,+1𝑙}(𝑡)(3.21) where 𝑌_𝑗(𝑡)=[00⋯𝑦_{𝑣𝑖+1,𝑗}(𝑡)]^𝑇, for 𝑗=1,2,…,𝑙.
Consequently, (3.20) is rewritten as follows:
𝑌₁^(𝑟)(𝑡)𝑌₂^(𝑟)(𝑡)⋯𝑌_𝑙^(𝑟)(𝑡)=𝐻_𝑣𝑖𝑌₁(𝑡)𝐻_𝑣𝑖𝑌₂(𝑡)⋯𝐻_𝑣𝑖𝑌_𝑙+𝑌(𝑡)₁(𝑡)𝑌₂(𝑡)⋯𝑌_𝑙(𝑡),(3.22) or, equivalently, 𝑌_𝑗^(𝑟)(𝑡)=𝐻_𝑣𝑖𝑌_𝑗(𝑡)+𝑌_𝑗(𝑡),(3.23) and eventually, as a scalar system, we obtain 𝑦^(𝑟)_1,𝑗(𝑡)=𝑦_2,𝑗(𝑡),𝑦^(𝑟)_2,𝑗(𝑡)=𝑦_3,𝑗𝑦(𝑡),…,_{𝑣(𝑟)𝑖−1,𝑗}(𝑡)=𝑦_{𝑣𝑖,𝑗}(𝑡),𝑦_𝑣^(𝑟)_𝑖,𝑗(𝑡)=𝑦_{𝑣𝑖+1,𝑗}(𝑡).(3.24)
Denote that element 𝑦_{𝑣𝑖+1,𝑗}(𝑡) is an arbitrary function; then the solution is given iteratively, as follows.
Firstly, we take the equation 𝑦_𝑣^(𝑟)_𝑖,𝑗(𝑡)=𝑦_{𝑣𝑖+1,𝑗}(𝑡) for every 𝑗=1,2,…,𝑙,
𝑦_𝑣^(𝑟−1)_𝑖,𝑗(𝑡)=𝑦_{𝑣𝑖+1,𝑗}(𝑡)𝑑𝑡+𝑐_𝑣𝑖,1𝑦,…,_{𝑣𝑖,𝑗}(𝑡)=⋯𝑦_{𝑣𝑖+1,𝑗}(𝑡)𝑑𝑡⋯𝑑𝑡_𝑟-times+^𝑟_𝜉=1𝑐_{𝑣𝑖,𝑟−𝜉+1}𝑡^𝜉−1.(𝜉−1)!(3.25) We continue the procedure, for 𝑦_𝑣^(𝑟)_{𝑖−1,𝑗}(𝑡)=𝑦_{𝑣𝑖,𝑗}(𝑡), and so forth. Thus, we finally obtain (3.15).

With the following remark, we obtain the solution of subsystem (3.6).

Remark 3.4. The solution of subsystem (3.6) is given by 𝑌_𝑣(𝑡)=𝑌_𝑣𝑔+1(𝑡)⊕⋯⊕𝑌_𝑣𝑖(𝑡)⊕⋯⊕𝑌_{𝑣𝑚−𝑟}(𝑡),(3.26) where the results of Proposition 3.3 are also considered.

Remark 3.5. Considering the solution (3.14), and therefore the system (3.6), it should be pointed out that the solution is not unique, since the last component of the solution vector is chosen arbitrary. Moreover, it is worth to be emphasized here that the solution of the singular system (1.3) is not unique.

Proposition 3.6. The system Λ^𝑇_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗(𝑡)=𝜆^𝑇_𝑢𝑗𝑌_𝑢𝑗(𝑡)(3.27) has only the zero solution.

Proof. Consider that system (3.27) can be rewritten as follows: 𝑒^𝑇_𝑢𝑗⋯𝐻_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗0(𝑡)=⋯^𝑇𝐼_𝑢𝑗𝑌_𝑢𝑗(𝑡)(3.28) for every 𝑗=ℎ+1,ℎ+2,…,𝑚−𝑟.
Afterwards, we obtain straightforwardly the following system:
𝑒^𝑇_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗𝐻(𝑡)_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗=0(𝑡)𝑌^𝑇_𝑢𝑗⇔𝑒(𝑡)^𝑇𝑢𝑗𝑌_𝑢^(𝑟)_𝑗(𝑡)=0^𝑇,𝐻_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗(𝑡)=𝑌_𝑢𝑗(𝑡).(3.29) Now, by successively taking 𝑟th derivatives with respect to 𝑡 on both sides of 𝐻_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗(𝑡)=𝑌_𝑢𝑗(𝑡),(3.30) and left multiplying by the matrix 𝐻_𝑢𝑗, 𝑢_𝑗−1 times (where 𝑢_𝑗 is the index of the nilpotent matrix 𝐻_𝑢𝑗, i.e., 𝐻^𝑢𝑗_𝑢𝑗=𝕆), we obtain the following equations: 𝐻²_𝑢𝑗𝑌_𝑢^(2𝑟)_𝑗(𝑡)=𝐻_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗(𝑡),…,𝐻^𝑢𝑗_𝑢𝑗𝑌^(𝑢𝑗_𝑢^𝑟)_𝑗(𝑡)=𝐻^𝑢𝑗_𝑢⁻¹_𝑗𝑌^((𝑢𝑗_𝑢^−1)𝑟)_𝑗(𝑡).(3.31) Thus, we conclude to the following expression: 𝑌_𝑢𝑗(𝑡)=𝐻_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗(𝑡)=𝐻²_𝑢𝑗𝑌_𝑢^(2𝑟)_𝑗(𝑡)=⋯=𝐻^𝑢𝑗_𝑢𝑗𝑌^(𝑢𝑗_𝑢^𝑟)_𝑗(𝑡)=𝕆.(3.32)

Remark 3.7. Consequently, the subsystem (3.7) has also the zero solution.

Proposition 3.8 (see [5]). (a) The analytic solution of the so-called slow subsystem (3.8) is given by 𝑌_𝑝(𝑡)=𝐿𝑅⊕^𝜅⊕_𝑗=1^𝑟−1_𝑘=0𝑒^{𝐽𝑗𝑘(𝜆𝑗𝑘)(𝑡−𝑡𝑜)}𝑅⁻¹𝐙(𝑡_𝑜),(3.33) where 𝐿=[𝐼_𝑝𝕆⋯𝕆]∈ℳ_𝑝,pr; 𝑅∈ℳ_prsuch that 𝐉=𝑅⁻¹𝐀𝑅.
Note that 𝐉∈ℳ_pr is the Jordan Canonical form of matrix
𝐀=𝕆𝐼_𝑝𝕆⋯𝕆𝕆𝕆𝐼_𝑝⋯𝕆⋮⋮⋮⋱⋮𝕆𝕆𝕆⋯𝐼_𝑝𝐽_𝑝𝕆𝕆⋯𝕆,(3.34) and 𝐙(𝑡_𝑜)=[𝑌^𝑇_𝑝(𝑡_𝑜)𝑌^′𝑇_𝑝(𝑡_𝑜)⋯𝑌_𝑝^{𝑇(𝑟−1)}(𝑡_𝑜)]^𝑇∈ℳ_pr,𝑙.
The eigenvalues of the matrix 𝐀 are given by
𝜆_𝑗𝑘=^𝑟|𝑎_𝑗|(cos2𝑘𝜋+𝜑_𝑗𝑟+𝑧sin2𝑘𝜋+𝜑_𝑗𝑟),(3.35) where 𝑎_𝑗=|𝑎_𝑗|(cos𝜑_𝑗+𝑧sin𝜑_𝑗) (𝑎_𝑗 finite elementary divisors) and 𝑧²=−1 for every 𝑗=1,2,…,𝜅 and 𝑘=0,1,2, …,𝑟−1.
(b) However, the relative fast subsystem (3.9) has only the zero solution.
It is worth to say that the results of Ben Taher and Rachidi [14] can be compared with the results of Proposition 3.8, which has been discussed extensively in [5].

Remark 3.9. The characteristic polynomial of 𝐀 is 𝜑(𝜆)=∏^𝜅_𝑗=1(𝜆^𝑟−𝑎_𝑗)^𝑝𝑗, with 𝑎_𝑖≠𝑎_𝑗 for 𝑖≠𝑗 and ∑^𝜅_𝑗=1𝑝_𝑗=𝑝. Without loss of generality, we define that 𝑑₁=𝜏₁,𝑑₂=𝜏₂,…,𝑑_𝑙=𝜏_𝑙,and𝑑_𝑙+1<𝜏_𝑙+1,…,𝑑_𝜅<𝜏_𝜅,(3.36) where 𝑑_𝑗,𝜏_𝑗,𝑗=1,2,…,𝜅, are the geometric and algebraic multiplicities of the given eigenvalues 𝑎_𝑗, respectively.
(i) Consequently, when 𝑑_𝑗=𝜏_𝑗, then
𝐽_𝑗𝑘(𝜆_𝑗𝜅𝜆)=^𝑗𝑘𝜆_𝑗𝑘⋱𝜆_𝑗𝑘∈ℳ_𝜏𝑗(3.37) is also a diagonal matrix with diagonal elements of the eigenvalue 𝜆_𝑗𝑘, for 𝑗=1,…,𝑙.
(ii) When 𝑑_𝑗<𝜏_𝑗, then
𝐽_{𝑗𝑘,𝑧𝑗}=𝜆_𝑗𝑘1𝜆_𝑗𝑘1𝜆_𝑗𝑘⋱𝜆⋱1_𝑗𝑘∈ℳ_𝑧𝑗(3.38) for 𝑗=𝑙+1,𝑙+2,…,𝜅, and 𝑧_𝑗=1,2,…,𝑑_𝑗.
Hence, the set of consistent initial conditions for system
𝐹_𝑘𝑌^(𝑟)(𝑡)=𝐺_𝑘𝑌(𝑡)(3.39) has the following form: 𝑌^(𝑘)(𝑡_𝑜)={𝑌_𝑔^(𝑘)𝑇(𝑡_𝑜)𝑌_𝑣^(𝑘)𝑇(𝑡_𝑜)𝕆^𝑇_𝑢𝑌_𝑝^(𝑘)𝑇(𝑡_𝑜)𝕆^𝑇_𝑞^𝑇;𝑘=0,…,𝑟−1}.(3.40) In more details, since we have considered (3.14) and we can denote 𝑄=𝑄_𝑛,𝑔𝑄_𝑛,𝑣𝑄_𝑛,𝑢𝑄_𝑛,𝑝𝑄_𝑛,𝑞,(3.41) then we can derive the following expression: 𝑋(𝑡_𝑜)=𝑄_𝑛,𝑔𝑄_𝑛,𝑣𝑄_𝑛,𝑢𝑄_𝑛,𝑝𝑄_𝑛,𝑞𝑌^𝑇_𝑔(𝑡_𝑜)𝑌^𝑇_𝑣(𝑡_𝑜)𝕆^𝑇_𝑢𝑌^𝑇_𝑝(𝑡_𝑜)𝕆^𝑇_𝑞^𝑇=𝑄_𝑛,𝑔𝑌_𝑔(𝑡_𝑜)+𝑄_𝑛,𝑣𝑌_𝑣(𝑡_𝑜)+𝑄_𝑛,𝑝𝑌_𝑝(𝑡_𝑜).(3.42) Then, the set of consistent initial conditions for (1.3) is given by {𝑄_𝑛,𝑔𝑌_𝑔(𝑡_𝑜)+𝑄_𝑛,𝑣𝑌_𝑣(𝑡_𝑜)+𝑄_𝑛,𝑝𝑌_𝑝(𝑡_𝑜)𝑄_𝑛,𝑔𝑌^′_𝑔(𝑡_𝑜)+𝑄_𝑛,𝑣𝑌^′_𝑣(𝑡_𝑜)+𝑄_𝑛,𝑝𝑌^′_𝑝(𝑡_𝑜𝑄)⋯_𝑛,𝑔𝑌_𝑔^(𝑟−1)(𝑡_𝑜)+𝑄_𝑛,𝑣𝑌_𝑣^(𝑟−1)(𝑡_𝑜)+𝑄_𝑛,𝑝𝑌_𝑝^(𝑟−1)(𝑡_𝑜)}.(3.43) Now, taking into consideration (3.2) and (3.43), we conclude to 𝑋(𝑡_𝑜)=𝑄_𝑛,𝑔𝑌_𝑔(𝑡_𝑜)+𝑄_𝑛,𝑣𝑌_𝑣(𝑡_𝑜)+𝑄_𝑛,𝑝𝑌_𝑝(𝑡_𝑜𝑋),_′(𝑡_𝑜)=𝑄_𝑛,𝑔𝑌^′_𝑔(𝑡_𝑜)+𝑄_𝑛,𝑣𝑌^′_𝑣(𝑡_𝑜)+𝑄_𝑛,𝑝𝑌^′_𝑝(𝑡_𝑜⋮𝑋),_(𝑟−1)(𝑡_𝑜)=𝑄_𝑛,𝑔𝑌_𝑔^(𝑟−1)(𝑡_𝑜)+𝑄_𝑛,𝑣𝑌_𝑣^(𝑟−1)(𝑡_𝑜)+𝑄_𝑛,𝑝𝑌_𝑝^(𝑟−1)(𝑡_𝑜).(3.44)

Theorem 3.10. The analytic solution of (3.2) is given by 𝑋(𝑡)=𝑄_𝑛,𝑔𝑌_𝑔(𝑡)+𝑄_𝑛,𝑣𝑌_𝑣(𝑡)+𝑄_𝑛,𝑝𝐿𝑅^𝜅_𝑗=1^𝑟−1_𝑘=0𝑒^{𝐽𝑗𝑘(𝜆𝑗𝑘)(𝑡−𝑡𝑜)}𝑅⁻¹𝐙(𝑡_𝑜)(3.45) for 𝑌_𝑣(𝑡)=𝑌_𝑣𝑔+1(𝑡)⊕⋯⊕𝑌_𝑣𝑖(𝑡)⊕⋯⊕𝑌_{𝑣𝑚−𝑟}(𝑡), where 𝑌_𝑣𝑖(𝑡)=[𝑦_𝜆,𝑗]_{𝜆=1,2,…,𝑣𝑖,𝑗=1,2,…,𝑙}, for 𝑖=𝑔+1,…,𝑚−𝑟 and 𝑦_𝜆,𝑗(𝑡)=⋯𝑦_𝜆+1,𝑗(𝑡)𝑑𝑡⋯𝑑𝑡_𝑟-times+^𝑟_𝜉=1𝑐_{𝜆,𝑟−𝜉+1}𝑡^𝜉−1(𝜉−1)!.(3.46) (Note that 𝑐_{𝜆,𝑟−𝜉+1} should be uniquely determined via the given initial conditions.)
The matrix 𝑌_𝑔(𝑡) is arbitrarily chosen. Moreover
𝐿=𝐼_𝑝𝕆⋯𝕆∈ℳ_𝑝,pr,𝑅∈ℳ_pr(3.47) such that 𝐉=𝑅⁻¹𝐀𝑅, where 𝐉∈ℳ_pr is the Jordan Canonical form of matrix 𝐀, and 𝐙(𝑡_𝑜)=𝑌^𝑇_𝑝(𝑡_𝑜)𝑌^′𝑇_𝑝(𝑡_𝑜)⋯𝑌_𝑝^{𝑇(𝑟−1)}(𝑡_𝑜)^𝑇∈ℳ_pr,𝑙.(3.48)

Proof. Using the results of Lemma 3.1, Propositions 3.2–3.8, Remarks 3.4 and 3.7 and (3.10) then we obtain 𝑋(𝑡)=𝑄𝑌(𝑡)=𝑄_𝑛,𝑔𝑄_𝑛,𝑣𝑄_𝑛,𝑢𝑄_𝑛,𝑝𝑄_𝑛,𝑞⋅𝑌^𝑇_𝑔(𝑡)𝑌^𝑇_𝑣(𝑡)𝕆^𝑇_𝑢𝑌^𝑇_𝑝(𝑡)𝕆^𝑇_𝑞^𝑇=𝑄_𝑛,𝑔𝑌_𝑔(𝑡)+𝑄_𝑛,𝑣𝑌_𝑣(𝑡)+𝑄_𝑛,𝑝𝑌_𝑝(𝑡).(3.49) Finally, (3.45) is derived.

The next remark connects the solution with the set of initial condition for the system (1.3).

Remark 3.11. If 𝑄_𝑛,𝑝 is the existing left inverse of 𝑄_𝑛,𝑝, then considering also (3.10) 𝐙(𝑡_𝑜𝑌)=^𝑝(𝑡_𝑜)𝑌^′_𝑝(𝑡_𝑜)⋮𝑌_𝑝^(𝑟−1)(𝑡_𝑜)=𝑄_𝑝,𝑛𝑋_𝑝(𝑡_𝑜)𝑄_𝑝,𝑛𝑋^′_𝑝(𝑡_𝑜)⋮𝑄_𝑝,𝑛𝑋_𝑝^(𝑟−1)(𝑡_𝑜)=𝑄_𝑝,𝑛𝑄_𝑝,𝑛⋱𝑄_𝑝,𝑛𝑋_𝑝(𝑡_𝑜)𝑋^′_𝑝(𝑡_𝑜)⋮𝑋_𝑝^(𝑟−1)(𝑡_𝑜)=𝑄Ψ(𝑡_𝑜).(3.50) Finally, the solution (3.45) is given by 𝑋(𝑡)=𝑄_𝑛,𝑔𝑌_𝑔(𝑡)+𝑄_𝑛,𝑣𝑌_𝑣(𝑡)+𝑄_𝑛,𝑝𝐿𝑅^𝜅_𝑗=1^𝑟−1_𝑘=0𝑒^{𝐽𝑗𝑘(𝜆𝑗𝑘)(𝑡−𝑡𝑜)}𝑅⁻¹𝑄Ψ(𝑡_𝑜),(3.51) where Ψ(𝑡_𝑜)=[_𝑋^𝑇_{𝑝(𝑡𝑜)𝑋}^′𝑇_{𝑝(𝑡𝑜)⋯𝑋𝑝}^{𝑇(𝑟−1)}_(𝑡𝑜)]^𝑇∈ℳ_pr,𝑙 and 𝑄_𝑛,𝑝 is the existing left inverse of 𝑄_𝑛,𝑝.

The following two expressions, that is, (3.52) and (3.54) are based on Lemmas 2.3 and 2.4. Thus, two new analytical formulas are derived which are practically very useful. Their proofs are straightforward exercise of Lemmas 2.3, 2.4, and (3.51).

Proposition 3.12. Considering the results of Lemma 2.3, one obtains the expression 𝑋(𝑡)=𝑄_𝑛,𝑔𝑌_𝑔(𝑡)+𝑄_𝑛,𝑣𝑌_𝑣(𝑡)+𝑄_𝑛,𝑝𝐿𝑅[(^𝑙_𝑗=0^𝑟−1_𝑘=0𝑒^{𝜆𝑗𝑘(𝑡−𝑡𝑜)}𝐼_𝜏𝑗𝑘)(^𝜅_𝑗=𝑙+1^𝑟−1𝑑_𝑘=0^𝑗_𝑧𝑗=1(𝑑_𝑘1𝑘2)_𝑧𝑗)]𝑅⁻¹𝑄Ψ(𝑡_𝑜),(3.52) where 𝑑_𝑘1𝑘2=𝑒^{𝜆𝑗𝑘(𝑡−𝑡𝑜)}(𝑡−𝑡_𝑜)^{𝑘2−𝑘1}(𝑘₂−𝑘₁)!,1≤𝑘₁≤𝑘₂≤𝑧_𝑗,0,otherwise(3.53) for 𝑗=𝑙+1,𝑙+2,…,𝜅, and 𝑧_𝑗=1,2,…,𝑑_𝑗.

Another expression for the exponential matrix of Jordan block, see (2.18), is provided by the following lemma.

Proposition 3.13. Considering the results of Lemma 2.4, one obtains the expression 𝑋(𝑡)=𝑄_𝑛,𝑔𝑌_𝑔(𝑡)+𝑄_𝑛,𝑣𝑌_𝑣(𝑡)+𝑄_𝑛,𝑝𝐿𝑅[(^𝑙_𝑗=0^𝑟−1_𝑘=0𝑒^{𝜆𝑗𝑘(𝑡−𝑡𝑜)}𝐼_𝜏𝑗𝑘)(^𝜅_𝑗=𝑙+1^𝑟−1𝑑_𝑘=0^𝑗_𝑧𝑗^𝑧₌₁^𝑗−1_𝑖=0𝑓_𝑖(𝑡−𝑡_𝑜)[𝐽_𝑧𝑗(𝜆_𝑗𝑘)]^𝑖)]×𝑅⁻¹𝑄Ψ(𝑡_𝑜),(3.54) where the 𝑓_𝑘(𝑡−𝑡_𝑜)^,𝑠 satisfy the following system of 𝑧_𝑗 (for 𝑧_𝑗=1,2,…,𝑑_𝑗) equations: ^𝑧𝑗−1_𝑖=𝑘𝑖𝑘𝑎_𝑗^𝑖−𝑘𝑓_𝑖(𝑡−𝑡_𝑜)=(𝑡−𝑡_𝑜)^𝑘𝑒𝑘!^{𝜆𝑗𝑘𝑡},𝑘=1,2,…,𝑧_𝑗(3.55) and [𝐽_𝑧𝑗(𝜆_𝑗𝑘)]^𝑖=[𝛽_𝑘^(𝑖)_1𝑘2]_𝑧𝑗, for 1≤𝑘₁,𝑘₂≤𝑧_𝑗 where 𝛽_𝑘^(𝑖)_1𝑘2=(^𝑖_{𝑘2−𝑘1})𝑎^{𝑖−(𝑘2−𝑘1)}_𝑗.

Analyzing more the results of this section, see Theorem 3.10 and Lemma 2.3 and 2.4, we can present briefly a symbolical algorithm for the solution of system (1.3).

Symbolical Algorithm
Step 1. Determine the pencil 𝑠𝐹−𝐺.Step 2. Calculate the expressions (3.3). Thus, we have to find the f.e.d, i.e.d., r.m.i, c.m.i, and so forth, (i.e., the complex Kronecker form 𝑠𝐹_𝑘−𝐺_𝑘 of the singular pencil 𝑠𝐹−𝐺 here; it should be noticed that this step is not an easy task; some parts are still under research). Step 3. Using the results of Step 2, determine the matrices Λ_𝑣,𝜆_𝑣, Λ^𝑇_𝑢, 𝜆^𝑇_𝑢, 𝐼_𝑝,𝐽_𝑝,𝐻_𝑞 and 𝐼_𝑞.Step 4. Determine 𝑄_𝑛,𝑔, 𝑄_𝑛,𝑣, 𝑄_𝑛,𝑝, 𝐿, 𝑅 (using the Jordan canonical form of matrix 𝐀), 𝑄 and Ψ(𝑡_𝑜) (see Remark 3.9).Step 5. Considering the transformation (3.10), that is, 𝑋(𝑡)=𝑄𝑌(𝑡), we obtain (3.54). Then the following.Sub step 5.1. Choose an arbitrary matrix 𝑌_𝑔(𝑡).Sub step 5.2. Determine the matrix 𝑌_𝑣(𝑡), that is, 𝑌_𝑣(𝑡)=𝑌_𝑣𝑔+1(𝑡)⊕…⊕𝑌_𝑣𝑖(𝑡)⊕…⊕𝑌_{𝑣𝑚−𝑟}(𝑡),(3.56) where 𝑌_𝑣𝑖(𝑡)=[𝑦_𝜆,𝑗]_{𝜆=1,2,…,𝑣𝑖,𝑗=1,2,…,𝑙}, for 𝑖=𝑔+1,…,𝑚−𝑟 and 𝑦_𝜆,𝑗(𝑡)=⋯𝑦_𝜆+1,𝑗(𝑡)𝑑𝑡⋯𝑑𝑡_𝑟-times+^𝑟_𝜉=1𝑐_{𝜆,𝑟−𝜉+1}𝑡^𝜉−1(𝜉−1)!.(3.57)Step fta
Following the results of Lemma 4, determine ^𝜅_𝑗=1^𝑟−1_𝑘=0𝑒^{𝐽𝑗𝑘(𝜆𝑗𝑘)(𝑡−𝑡𝑜)}=(^𝑙_𝑗=0^𝑟−1_𝑘=0𝑒^{𝜆𝑗𝑘(𝑡−𝑡𝑜)}𝐼_𝜏𝑗𝑘)(^𝜅_𝑗=𝑙+1^𝑟−1𝑑_𝑘=0^𝑗_𝑧𝑗=1(𝑑_𝑘1𝑘2)_𝑧𝑗),(3.58) where 𝑑_𝑘1𝑘2=𝑒^{𝜆𝑗𝑘(𝑡−𝑡𝑜)}(𝑡−𝑡_𝑜)^{𝑘2−𝑘1}(𝑘₂−𝑘₁)!,1≤𝑘₁≤𝑘₂≤𝑧_𝑗,0,otherwise(3.59) for 𝑗=𝑙+1,𝑙+2,…,𝜅, and 𝑧_𝑗=1,2,…,𝑑_𝑗. Step ftb
Following the results of Lemma 5, determine ^𝜅_𝑗=1^𝑟−1_𝑘=0𝑒^{𝐽𝑗𝑘(𝜆𝑗𝑘)(𝑡−𝑡𝑜)}=(^𝑙_𝑗=0^𝑟−1_𝑘=0𝑒^{𝜆𝑗𝑘(𝑡−𝑡𝑜)}𝐼_𝜏𝑗𝑘)⊕(^𝜅_𝑗=𝑙+1^𝑟−1𝑑_𝑘=0^𝑗_𝑧𝑗^𝑧₌₁^𝑗−1_𝑖=0𝑓_𝑖(𝑡−𝑡_𝑜)[𝐽_𝑧𝑗(𝜆_𝑗𝑘)]^𝑖),(3.60) where 𝑓_𝑘(𝑡−𝑡_𝑜)^,𝑠 satisfy the following system of 𝑧_𝑗 (for 𝑧_𝑗=1,2,…,𝑑_𝑗) equations: ^𝑧𝑗−1_𝑖=𝑘𝑖𝑘𝑎_𝑗^𝑖−𝑘𝑓_𝑖(𝑡−𝑡_𝑜)=(𝑡−𝑡_𝑜)^𝑘𝑒𝑘!^{𝜆𝑗𝑘𝑡},𝑘=1,2,…,𝑧_𝑗,(3.61) and [𝐽_𝑧𝑗(𝜆_𝑗𝑘)]^𝑖=[𝛽_𝑘^(𝑖)_1𝑘2]_𝑧𝑗, for 1≤𝑘₁,𝑘₂≤𝑧_𝑗, and 𝛽_𝑘^(𝑖)_1𝑘2=(^𝑖_{𝑘2−𝑘1})𝑎^{𝑖−(𝑘2−𝑘1)}_𝑗.

Example 3.14 (with consistent initial condition). Consider the 2nd order system 𝐹𝑋̈(𝑡)=𝐺𝑋(𝑡),(3.62) where 𝐹,𝐺∈ℳ(11×12;ℝ), and 𝑋∈𝒞^∞(ℝ,ℳ(12×5;ℝ)), with 𝐹=000000000000011−10−10−100000001000000000−1001100000000−11−101100000−100010100000000000000000100−1−10010000−100010001000−10011−10000101000−1000−100∈ℳ_11,12,𝐺=0000000000000−100110000000000000000000000010000000100−1−20000000000001000000−100010100000000−100011000−1000100010000000100001−101000−1000−101∈ℳ_11,12,(3.63)with known consistent initial conditions 𝑋(𝑡_𝑜),𝑋^′(𝑡_𝑜).
From the singularity of 𝑠𝐹−𝐺, there exist nonsingular matrices
𝑃=1000000000001100000000001000000000001000000001011000000000001000000000001000000000001000000000001000100100001000000000101∈ℳ₁₁,𝑄=110000001000010001000000001000010000000100000000010010000000000001000000000000100000010000010000000010001000010000000100000000000011000000000001∈ℳ₁₂(3.64) Then, using (3.3), we obtain 𝑃𝐹𝑄=𝐹_𝑘=𝕆_1,2⊕Λ₃⊕Λ^𝑇₂⊕𝐼₂⊕𝐻₂,𝑃𝐺𝑄=𝐺_𝑘=𝕆_1,2⊕𝜆₃⊕𝜆^𝑇₂⊕𝐽₂⊕𝐼₂,(3.65) where Λ₂,𝜆₂, Λ^𝑇₂, 𝜆^𝑇₂, 𝐼₂,𝐽₂, and 𝐻₂ are given by Λ₃=[𝐼₃⋮0]=100001000010∈ℳ_3,4,𝜆₃=[𝐻₃⋮𝜀₃]=001000000001∈ℳ_2,3,Λ^𝑇₂=𝑒^𝑇₂⋯𝐻₂=100100∈ℳ_3,2,𝜆^𝑇₂=0^𝑇⋯𝐼₂=001001∈ℳ_3,2,𝐼₂=1001,𝐽₂(1)=1101,𝐻₂=.0100(3.66) Considering the transformation (3.10), that is, 𝑋(𝑡)=𝑄𝑌(𝑡), we obtain the results of Lemma 2.3 (see also 𝐉, below): 𝑋(𝑡)=𝑄_12,𝑔=2𝑌_𝑔=2(𝑡)+𝑄_12,𝑣=2𝑌_𝑣=2(𝑡)+𝑄_12,𝑝=2𝐿𝑅[²²_{𝑗=1𝑧𝑗=1}(𝑑_𝑘1𝑘2)_𝑧𝑗]𝑅⁻¹𝑄Ψ(𝑡_𝑜),(3.67) where 𝑄_12,𝑔=2=,𝑄110000001000010001000000_12,𝑣=2=,𝑄001000010000000100000000_12,𝑝=2=.000010001000010000000100(3.68)
(i) 𝑌_𝑔=2(𝑡)∈ℳ_2,5 is an arbitrarily chosen matrix.
(ii) 𝑌_𝑣=2(𝑡)∈ℳ_2,5 with 𝑌₂(𝑡)=[𝑦_𝜆,𝑗]_{𝜆=1,2,𝑗=1,2,…,5}, and
𝑦_𝜆,𝑗(𝑡)=𝑦_𝜆+1,𝑗(𝑡)𝑑𝑡𝑑𝑡+²_𝜉=1𝑐_{𝜆,3−𝜉}𝑡^𝜉−1(𝜉−1)!.(3.69)
(iii)
𝐿=𝐼₂𝕆₂=10000100∈ℳ_2,4.(3.70)
(iv)
1𝑅=41414−1410−2012−140140012012,𝑅⁻¹=,21−200−10121200101𝐉=𝑅⁻¹𝐀𝑅=.−11000−10000110001(3.71) (We have only two eigenvalues, 𝜆₁=−1 and 𝜆₂=1.)
(v)
𝑄Ψ(𝑡_𝑜𝑄)=^𝑝=2,12𝑄_𝑝=2,12𝑋_𝑝(𝑡_𝑜)𝑋^′_𝑝(𝑡_𝑜),(3.72) where 𝑋(𝑡_𝑜),𝑋^′(𝑡_𝑜) are known, and 𝑄_𝑝=2,12=0100−1−10010000−10001000100.(3.73)
(vi) With
(3.74)
Combining the above arithmetic results, the analytic solution of system (3.62) is given by considering (3.58).

4. Solution Space Form of Nonconsistent Initial Conditions

In this short section, we would like to describe briefly the impulse behaviour of the solution of the original system (1.3), at time 𝑡_𝑜, see also [11–13]. In that case, we reformulate Proposition 3.6, so that the impulse solution is finally obtained. Note that in this part of the paper the condition that 𝑋∈𝒞^∞(𝔽,ℳ(𝑚×𝑙;𝔽)) does not hold anymore, since we are interesting for solutions with impulsive behaviour (again 𝔽 can be either ℝ or ℂ).

Moreover, we assume that the space of nonconsistent initial conditions is denoted by 𝒞^∗₀, which is called also redundancy space. Then, considering also Lemma 3.1, and especially (3.7) and (3.9), we have the nonconsistent initial condition that is

𝑌_𝑢^(𝑘)_𝑗(𝑡_𝑜)≠𝑌_{𝑢𝑗,𝑜}𝑌,for𝑗=ℎ+1,ℎ+2,…,𝑚−𝑟,_{𝑞(𝑘)𝑧}(𝑡_𝑜)≠𝑌_{𝑞𝑧,𝑜},for𝑧=1,2,…,𝜎,(4.1) for 𝑘=0,1,…,𝑟−1.

In order to be able to find a solution, we use the classical method of Laplace transformation. This method has been applied several times in descriptor system theory; see for instance, [4, 5, 11].

Proposition 4.1. The analytic solution of the system (3.27) is given by 𝑌_𝑢𝑗(𝑡)=−^𝑟−1𝑢_𝑘=0^𝑗−2_𝜁=0𝛿^{(𝑟𝜁+𝑟−1−𝑘)}(𝑡)𝐻_𝑢^𝜁+1_𝑗𝑌_𝑢^(𝑘)_𝑗,𝑜,(4.2) where 𝛿 and 𝛿^(𝑘) are the delta function of Dirac and its derivatives, respectively.

Proof. Let us start by observing that—as it is well known—there exists a 𝑢_𝑗∈ℕ such that 𝐻^𝑢𝑗_𝑢𝑗=𝕆, that is, the index of nilpotency equals 𝑢_𝑗 of 𝐻_𝑢𝑗.
Moreover, system (3.27) can be rewritten as follows; see also proof of Proposition 3.6,
𝑒^𝑇_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗𝐻(𝑡)_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗=0(𝑡)𝑌^𝑇_𝑢𝑗⇔𝑒(𝑡)^𝑇𝑢𝑗𝑌_𝑢^(𝑟)_𝑗(𝑡)=0^𝑡,𝐻_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗(𝑡)=𝑌_𝑢𝑗(𝑡).(4.3) Where by taking the Laplace transformation of 𝐻_𝑢𝑗𝑌_𝑢^(𝑟)_𝑗(𝑡)=𝑌_𝑢𝑗(𝑡), the following expression derives: 𝐻_𝑢𝑗ℑ{𝑌_𝑢^(𝑟)_𝑗(𝑡)}=ℑ{𝑌_𝑢𝑗(𝑡)},(4.4) and by defining ℑ{𝑌_𝑢𝑗(𝑡)}=𝒳_𝑢𝑗(𝑠), we obtain (𝑠^𝑟𝐻_𝑢𝑗−𝐼_𝑢𝑗)𝒳_𝑢𝑗(𝑠)=𝐻_𝑢𝑗^𝑟−1_𝑘=0𝑠^{𝑟−1−𝑘}𝑌_𝑢^(𝑘)_𝑗,𝑜.(4.5) Since 𝑢_𝑗 is the index nilpotency of 𝐻_𝑢𝑗, it is known that (𝑠^𝑟𝐻_𝑢𝑗−𝐼_𝑢𝑗)⁻¹=−^𝑢𝑗−1_𝜁=0(𝑠^𝑟𝐻_𝑢𝑗)^𝜁,(4.6) where 𝐻⁰_𝑢𝑗=𝐼_𝑢𝑗; see, for instance [4, 10]. Thus, substituting the above expression into (4.5), the following equation is being taken: 𝒳_𝑢𝑗(𝑠)=−^𝑟−1𝑢_𝑘=0^𝑗−2_𝜁=0𝑠^{𝑟𝜁+𝑟−1−𝑘}𝐻_𝑢^𝜁+1_𝑗𝑌_𝑢^(𝑘)_𝑗,𝑜.(4.7) Since ℑ{𝛿^(𝑘)(𝑡)}=𝑠^𝑘, (4.7) is transformed into (4.8): 𝒳_𝑢𝑗(𝑠)=−^𝑟−1𝑢_𝑘=0^𝑗−2_𝜁=0ℑ{𝛿^{(𝑟𝜁+𝑟−1−𝑘)}}(𝑡)𝐻_𝑢^𝜁+1_𝑗𝑌_𝑢^(𝑘)_𝑗,𝑜.(4.8) Now, by applying the inverse Laplace transformation into (4.8), the equation (4.2) is derived.

Remark 4.2. The analytic solution of the subsystem (3.7) is given by 𝑌_𝑢(𝑡)=𝑌_𝑢ℎ+1(𝑡)⊕⋯⊕𝑌_𝑢𝑗(𝑡)⊕⋯⊕𝑌_{𝑢𝑛−𝑟}(𝑡),(4.9) where the results of Proposition 4.1 are also considered.

Similarly to Proposition 4.1, we can prove the following proposition.

Proposition 4.3. The analytic solution of the system (3.9) is given by 𝑌_𝑞(𝑠)=−^𝑟−1𝑞_𝑘=0^∗−2_𝑧=0𝛿^{(𝑟𝑧+𝑟−1−𝑘)}(𝑡)𝐻_𝑞^𝑧+1𝑌^(𝑘)_𝑞,𝑜.(4.10)

Theorem 4.4. The analytic solution of (1.3) is given by 𝑋(𝑡)=𝑄_𝑛,𝑔𝑌_𝑔(𝑡)+𝑄_𝑛,𝑣𝑌_𝑣(𝑡)+𝑄_𝑛,𝑢𝑌_𝑢(𝑡)+𝑄_𝑛,𝑝𝐿𝑅⊕^𝜅⊕_𝑗=0^𝑟−1_𝑘=0𝑒^{𝐽𝑗𝑘(𝜆𝑗𝑘)(𝑡−𝑡𝑜)}𝑅⁻¹𝑍(𝑡_𝑜)−𝑄_𝑛,𝑞^𝑟−1𝑞_𝑘=0^∗−2_𝑧=0𝛿^{(𝑟𝑧+𝑟−1−𝑘)}(𝑡)𝐻_𝑞^𝑧+1𝑌^(𝑘)_𝑞,𝑜.(4.11) The matrix 𝑌_𝑔(𝑡) is arbitrarily chosen. For 𝑌_𝑣(𝑡)=𝑌_𝑣𝑔+1(𝑡)⊕⋯⊕𝑌_𝑣𝑖(𝑡)⊕⋯⊕𝑌_{𝑣𝑚−𝑟}(𝑡)where 𝑌_𝑣𝑖(𝑡)=[𝑦_𝜆,𝑗]_{𝜆=1,2,…,𝑣𝑖,𝑗=1,2,…,𝑙}, for 𝑖=𝑔+1,…,𝑚−𝑟, and 𝑦_𝜆,𝑗(𝑡)=⋯𝑦_𝜆+1,𝑗(𝑡)𝑑𝑡⋯𝑑𝑡_𝑟-times+^𝑟_𝜉=1𝑐_{𝜆,𝑟−𝜉+1}𝑡^𝜉−1(𝜉−1)!.(4.12) For 𝑌_𝑢(𝑡)=𝑌_𝑢ℎ+1(𝑡)⊕⋯⊕𝑌_𝑢𝑗(𝑡)⊕⋯⊕𝑌_{𝑢𝑛−𝑟}(𝑡),(4.13) where 𝑌_𝑢𝑗(𝑡)=−∑^𝑟−1_𝑘=0∑^𝑢𝑗−2_𝜁=0𝛿^{(𝑟𝜁+𝑟−1−𝑘)}(𝑡)𝐻_𝑢^𝜁+1_𝑗𝑌_𝑢^(𝑘)_𝑗,𝑜.
Moreover 𝐿=[𝐼_𝑝𝕆⋯𝕆]∈ℳ_𝑝,pr; 𝑅∈ℳ_pr such that 𝐉=𝑅⁻¹𝐀𝑅, where 𝐉∈ℳ_pr is the Jordan Canonical form of matrix 𝐀, and
𝐙(𝑡_𝑜)=𝑌^𝑡_𝑞(𝑡_𝑜)𝑌^′𝑡_𝑞(𝑡_𝑜)⋯𝑌_𝑝^{(𝑟−1)𝑡}(𝑡_𝑜)^𝑡∈ℳ_pr,𝑙.(4.14)

Proof. Combining the results of Theorem 3.10 and the above discussion, the solution is provided by (4.11).

Remark 4.5. For 𝑡>𝑡_𝑜, it is obvious that (3.45) is satisfied. Thus, we should stress that the system (1.3) has the above impulse behaviour at time instant where a nonconsistent initial value is assumed, while it returns to smooth behaviour at time instant 𝑡_𝑜.

5. Conclusions

In this paper, we study the class of LTI descriptor (singular) matrix descriptor differential equations of higher order whose coefficients are rectangular constant matrices. By taking into consideration that the relevant pencil is singular, we get affected by the Kronecker canonical form in order to decompose the differential system into five subsystems. Afterwards, we provide analytical formulas for this general class of Apostol-Kolodner type of equations when we have consistent and nonconsistent initial conditions.

As a future research, we would like to compare more our results with those of the preceding papers of Kolodner [9], Ben Taher and Rachidi [8, 14], and Geerts [12, 13]. Analytically, following the work in [9], we want to investigate whether it is possible to apply our results to find the dynamical solution of LTI matrix descriptor (singular) differential systems. Moreover, it also our wish to apply some combinatorial method for computing the matrix powers and exponential, as it is appeared in [8]. Finally, it is worth to extend the results of Geerts [13] to the general class of LTI matrix descriptor (regular and singular) differential systems.

Acknowledgments

The authors are very grateful to the anonymous referees' comments, suggestions, and remarks which highly improved the quality of the paper. They would like to also thank them for the very interesting future research directions that they have proposed to us.