Abstract and Applied Analysis
Volume 2010 (2010), Article ID 897301, 24 pages
doi:10.1155/2010/897301
Research Article

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

Athanasios A. Pantelous,1 Athanasios D. Karageorgos,2 Grigoris I. Kalogeropoulos,2 and Kostas G. Arvanitis3

1Department of Mathematical Sciences, University of Liverpool, Peach Street, L69 7ZL Liverpool, UK
2Department of Mathematics, University of Athens, GR-15784, Greece
3Department of Natural Resources Management and Agricultural Engineering, Agricultural University of Athens, GR-11855, Greece

Received 27 July 2009; Revised 11 December 2009; Accepted 27 January 2010

Academic Editor: Ağacık Zafer

Copyright © 2010 Athanasios A. Pantelous et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

𝐹 𝑋 ( 𝑟 ) ( 𝑡 ) = 𝐴 𝑋 ( 𝑡 ) + 𝐵 𝑈 ( 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 ) = − 𝐹 𝑋 ( 𝑡 ) , ( 1 . 2 ) we can obtain (1.3), where 𝐺 = 𝐴 − 𝐵 𝐹 .

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,

𝐹 𝑋 ( 𝑟 ) ( 𝑡 ) = 𝐺 𝑋 ( 𝑡 ) , ( 1 . 3 ) where, for (1.1), (1.2), and (1.3), 𝑟 th 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 𝑋 0 admits a functional solution. Thus, we shall call 𝑋 0 a 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 ) = 𝑋 0 .

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 ) ≠ 𝑋 0 .

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 𝑛 𝑚 and an indeterminate 𝑠 ∈ 𝔽 , the matrix pencil 𝑠 𝐹 − 𝐺 is called regular when 𝑚 = 𝑛 and d e t ( 𝑠 𝐹 − 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 𝑠 𝐹 − 𝐺 is said to be strictly equivalent to the pencil 𝑠 𝐹 − 𝐺 if and only if there exist nonsingular 𝑃 ∈ ℳ 𝑛 and 𝑄 ∈ ℳ 𝑚 such that 𝑃 ( 𝑠 𝐹 − 𝐺 ) 𝑄 = 𝑠 𝐹 − 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 = r a n k 𝔽 ( 𝑠 ) ( 𝑠 𝐹 − 𝐺 ) , where 𝔽 ( 𝑠 ) denotes the field of rational functions in 𝑠 having coefficients in the field 𝔽 . The equations

( 𝑠 𝐹 − 𝐺 ) 𝑥 ( 𝑠 ) = 0 , 𝜓 𝑇 ( 𝑠 ) ( 𝑠 𝐹 − 𝐺 ) = 0 𝑇 ( 2 . 2 ) have nonzero solutions 𝑥 ( 𝑠 ) and 𝜓 ( 𝑠 ) which are vectors in the rational vector spaces

𝒩 r i g h t ( 𝑠 ) ≜ 𝒩 r i g h t ( 𝑠 𝐹 − 𝐺 ) , 𝒩 l e f t ( 𝑠 ) ≜ 𝒩 l e f t ( 𝑠 𝐹 − 𝐺 ) , ( 2 . 3 ) respectively, where

𝒩 r i g h t ( 𝑠 ) ≜ 𝑥 ( 𝑠 ) ∈ 𝔽 ( 𝑠 ) 𝑚 ∶ ( 𝑠 𝐹 − 𝐺 ) 𝑥 ( 𝑠 ) = 0 𝑛 , 𝒩 l e f t ( 𝑠 ) = 𝜓 ( 𝑠 ) ∈ 𝔽 ( 𝑠 ) 𝑛 ∶ 𝜓 𝑇 ( 𝑠 ) ( 𝑠 𝐹 − 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 𝑠 𝐹 − 𝐺 , respectively. Furthermore, if 𝑟 = r a n k 𝔽 ( 𝑠 ) ( 𝑠 𝐹 − 𝐺 ) , it is evident such that

𝑟 = 𝑚 − 𝑟 𝑖 = 𝑔 + 1 𝑣 𝑖 + 𝑛 − 𝑟 𝑗 = ℎ + 1 𝑢 𝑗 + r a n k 𝔽 ( 𝑠 ) 𝑠 𝐹 𝑤 − 𝐺 𝑤 , ( 2 . 5 ) where 𝑠 𝐹 𝑤 − 𝐺 𝑤 is the complex Weierstrass canonical form; see [3].

Let 𝐵 1 , 𝐵 2 , . . . , 𝐵 𝑛 be elements of ℳ 𝑛 .

The direct sum of them denoted by 𝐵 1 ⊕ 𝐵 2 ⊕ ⋯ ⊕ 𝐵 𝑛 is the b l o c k d i a g { 𝐵 1 , 𝐵 2 , … , 𝐵 𝑛 } .

Thus, there exists 𝑃 ∈ ℳ 𝑚 and 𝑄 ∈ ℳ 𝑛 such that the complex Kronecker form 𝑠 𝐹 𝑘 − 𝐺 𝑘 of the singular pencil 𝑠 𝐹 − 𝐺 is defined as follows:

𝑠 𝐹 𝑘 − 𝐺 𝑘 ≜ 𝕆 ℎ , 𝑔 ⊕ 𝑠 Λ 𝑣 − 𝜆 𝑣 ⊕ 𝑠 Λ 𝑇 𝑢 − 𝜆 𝑇 𝑢 ⊕ 𝑠 𝐼 𝑝 − 𝐽 𝑝 ⊕ 𝑠 𝐻 𝑞 − 𝐼 𝑞 , ( 2 . 6 ) where 𝑣 = ∑ 𝑚 − 𝑟 𝑖 = 𝑔 + 1 𝑣 𝑖 , 𝑢 = ∑ 𝑛 − 𝑟 𝑗 = ℎ + 1 𝑢 𝑗 , 𝑝 = ∑ 𝜅 𝑗 = 1 𝑝 𝑗 , and 𝑞 = ∑ 𝜎 𝑗 = 1 𝑞 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 𝑠 Λ 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 𝑠 𝐹 − 𝑄 and has the form

𝑠 Λ 𝑣 − 𝜆 𝑣 ≜ 𝑠 Λ 𝑣 𝑔 + 1 − 𝜆 𝑣 𝑔 + 1 ⊕ ⋯ ⊕ 𝑠 Λ 𝑣 𝑖 − 𝜆 𝑣 𝑖 ⊕ ⋯ ⊕ 𝑠 Λ 𝑣 𝑚 − 𝑟 − 𝜆 𝑣 𝑚 − 𝑟 , ( 2 . 7 ) where Λ 𝑣 𝑖 = [ 𝐼 𝑣 𝑖 ⋮ 0 ] ∈ ℳ 𝑣 𝑖 , 𝑣 𝑖 + 1 , 𝜆 𝑣 𝑖 = [ 𝐻 𝑣 𝑖 ⋮ 𝜀 𝑣 𝑖 ] ∈ ℳ 𝑣 𝑖 , 𝑣 𝑖 + 1 for every 𝑖 = 𝑔 + 1 , 𝑔 + 2 , . . . , 𝑚 − 𝑟 , and 𝐼 𝑣 𝑖 and 𝐻 𝑣 𝑖 denote the 𝑣 𝑖 × 𝑣 𝑖 identity and the nilpotent (with index of nilpotency 𝑣 𝑖 ) matrix, respectively. 0 and 𝜀 𝑣 𝑖 = [ 0 ⋯ 0 1 ] 𝑇 are the zero column and the column with element 1 at the 𝑣 𝑖 place, respectively.

(S3) The third normal block 𝑠 Λ 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 𝑠 𝐹 − 𝐺 and has the form

𝑠 Λ 𝑇 𝑢 − 𝜆 𝑇 𝑢 ≜ 𝑠 Λ 𝑇 𝑢 ℎ + 1 − 𝜆 𝑇 𝑢 ℎ + 1 ⊕ ⋯ ⊕ 𝑠 Λ 𝑇 𝑢 𝑗 − 𝜆 𝑇 𝑢 𝑗 ⊕ ⋯ ⊕ 𝑠 Λ 𝑇 𝑢 𝑛 − 𝑟 − 𝜆 𝑇 𝑢 𝑛 − 𝑟 , ( 2 . 8 ) where Λ 𝑇 𝑢 𝑗 = [ 𝑒 𝑇 𝑢 𝑗 ⋯ 𝐻 𝑢 𝑗 ] ∈ ℳ 𝑢 𝑗 + 1 , 𝑢 𝑗 , 𝜆 𝑇 𝑢 𝑗 = [ 0 𝑇 ⋯ 𝐼 𝑢 𝑗 ] ∈ ℳ 𝑢 𝑗 + 1 , 𝑢 𝑗 for every 𝑗 = ℎ + 1 , ℎ + 2 , … , 𝑚 − 𝑟 , and 𝐼 𝑢 𝑗 and 𝐻 𝑢 𝑗 denote the 𝑢 𝑗 × 𝑢 𝑗 identity and nilpotent (with index of nilpotency 𝑢 𝑗 ) matrix, respectively. 0 and 𝑒 𝑢 𝑗 = [ 1 ⋯ 0 0 ] 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 𝑠 𝐹 𝑤 − 𝐺 𝑤 of the singular pencil 𝑠 𝐹 − 𝐺 which is defined by

𝑠 𝐹 𝑤 − 𝑄 𝑤 ≜ 𝑠 𝐼 𝑝 − 𝐽 𝑝 ⊕ 𝑠 𝐻 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 − 𝑎 1 𝑝 1 , … , 𝑠 − 𝑎 𝜅 𝑝 𝜅 , 𝜅 𝑗 = 1 𝑝 𝑗 = 𝑝 ( 2 . 1 0 ) of 𝑠 𝐹 − 𝐺 and has the form

𝑠 𝐼 𝑝 − 𝐽 𝑝 ≜ 𝑠 𝐼 𝑝 1 − 𝐽 𝑝 1 𝑎 1 ⊕ ⋯ ⊕ 𝑠 𝐼 𝑝 𝜅 − 𝐽 𝑝 𝜅 𝑎 𝜅 . ( 2 . 1 1 ) And also the 𝑞 blocks of the second uniquely defined block 𝑠 𝐻 𝑞 − 𝐼 𝑞 correspond to the infinite elementary divisors (i.e.d.)

̂ 𝑠 𝑞 1 , … , ̂ 𝑠 𝑞 𝜎 , 𝜎 𝑗 = 1 𝑞 𝑗 = 𝑞 ( 2 . 1 2 ) of 𝑠 𝐹 − 𝐺 and have the form

𝑠 𝐻 𝑞 − 𝐼 𝑞 ≜ 𝑠 𝐻 𝑞 1 − 𝐼 𝑞 1 ⊕ ⋯ ⊕ 𝑠 𝐻 𝑞 𝜎 − 𝐼 𝑞 𝜎 . ( 2 . 1 3 )

Thus 𝐻 𝑞 is a nilpotent element of ℳ 𝑛 with index ̃ 𝑞 = m a x { 𝑞 𝑗 ∶ 𝑗 = 1 , 2 , … , 𝜎 } , where

𝐻 ̃ 𝑞 𝑞 = 𝕆 , ( 2 . 1 4 ) and 𝐼 𝑝 𝑗 , 𝐽 𝑝 𝑗 ( 𝑎 𝑗 ) , 𝐻 𝑞 𝑗 are the matrices

𝐼 𝑝 𝑗 = ⎡ ⎢ ⎢ ⎢ ⎣ 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 ⎤ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝑝 𝑗 , 𝐽 𝑝 𝑗 𝑎 𝑗 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑎 𝑗 1 0 ⋯ 0 0 𝑎 𝑗 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 𝑎 𝑗 1 0 0 0 0 𝑎 𝑗 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝑝 𝑗 , 𝐻 𝑞 𝑗 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 ⋯ 0 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 0 1 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝑞 𝑗 . ( 2 . 1 5 )

In the last part of this introductive section, some elements for the analytic computation of 𝑒 𝐴 ( 𝑡 − 𝑡 𝑜 ) , 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 𝐽 𝑝 𝑗 ( 𝑎 𝑗 ) ( 𝑡 − 𝑡 𝑜 ) = 𝑑 𝑘 1 𝑘 2 𝑝 𝑗 , ( 2 . 1 6 ) where 𝑑 𝑘 1 𝑘 2 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑒 𝑎 𝑗 ( 𝑡 − 𝑡 𝑜 ) 𝑡 − 𝑡 𝑜 𝑘 2 − 𝑘 1 𝑘 2 − 𝑘 1 ! , 1 ≤ 𝑘 1 ≤ 𝑘 2 ≤ 𝑝 𝑗 , 0 , o t h e r w i s e . ( 2 . 1 7 )

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

Lemma 2.4 (see [15]). 𝑒 𝐽 𝑝 𝑗 ( 𝑎 𝑗 ) ( 𝑡 − 𝑡 𝑜 ) = 𝑝 𝑗 − 1 𝑖 = 0 𝑓 𝑖 𝑡 − 𝑡 𝑜 𝐽 𝑝 𝑗 𝑎 𝑗 𝑖 , ( 2 . 1 8 ) where the 𝑓 𝑘 ( 𝑡 − 𝑡 𝑜 ) , 𝑠 satisfy the following system of 𝑝 𝑗 equations: 𝑝 𝑗 − 1 𝑖 = 𝑘 𝑖 𝑘 𝑎 𝑖 − 𝑘 𝑗 𝑓 𝑖 𝑡 − 𝑡 𝑜 = 𝑡 − 𝑡 𝑜 𝑘 𝑘 ! 𝑒 𝑎 𝑗 𝑡 , 𝑘 = 1 , 2 , . . . , 𝑝 𝑗 , 𝐽 𝑝 𝑗 𝑎 𝑗 𝑖 = 𝛽 ( 𝑖 ) 𝑘 1 𝑘 2 𝑝 𝑗 , f o r 1 ≤ 𝑘 1 , 𝑘 2 ≤ 𝑝 𝑗 , ( 2 . 1 9 ) where 𝛽 ( 𝑖 ) 𝑘 1 𝑘 2 = ( 𝑖 𝑘 2 − 𝑘 1 ) 𝑎 𝑖 − ( 𝑘 2 − 𝑘 1 ) 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

𝐹 𝑋 ( 𝑟 ) ( 𝑡 ) = 𝐺 𝑋 ( 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 𝑟 𝐹 − 𝐺 , there exist nonsingular matrices 𝑃 ∈ ℳ 𝑛 and 𝑄 ∈ ℳ 𝑚 such that (see also Section 2)

𝑃 𝐹 𝑄 = 𝐹 𝑘 = 𝕆 ℎ , 𝑔 ⊕ Λ 𝑣 ⊕ Λ 𝑇 𝑢 ⊕ 𝐼 𝑝 ⊕ 𝐻 𝑞 , 𝑃 𝐺 𝑄 = 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 𝑣 𝑔 + 1 ⊕ ⋯ ⊕ Λ 𝑣 𝑖 ⊕ ⋯ ⊕ Λ 𝑣 𝑚 − 𝑟 , 𝜆 𝑣 ≜ 𝜆 𝑣 𝑔 + 1 ⊕ ⋯ ⊕ 𝜆 𝑣 𝑖 ⊕ ⋯ ⊕ 𝜆 𝑣 𝑚 − 𝑟 , Λ 𝑇 𝑢 ≜ Λ 𝑇 𝑢 ℎ + 1 ⊕ ⋯ ⊕ Λ 𝑇 𝑢 𝑗 ⊕ ⋯ ⊕ Λ 𝑇 𝑢 𝑛 − 𝑟 , 𝜆 𝑇 𝑢 ≜ 𝜆 𝑇 𝑢 ℎ + 1 ⊕ ⋯ ⊕ 𝜆 𝑇 𝑢 𝑗 ⊕ ⋯ ⊕ 𝜆 𝑇 𝑢 𝑛 − 𝑟 , 𝐼 𝑝 = 𝐼 𝑝 1 ⊕ ⋯ ⊕ 𝐼 𝑝 𝜅 , 𝐽 𝑝 = 𝐽 𝑝 1 𝑎 1 ⊕ ⋯ ⊕ 𝐽 𝑝 𝜅 𝑎 𝜅 , 𝐻 𝑞 ≜ 𝐻 𝑞 1 ⊕ ⋯ ⊕ 𝐻 𝑞 𝜎 , 𝐼 𝑞 = 𝐼 𝑞 1 ⊕ ⋯ ⊕ 𝐼 𝑞 𝜎 . ( 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 , 𝑔 𝑌 ( 𝑟 ) 𝑔 ( 𝑡 ) = 𝕆 ℎ , 𝑔 𝑌 𝑔 ( 𝑡 ) , ( 3 . 5 ) Λ 𝑣 𝑌 ( 𝑟 ) 𝑣 ( 𝑡 ) = 𝜆 𝑣 𝑌 𝑣 ( 𝑡 ) , ( 3 . 6 ) Λ 𝑇 𝑢 𝑌 ( 𝑟 ) 𝑢 ( 𝑡 ) = 𝜆 𝑇 𝑢 𝑌 𝑢 ( 𝑡 ) , ( 3 . 7 ) the so-called slow subsystem 𝑌 ( 𝑟 ) 𝑝 ( 𝑡 ) = 𝐽 𝑝 𝑌 𝑝 ( 𝑡 ) , ( 3 . 8 ) and the relative fast subsystem 𝐻 𝑞 𝑌 ( 𝑟 ) 𝑞 ( 𝑡 ) = 𝑌 𝑞 ( 𝑡 ) . ( 3 . 9 )

Proof. Consider the transformation 𝑋 ( 𝑡 ) = 𝑄 𝑌 ( 𝑡 ) , ( 3 . 1 0 ) where 𝑄 ∈ ℳ 𝑚 and 𝑌 ∈ 𝒞 ∞ ( 𝔽 , ℳ 𝑚 , 𝑙 ) . Substituting the previous expression into (1.3), we obtain 𝐹 𝑄 𝑌 ( 𝑟 ) ( 𝑡 ) = 𝐺 𝑄 𝑌 ( 𝑡 ) . ( 3 . 1 1 ) Whereby, multiplying by 𝑃 , we arrive at 𝐹 𝑘 𝑌 ( 𝑟 ) ( 𝑡 ) = 𝐺 𝑘 𝑌 ( 𝑡 ) . ( 3 . 1 2 ) 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 . 1 3 ) 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 + 1 𝑣 𝑖 , 𝑢 = ∑ 𝑛 − 𝑟 𝑗 = ℎ + 1 𝑢 𝑗 , 𝑝 = ∑ 𝜅 𝑗 = 1 𝑝 𝑗 , and 𝑞 = ∑ 𝜎 𝑗 = 1 𝑞 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 𝑣 𝑖 𝑌 ( 𝑟 ) 𝑣 𝑖 ( 𝑡 ) = 𝜆 𝑣 𝑖 𝑌 𝑣 𝑖 ( 𝑡 ) ( 3 . 1 4 ) is given by the expression 𝑌 𝑣 𝑖 ( 𝑡 ) = 𝑌 1 ( 𝑡 ) 𝑌 2 ( 𝑡 ) ⋯ 𝑌 𝑙 ( 𝑡 ) = 𝑦 𝜆 , 𝑗 ( 𝑡 ) 𝜆 = 1 , 2 , … , 𝑣 𝑖 𝑗 = 1 , 2 , … , 𝑙 , ( 3 . 1 5 ) where 𝑦 𝜆 , 𝑗 ( 𝑡 ) = ⋯ 𝑦 𝜆 + 1 , 𝑗 ( 𝑡 ) 𝑑 𝑡 ⋯ 𝑑 𝑡 𝑟 - t i m e s + 𝑟 𝜉 = 1 𝑐 𝜆 , 𝑟 − 𝜉 + 1 𝑡 𝜉 − 1 ( 𝜉 − 1 ) ! , ( 3 . 1 6 ) 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 𝑣 𝑖 ⋮ 0 𝑌 ( 𝑟 ) 𝑣 𝜄 ( 𝑡 ) = 𝐻 𝑣 𝑖 ⋮ 𝜀 𝑣 𝑖 𝑌 𝑣 𝜄 ( 𝑡 ) , ( 3 . 1 7 ) for every 𝑖 = 𝑔 + 1 , 𝑔 + 2 , … , 𝑚 − 𝑟 . Now, we denote 𝑌 𝑣 𝑖 ( 𝑡 ) = Ψ 𝑣 𝑖 ( 𝑡 ) 𝑦 1 ( 𝑡 ) , ( 3 . 1 8 ) where Ψ 𝑣 𝑖 ( 𝑡 ) ∈ ℳ 𝑣 𝑖 𝑙 , Ψ 𝑣 𝑖 ( 𝑡 ) = [ 𝑌 1 ( 𝑡 ) 𝑌 2 ( 𝑡 ) ⋯ 𝑌 𝑙 ( 𝑡 ) ] with 𝑌 𝑗 ( 𝑡 ) = [ 𝑦 1 , 𝑗 ( 𝑡 ) 𝑦 2 , 𝑗 ( 𝑡 ) ⋯ 𝑦 𝑣 𝑖 , 𝑗 ( 𝑡 ) ] 𝑇 , and 𝑦 1 ( 𝑡 ) ∈ ℳ 1 𝑙 (vector, 1 × 𝑙 ).
Thus,
𝐼 𝑣 𝑖 ⋮ 0 ⎡ ⎢ ⎣ Ψ ( 𝑟 ) 𝑣 𝑖 ( 𝑡 ) 𝑦 ( 𝑟 ) 1 ( 𝑡 ) ⎤ ⎥ ⎦ = 𝐻 𝑣 𝑖 ⋮ 𝜀 𝑣 𝑖 Ψ 𝑣 𝑖 ( 𝑡 ) 𝑦 1 ( 𝑡 ) , ( 3 . 1 9 ) or, equivalently, we obtain Ψ ( 𝑟 ) 𝑣 𝑖 ( 𝑡 ) = 𝐻 𝑣 𝑖 Ψ 𝑣 𝑖 ( 𝑡 ) + 𝜀 𝑣 𝑖 𝑦 1 ( 𝑡 ) . ( 3 . 2 0 ) Note that 𝜀 𝑣 𝑖 𝑦 1 ( 𝑡 ) is a matrix with 𝑣 𝑖 × 𝑣 𝑖 -elements as follows 𝜀 𝑣 𝑖 𝑦 1 ( 𝑡 ) = ̃ ‌ 𝑌 1 ( 𝑡 ) ̃ ‌ 𝑌 2 ( 𝑡 ) ⋯ ̃ ‌ 𝑌 𝑙 ( 𝑡 ) ≜ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝕆 𝑣 𝑖 − 1 , 𝑙 𝑦 𝑣 𝑖 + 1 , 1 ( 𝑡 ) 𝑦 𝑣 𝑖 + 1 , 2 ( 𝑡 ) ⋯ 𝑦 𝑣 𝑖 , + 1 𝑙 ( 𝑡 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ( 3 . 2 1 ) where ̃ ‌ 𝑌 𝑗 ( 𝑡 ) = [ 0 0 ⋯ 𝑦 𝑣 𝑖 + 1 , 𝑗 ( 𝑡 ) ] 𝑇 , for 𝑗 = 1 , 2 , … , 𝑙 .
Consequently, (3.20) is rewritten as follows:
𝑌 ( 𝑟 ) 1 ( 𝑡 ) 𝑌 ( 𝑟 ) 2 ( 𝑡 ) ⋯ 𝑌 ( 𝑟 ) 𝑙 ( 𝑡 ) = 𝐻 𝑣 𝑖 𝑌 1 ( 𝑡 ) 𝐻 𝑣 𝑖 𝑌 2 ( 𝑡 ) ⋯ 𝐻 𝑣 𝑖 𝑌 𝑙 ( 𝑡 ) + ̃ ‌ 𝑌 1 ( 𝑡 ) ̃ ‌ 𝑌 2 ( 𝑡 ) ⋯ ̃ ‌ 𝑌 𝑙 ( 𝑡 ) , ( 3 . 2 2 ) or, equivalently, 𝑌 ( 𝑟 ) 𝑗 ( 𝑡 ) = 𝐻 𝑣 𝑖 𝑌 𝑗 ( 𝑡 ) + ̃ ‌ 𝑌 𝑗 ( 𝑡 ) , ( 3 . 2 3 ) and eventually, as a scalar system, we obtain 𝑦 ( 𝑟 ) 1 , 𝑗 ( 𝑡 ) = 𝑦 2 , 𝑗 ( 𝑡 ) , 𝑦 ( 𝑟 ) 2 , 𝑗 ( 𝑡 ) = 𝑦 3 , 𝑗 ( 𝑡 ) , … , 𝑦 ( 𝑟 ) 𝑣 𝑖 − 1 , 𝑗 ( 𝑡 ) = 𝑦 𝑣 𝑖 , 𝑗 ( 𝑡 ) , 𝑦 ( 𝑟 ) 𝑣 𝑖 , 𝑗 ( 𝑡 ) = 𝑦 𝑣 𝑖 + 1 , 𝑗 ( 𝑡 ) . ( 3 . 2 4 )
Denote that element 𝑦 𝑣 𝑖 + 1 , 𝑗 ( 𝑡 ) is an arbitrary function; then the solution is given iteratively, as follows.
Firstly, we take the equation 𝑦 ( 𝑟 ) 𝑣 𝑖 , 𝑗 ( 𝑡 ) = 𝑦 𝑣 𝑖 + 1 , 𝑗 ( 𝑡 ) for every 𝑗 = 1 , 2 , … , 𝑙 ,
𝑦 ( 𝑟 − 1 ) 𝑣 𝑖 , 𝑗 ( 𝑡 ) = 𝑦 𝑣 𝑖 + 1 , 𝑗 ( 𝑡 ) 𝑑 𝑡 + 𝑐 𝑣 𝑖 , 1 , … , 𝑦 𝑣 𝑖 , 𝑗 ( 𝑡 ) = ⋯ 𝑦 𝑣 𝑖 + 1 , 𝑗 ( 𝑡 ) 𝑑 𝑡 ⋯ 𝑑 𝑡 𝑟 - t i m e s + 𝑟 𝜉 = 1 𝑐 𝑣 𝑖 , 𝑟 − 𝜉 + 1 𝑡 𝜉 − 1 ( 𝜉 − 1 ) ! . ( 3 . 2 5 ) We continue the procedure, for 𝑦 ( 𝑟 ) 𝑣 𝑖 − 1 , 𝑗 ( 𝑡 ) = 𝑦 𝑣 𝑖 , 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 𝑣 𝑔 + 1 ( 𝑡 ) ⊕ ⋯ ⊕ 𝑌 𝑣 𝑖 ( 𝑡 ) ⊕ ⋯ ⊕ 𝑌 𝑣 𝑚 − 𝑟 ( 𝑡 ) , ( 3 . 2 6 ) 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 𝑢 𝑗 𝑌 ( 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = 𝜆 𝑇 𝑢 𝑗 𝑌 𝑢 𝑗 ( 𝑡 ) ( 3 . 2 7 ) has only the zero solution.

Proof. Consider that system (3.27) can be rewritten as follows: ⎡ ⎢ ⎢ ⎣ 𝑒 𝑇 𝑢 𝑗 ⋯ 𝐻 𝑢 𝑗 ⎤ ⎥ ⎥ ⎦ 𝑌 ( 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = ⎡ ⎢ ⎢ ⎣ 0 𝑇 ⋯ 𝐼 𝑢 𝑗 ⎤ ⎥ ⎥ ⎦ 𝑌 𝑢 𝑗 ( 𝑡 ) ( 3 . 2 8 ) for every 𝑗 = ℎ + 1 , ℎ + 2 , … , 𝑚 − 𝑟 .
Afterwards, we obtain straightforwardly the following system:
⎡ ⎢ ⎣ 𝑒 𝑇 𝑢 𝑗 𝑌 ( 𝑟 ) 𝑢 𝑗 ( 𝑡 ) 𝐻 𝑢 𝑗 𝑌 ( 𝑟 ) 𝑢 𝑗 ( 𝑡 ) ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ 0 𝑇 𝑌 𝑢 𝑗 ( 𝑡 ) ⎤ ⎥ ⎦ ⟺ ⎧ ⎪ ⎨ ⎪ ⎩ 𝑒 𝑇 𝑢 𝑗 𝑌 ( 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = 0 𝑇 , 𝐻 𝑢 𝑗 𝑌 ( 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = 𝑌 𝑢 𝑗 ( 𝑡 ) . ( 3 . 2 9 ) Now, by successively taking 𝑟 th derivatives with respect to 𝑡 on both sides of 𝐻 𝑢 𝑗 𝑌 ( 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = 𝑌 𝑢 𝑗 ( 𝑡 ) , ( 3 . 3 0 ) and left multiplying by the matrix 𝐻 𝑢 𝑗 , 𝑢 𝑗 − 1 times (where 𝑢 𝑗 is the index of the nilpotent matrix 𝐻 𝑢 𝑗 , i.e., 𝐻 𝑢 𝑗 𝑢 𝑗 = 𝕆 ), we obtain the following equations: 𝐻 2 𝑢 𝑗 𝑌 ( 2 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = 𝐻 𝑢 𝑗 𝑌 ( 𝑟 ) 𝑢 𝑗 ( 𝑡 ) , … , 𝐻 𝑢 𝑗 𝑢 𝑗 𝑌 ( 𝑢 𝑗 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = 𝐻 𝑢 𝑗 − 1 𝑢 𝑗 𝑌 ( ( 𝑢 𝑗 − 1 ) 𝑟 ) 𝑢 𝑗 ( 𝑡 ) . ( 3 . 3 1 ) Thus, we conclude to the following expression: 𝑌 𝑢 𝑗 ( 𝑡 ) = 𝐻 𝑢 𝑗 𝑌 ( 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = 𝐻 2 𝑢 𝑗 𝑌 ( 2 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = ⋯ = 𝐻 𝑢 𝑗 𝑢 𝑗 𝑌 ( 𝑢 𝑗 𝑟 ) 𝑢 𝑗 ( 𝑡 ) = 𝕆 . ( 3 . 3 2 )

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 ) = 𝐿 𝑅 𝜅 ⊕ 𝑗 = 1 𝑟 − 1 ⊕ 𝑘 = 0 𝑒 𝐽 𝑗 𝑘 ( 𝜆 𝑗 𝑘 ) ( 𝑡 − 𝑡 𝑜 ) 𝑅 − 1 𝐙 𝑡 𝑜 , ( 3 . 3 3 ) where 𝐿 = [ 𝐼 𝑝 𝕆 ⋯ 𝕆 ] ∈ ℳ 𝑝 , p r ; 𝑅 ∈ ℳ p r such that 𝐉 = 𝑅 − 1 𝐀 𝑅 .
Note that 𝐉 ∈ ℳ p r is the Jordan Canonical form of matrix
𝐀 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝕆 𝐼 𝑝 𝕆 ⋯ 𝕆 𝕆 𝕆 𝐼 𝑝 ⋯ 𝕆 ⋮ ⋮ ⋮ ⋱ ⋮ 𝕆 𝕆 𝕆 ⋯ 𝐼