Journal of Control Science and Engineering
Volume 2009 (2009), Article ID 562484, 29 pages
doi:10.1155/2009/562484
Research Article

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

Athanasios D. Karageorgos,1 Athanasios A. Pantelous,1,2,3 and Grigoris I. Kalogeropoulos1

1Department of Mathematics, University of Athens, Athens, Greece
2School of Engineering and Mathematical Sciences, City University, London, UK
3Department of Mathematical Sciences, University of Liverpool, Liverpool, UK

Received 4 March 2009; Accepted 19 June 2009

Academic Editor: Amit Bhaya

Copyright © 2009 Athanasios D. Karageorgos 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 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,

𝐹 𝑥 ( 𝑟 ) ( 𝑡 ) = 𝐺 𝑥 ( 𝑡 ) + 𝑏 𝑢 ( 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 d e t 𝑀 = 0 ( 0 is 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 𝑛 𝑚 = 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 𝑛 t h -order linear differential system, see (1), the input can be a linear combination of Dirac 𝛿 -function and its first ( 𝑛 − 1 ) t h -derivatives as follows:

𝑢 𝑜 ( 𝑡 ) = 𝑎 𝑜 𝛿 ( 𝑡 ) + 𝑎 1 𝛿 ( 1 ) ( 𝑡 ) + 𝑎 2 𝛿 ( 2 ) ( 𝑡 ) + ⋯ + 𝑎 𝑛 − 1 𝛿 ( 𝑛 − 1 ) ( 𝑡 ) = 𝑛 − 1 𝑘 = 1 𝑎 𝑘 𝛿 ( 𝑘 ) ( 𝑡 ) , ( 3 ) where 𝛿 ( 𝑘 ) ( 𝑡 ) or 𝑑 𝑘 𝛿 ( 𝑡 ) / 𝑑 𝑡 𝑘 is the 𝑘 t h -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 − ) = [ 0 0 ⋯ 0 ] 𝑡 , ( 4 ) and at time 0 + , it achieves

𝑥 0 + = 𝑥 0 1 𝑥 0 2 ⋯ 𝑥 0 𝑛 𝑡 , 𝑥 0 + = 𝑥 1 1 𝑥 1 2 ⋯ 𝑥 1 𝑛 𝑡 , … , 𝑥 ( 𝑟 − 1 ) 0 + = 𝑥 𝑟 − 1 1 𝑥 𝑟 − 1 2 ⋯ 𝑥 𝑟 − 1 𝑛 𝑡 . ( 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 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 . In any other case, the pencil will be called singular.

Definition 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 𝑚 and 𝑄 ∈ ℳ 𝑛 such as 𝑃 ( 𝑠 𝐹 − 𝐺 ) 𝑄 = 𝑠 𝐹 − 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 𝑠 𝐹 − 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 𝑞 are called infinite elementary divisors (i.e.d.).

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 } .

Then, the complex Weierstrass form 𝑠 𝐹 𝑤 − 𝑄 𝑤 of the regular pencil 𝑠 𝐹 − 𝐺 is defined by

𝑠 𝐹 𝑤 − 𝑄 𝑤 ≜ 𝑠 𝐼 𝑝 − 𝐽 𝑝 𝑠 𝐻 𝑞 − 𝐼 𝑞 . ( 7 ) Now, the Jordan type element, that is, 𝐽 𝑝 , is uniquely defined by the set of f.e.d.

𝑠 − 𝑎 1 𝑝 1 , … , 𝑠 − 𝑎 𝜈 𝑝 𝜈 , 𝜈 𝑗 = 1 𝑝 𝑗 = 𝑝 , ( 8 ) 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 𝜈 − 𝐽 𝑝 𝜈 𝑎 𝜈 , ( 9 ) where

𝐼 𝑝 ≜ 𝐼 𝑝 1 𝐼 𝑝 2 ⋯ 𝐼 𝑝 𝜈 , 𝐽 𝑝 ≜ 𝐽 𝑝 1 𝑎 1 𝐽 𝑝 2 𝑎 2 ⋯ 𝐽 𝑝 𝜈 𝑎 𝜈 . ( 1 0 ) The 𝑞 blocks of the second, that is, 𝑠 𝐻 𝑞 − 𝐼 𝑞 , see (7), are uniquely defined by the set of i.e.d.

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

𝑠 𝐻 𝑞 − 𝐼 𝑞 ≜ 𝑠 𝐻 𝑞 1 − 𝐼 𝑞 1 ⋯ 𝑠 𝐻 𝑞 𝜎 − 𝐼 𝑞 𝜎 , ( 1 2 ) where

𝐼 𝑞 ≜ 𝐼 𝑞 1 𝐼 𝑞 2 ⋯ 𝐼 𝑞 𝜎 , 𝐻 𝑞 ≜ 𝐻 𝑞 1 𝐻 𝑞 2 ⋯ 𝐻 𝑞 𝜎 . ( 1 3 ) Furthermore, 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 , ( 1 4 ) and 𝐼 𝑝 𝑗 , 𝐽 𝑝 𝑗 ( 𝑎 𝑗 ) , 𝐻 𝑞 𝑗 are defined as

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 . ( 1 5 ) Moreover, for the matrices 𝐹 and 𝐺 , we have the parameterization

𝑃 𝐹 𝑄 = 𝐹 𝑤 = 𝐼 𝑝 𝐻 𝑞 , ( 1 6 ) 𝑃 𝐺 𝑄 = 𝐺 𝑤 = 𝐽 𝑝 𝐼 𝑞 . ( 1 7 ) 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 ) = l i m 𝑎 → 0 𝛿 𝑎 ( 𝑡 ) , ( 1 8 ) where 𝛿 𝑎 ( 𝑡 ) is called a nascent delta function. This limit is in the sense that

l i m 𝑎 → 0 + ∞ − ∞ 𝛿 𝑎 ( 𝑡 ) 𝑓 ( 𝑡 ) 𝑑 𝑡 = 𝑓 ( 0 ) . ( 1 9 )

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 ) = l i m 𝜎 → 0 1 𝜎 √ 2 𝜋 𝑒 − 𝑡 2 / 2 𝜎 2 = l i m 𝜎 → 0 1 𝜎 Φ 𝑡 𝜎 , ( 2 0 ) where Φ ( 𝑥 ) = ( 1 / √ 2 𝜋 ) 𝑒 − 𝑥 2 / 2 .

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

𝑢 ( 𝑡 ) = 1 𝜎 Φ 𝑡 𝜎 𝑎 𝑜 + 1 𝜎 2 Φ ( 1 ) 𝑡 𝜎 𝑎 1 + ⋯ + 1 𝜎 𝑛 Φ ( 𝑛 − 1 ) 𝑡 𝜎 𝑎 𝑛 − 1 = 𝑛 − 1 𝑘 = 0 1 𝜎 𝑘 + 1 Φ ( 𝑘 ) 𝑡 𝜎 𝑎 𝑘 . ( 2 1 ) Then, we take the limit

𝑢 𝑜 ( 𝑡 ) = l i m 𝜎 → 0 𝑢 ( 𝑡 ) . ( 2 2 ) 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 ( 𝑟 ) 𝑝 ( 𝑡 ) = 𝐽 𝑝 𝑦 𝑝 ( 𝑡 ) + ̃ ‌ 𝑏 𝑝 𝑢 𝑜 ( 𝑡 ) , ( 2 3 ) 𝐻 𝑞 𝑦 ( 𝑟 ) 𝑞 ( 𝑡 ) = 𝑦 𝑞 ( 𝑡 ) + ̃ ‌ 𝑏 𝑞 𝑢 𝑜 ( 𝑡 ) . ( 2 4 )

Proof. Consider the transformation 𝑥 ( 𝑡 ) = 𝑄 𝑦 ( 𝑡 ) . ( 2 5 ) Substituting the previous expression into (1) and considering also (3), we obtain 𝐹 𝑄 𝑦 ( 𝑟 ) ( 𝑡 ) = 𝐺 𝑄 𝑦 ( 𝑡 ) + 𝑏 𝑢 𝑜 ( 𝑡 ) . ( 2 6 ) Multiplying by 𝑃 , we arrive at 𝐹 𝑤 𝑦 ( 𝑟 ) ( 𝑡 ) = 𝐺 𝑤 𝑦 ( 𝑡 ) + 𝑃 𝑏 𝑢 𝑜 ( 𝑡 ) . ( 2 7 ) Now, we denote 𝑦 ( 𝑡 ) = ⎡ ⎢ ⎣ 𝑦 𝑝 ( 𝑡 ) 𝑦 𝑞 ( 𝑡 ) ⎤ ⎥ ⎦ , 𝑃 𝑏 ≜ ̃ ‌ 𝑏 = ⎡ ⎢ ⎣ ̃ ‌ 𝑏 𝑝 ̃ ‌ 𝑏 𝑞 ⎤ ⎥ ⎦ , ( 2 8 ) where 𝑝 + 𝑞 = 𝑛 .
Taking into account the following expressions, that is,
𝐼 𝑝 𝕆 𝑝 , 𝑞 𝕆 𝑞 , 𝑝 𝐻 𝑞 ⎡ ⎢ ⎣ 𝑦 ( 𝑟 ) 𝑝 ( 𝑡 ) 𝑦 ( 𝑟 ) 𝑞 ( 𝑡 ) ⎤ ⎥ ⎦ = 𝐽 𝑝 𝕆 𝑝 , 𝑞 𝕆 𝑞 , 𝑝 𝐼 𝑞 ⎡ ⎢ ⎣ 𝑦 𝑝 ( 𝑡 ) 𝑦 𝑞 ( 𝑡 ) ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ ̃ ‌ 𝑏 𝑝 ̃ ‌ 𝑏 𝑞 ⎤ ⎥ ⎦ 𝑢 𝑜 ( 𝑡 ) , ( 2 9 ) 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

𝑧 1 ( 𝑡 ) = 𝑦 𝑝 ( 𝑡 ) , 𝑧 2 ( 𝑡 ) = 𝑦 𝑝 ( 𝑡 ) , ⋮ 𝑧 𝑟 ( 𝑡 ) = 𝑦 ( 𝑟 − 1 ) 𝑝 ( 𝑡 ) . ( 3 0 ) Then, we have the following system of ordinary differential equations:

𝑧 1 ( 𝑡 ) = 𝑧 2 ( 𝑡 ) , 𝑧 2 ( 𝑡 ) = 𝑧 3 ( 𝑡 ) , ⋮ 𝑧 𝑟 ( 𝑡 ) = 𝐽 𝑝 𝑧 1 ( 𝑡 ) + ̃ ‌ 𝑏 𝑝 𝑢 𝑜 ( 𝑡 ) . ( 3 1 ) Finally, (31) can be expressed by using vector-matrix equations

𝐳 ( 𝑡 ) = 𝐑 𝐳 ( 𝑡 ) + 𝐋 𝑢 𝑜 ( 𝑡 ) , ( 3 2 ) where 𝐳 ( 𝑡 ) = [ 𝑧 𝑡 1 ( 𝑡 ) 𝑧 𝑡 2 ( 𝑡 ) ⋯ 𝑧 𝑡 𝑟 ( 𝑡 ) ] 𝑡 ( ( ) 𝑡 is the transpose tensor) and the coefficient matrices 𝐑 and 𝐋 are given by

𝐑 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝕆 𝑝 𝐼 𝑝 𝕆 𝑝 ⋯ 𝕆 𝑝 𝕆 𝑝 𝕆 𝑝 𝐼 𝑝 ⋯ 𝕆 𝑝 ⋮ ⋮ ⋮ ⋱ ⋮ 𝕆 𝑝 𝕆 𝑝 𝕆 𝑝 ⋯ 𝐼 𝑝 𝐽 𝑝 𝕆 𝑝 𝕆 𝑝 ⋯ 𝕆 𝑝 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝑟 𝑝 , 𝐋 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 𝑝 0 𝑝 ⋮ ̃ ‌ 𝑏 𝑝 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ ℳ ( 𝑟 𝑝 ) 1 , ( 3 3 ) with corresponding dimension of 𝐳 ( 𝑡 ) , 𝑟 𝑝 × 1 .

The state of system (1) at time 0 − is 𝑥 ( 0 − ) = [ 0 0 ⋯ 0 ] 𝑡 and at time 0 + it achieves 𝑥 ( 0 + ) = [ 𝑥 1 𝑥 2 ⋯ 𝑥 𝑛 ] 𝑡 .

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

𝑦 ( 0 − ) = 𝑄 − 1 𝑥 ( 0 − ) = 𝑦 ( 0 − ) = 𝑄 − 1 𝑥 ( 0 − ) = ⋯ = 𝑦 ( 𝑟 − 1 ) ( 0 − ) = 𝑄 − 1 𝑥 ( 𝑟 − 1 ) ( 0 − ) = 0 𝑡 , ( 3 4 ) and at time 0 + we obtain

𝑦 0 + = 𝑄 − 1 𝑥 0 + = 𝜒 0 1 𝜒 0 2 ⋯ 𝜒 0 𝑛 𝑡 , 𝑦 0 + = 𝑄 − 1 𝑥 0 + = 𝜒 1 1 𝜒 1 2 ⋯ 𝜒 1 𝑛 𝑡 , … , 𝑦 ( 𝑟 − 1 ) 0 + = 𝑄 − 1 𝑥 ( 𝑟 − 1 ) 0 + = 𝜒 𝑟 − 1 1 𝜒 𝑟 − 1 2 ⋯ 𝜒 𝑟 − 1 𝑛 𝑡 . ( 3 5 ) Moreover, 𝑦 ( 𝑖 ) 𝑝 ( 0 − ) = 0 𝑡 𝑝 ∈ ℳ 𝑝 , 𝑦 ( 𝑖 ) 𝑝 0 + = 𝜒 𝑖 1 𝜒 𝑖 2 ⋯ 𝜒 𝑖 𝑝 𝑡 ∈ ℳ 𝑝 , 𝑦 ( 𝑖 ) 𝑞 ( 0 − ) = 0 𝑡 𝑞 ∈ ℳ 𝑞 , 𝑦 ( 𝑖 ) 𝑞 0 + = 𝜒 𝑖 𝑝 + 1 𝜒 𝑖 𝑝 + 2 ⋯ 𝜒 𝑖 𝑛 𝑡 ∈ ℳ 𝑞 , ( 3 6 ) where 𝜒 𝑖 𝑗 ∈ 𝔽 for 𝑖 = 0 , 1 , 2 , … , 𝑟 − 1 , and 𝑗 = 1 , 2 , … , 𝑛 .

Furthermore,

𝐳 ( 0 − ) = 𝑧 𝑡 1 ( 0 − ) 𝑧 𝑡 2 ( 0 − ) ⋯ 𝑧 𝑡 𝑟 ( 0 − ) 𝑡 = 𝑦 𝑡 𝑝 ( 0 − ) 𝑦 𝑡 𝑝 ( 0 − ) ⋯ 𝑦 ( 𝑟 − 1 ) 𝑡 𝑝 ( 0 − ) 𝑡 = 0 𝑡 𝑟 𝑝 , 𝐳 0 + = 𝑧 𝑡 1 0 + 𝑧 𝑡 2 0 + ⋯ 𝑧 𝑡 𝑟 0 + 𝑡 = 𝑦 𝑡 𝑝 0 + 𝑦 𝑡 𝑝 0 + ⋯ 𝑦 ( 𝑟 − 1 ) 𝑡 𝑝 0 + 𝑡 = 𝜒 0 1 𝜒 0 2 ⋯ 𝜒 0 𝑝 𝜒 1 1 𝜒 1 2 ⋯ 𝜒 1 𝑝 ⋯ 𝜒 𝑟 − 1 1 𝜒 𝑟 − 1 2 ⋯ 𝜒 𝑟 − 1 𝑝 𝑡 . ( 3 7 )

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 − 𝜆 1 𝜏 𝑖 , ( 3 8 ) with 𝜆 𝑖 ≠ 𝜆 𝑗 for 𝑖 ≠ 𝑗 and ∑ 𝜅 𝑖 = 1 𝜏 𝑖 = 𝑟 𝑝 . Without loss of generality, we assume that 𝑑 1 = 𝜏 1 , 𝑑 2 = 𝜏 2 , … , 𝑑 𝑙 = 𝜏 𝑙 , f o r 1 < 𝑙 < 𝜅 , 𝑑 𝑙 + 1 < 𝜏 𝑙 + 1 , … , 𝑑 𝑘 < 𝜏 𝜅 , ( 3 9 ) 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 𝑟 𝑝 linearly independent vectors 𝑣 1 , 𝑣 2 , … , 𝑣 𝑟 𝑝 and 𝑟 𝑝 × 𝑟 𝑝 similarity transformation 𝐶 = [ 𝑣 1 , 𝑣 2 , … , 𝑣 𝑟 𝑝 ] that takes 𝐑 into the Jordan canonical form, as the following definition clarifies.

Definition 5. There exists an invertible matrix 𝐶 ∈ ℳ 𝑟 𝑝 such as 𝐉 = 𝐶 − 1 𝐑 𝐶 , 𝐉 ∈ ℳ 𝑟 𝑝 ; is it Jordan canonical form of matrix 𝐑 . Analytically, 𝐉 = 𝐽 𝑜 𝐽 𝑙 + 1 𝐽 𝑙 + 2 ⋯ 𝐽 𝜅 . ( 4 0 ) (i)The block diagonal matrix 𝐽 𝑜 = 𝐽 1 ⨁ 𝐽 2 ⨁ ⋯ ⨁ 𝐽 𝑙 , where 𝐽 𝑖 = ⎡ ⎢ ⎢ ⎢ ⎣ 𝜆 𝑖 𝜆 𝑖 ⋱ 𝜆 𝑖 ⎤ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝜏 𝑖 , ( 4 1 ) 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 = 1 𝜏 𝑖 .(ii) Also, the block matrix 𝐽 𝑗 = 𝐽 𝑗 , 1 ⨁ 𝐽 𝑗 , 2 ⨁ ⋯ ⨁ 𝐽 𝑗 , 𝑑 𝑗 ∈ ℳ 𝜏 𝑗 , where 𝐽 𝑗 , 𝑧 𝑗 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜆 𝑗 1 𝜆 𝑗 1 𝜆 𝑗 ⋱ ⋱ 1 𝜆 𝑗 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝑧 𝑗 f o r 𝑗 = 𝑙 + 1 , 𝑙 + 2 , … , 𝑘 , 𝑧 𝑗 = 1 , 2 , … , 𝑑 𝑗 . ( 4 2 )

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 ) 𝐛 𝑢 𝑜 ( 𝜏 ) 𝑑 𝜏 , ( 4 3 ) with initial condition 𝐳 ( 0 − ) = 0 𝑡 𝑟 𝑝 .

Proof. Consider the transformation 𝐳 ( 𝑡 ) = 𝐶 𝜁 ( 𝑡 ) , ( 4 4 ) where nonsingular 𝐶 ∈ ℳ 𝑟 𝑝 , and 𝜁 ( 0 − ) = 𝐶 − 1 𝐳 ( 0 − ) = 0 𝑡 𝑟 𝑝 .
Substituting (44) into (43), we obtain
𝐶 𝜁 ( 𝑡 ) = 𝐑 𝐶 𝜁 ( 𝑡 ) + 𝐋 𝑢 𝑜 ( 𝑡 ) ⟺ 𝜁 ( 𝑡 ) = 𝐶 − 1 𝐑 𝐶 𝜁 ( 𝑡 ) + 𝐶 − 1 𝐋 𝑢 𝑜 ( 𝑡 ) . ( 4 5 ) Furthermore, we define 𝐛 ≜ 𝐶 − 1 𝐋 , such as the last equation is transformed into 𝜁 ( 𝑡 ) = 𝐉 𝜁 ( 𝑡 ) + 𝐛 𝑢 𝑜 ( 𝑡 ) . ( 4 6 ) 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 𝐽 𝑜 ( 𝑡 − 𝜏 ) 𝑒 𝐽 𝑙 + 1 ( 𝑡 − 𝜏 ) 𝑒 𝐽 𝑙 + 2 ( 𝑡 − 𝜏 ) ⋯ 𝑒 𝐽 𝜅 ( 𝑡 − 𝜏 ) ∈ ℳ 𝑟 𝑝 , ( 4 7 ) where 𝑒 𝐽 𝑜 ( 𝑡 − 𝜏 ) = 𝑒 𝐽 1 ( 𝑡 − 𝜏 ) 𝑒 𝐽 2 ( 𝑡 − 𝜏 ) ⋯ 𝑒 𝐽 𝑙 ( 𝑡 − 𝜏 ) ∈ ℳ 𝑠 , 𝑒 𝐽 𝑖 ( 𝑡 − 𝜏 ) = 𝑒 𝜆 𝑖 ( 𝑡 − 𝜏 ) 𝑒 𝜆 𝑖 ( 𝑡 − 𝜏 ) ⋯ 𝑒 𝜆 𝑖 ( 𝑡 − 𝜏 ) ∈ ℳ 𝜏 𝑖 , ( 4 8 ) for 𝑖 = 1 , … , 𝑙 .
Furthermore,
𝑒 𝐽 𝑗 ( 𝑡 − 𝜏 ) = 𝑒 𝜆 𝑗 1 ( 𝑡 − 𝜏 ) 𝑒 𝜆 𝑗 2 ( 𝑡 − 𝜏 ) ⋯ 𝑒 𝜆 𝑗 𝑑 𝑗 ( 𝑡 − 𝜏 ) ∈ ℳ 𝜏 𝑗 , ( 4 9 ) where 𝑒 𝐽 𝑗 , 𝑧 𝑗 ( 𝑡 − 𝜏 ) = 𝑒 𝜆 𝑗 ( 𝑡 − 𝜏 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 𝑡 − 𝜏 ⋯ ( 𝑡 − 𝜏 ) 𝜇 𝑧 𝑗 − 1 𝜇 𝑧 𝑗 − 1 ! 0 1 ⋯ ( 𝑡 − 𝜏 ) 𝜇 𝑧 𝑗 − 2 𝜇 𝑧 𝑗 − 2 ! ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝜇 𝑧 𝑗 , ( 5 0 ) where 𝜇 𝑧 𝑗 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 = m a x 𝑧 𝑖 = 1 , 2 , … , 𝑑 𝑗 𝜇 𝑧 𝑗 is the index of annihilation for the eigenvalue 𝜆 𝑗 .
Now, taking into consideration (21) the solution (22) is transposed into
𝐳 ( 𝑡 ) = l i m 𝜎 → 0 𝐶 𝑒 𝐉 𝑡 𝑡 − ∞ 𝑒 − 𝐉 𝜏 𝐛 𝑢 ( 𝜏 ) 𝑑 𝜏 , ( 5 1 ) or equivalently 𝐳 ( 𝑡 ) = 𝐶 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ l i m 𝜎 → 0 𝑒 𝐽 𝑜 𝑡 𝑡 − ∞ 𝑒 − 𝐽 𝑜 𝜏 𝐛 𝑠 𝑢 ( 𝜏 ) 𝑑 𝜏 l i m 𝜎 → 0 𝑒 𝐽 𝑙 + 1 𝑡 𝑡 − ∞ 𝑒 − 𝐽 𝑙 + 1 𝜏 𝐛 𝜏 𝑙 + 1 𝑢 ( 𝜏 ) 𝑑 𝜏 ⋮ l i m 𝜎 → 0 𝑒 𝐽 𝑘 𝑡 𝑡 − ∞ 𝑒 − 𝐽 𝜅 𝜏 𝐛 𝜏 𝜅 𝑢 ( 𝜏 ) 𝑑 𝜏 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ( 5 2 ) where 𝐛 = [ 𝐛 𝑡 𝑠 𝐛 𝑡 𝜏 𝑙 + 1 ⋯ 𝐛 𝑡 𝜏 𝜅 ] 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, Φ 𝑡 𝜎 𝑡 = 𝐾 𝜎 = Φ ( 𝐾 ) ⟶ 0 , ( 5 3 ) and its derivatives Φ ( 𝑖 ) ( 𝑡 / 𝜎 ) 𝑡 = 𝐾 𝜎 = Φ ( 𝑖 ) ( 𝐾 ) 𝐾 → ∞ → 0 , for 𝑖 = 0 , 1 , 2 , … , 𝑘 − 1 .

Now, using the Remark 1, we obtain that l i m 𝜎 → 0 𝑒 𝐽 𝑖 𝑡 𝑡 = 𝐾 𝜎 = 𝐼 , for 𝑖 = 1 , 2 , … , 𝑙 , 𝑙 + 1 , 𝑙 + 2 , … , 𝑘 , and Φ − 1 ( 𝐾 + 𝜆 𝑖 𝜎 ) → Φ − 1 ( 𝐾 ) , as well. Moreover, for the analytic determination of vector (3), we need to calculate the integral ∫ 𝑡 − ∞ 𝑒 − 𝐽 𝜏 𝐛 𝑢 ( 𝜏 ) 𝑑 𝜏 .

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): l i m 𝜎 → 0 𝑡 − ∞ 𝑒 − 𝐽 0 𝑡 𝐛 𝑠 𝑢 ( 𝜏 ) 𝑑 𝜏 = Φ − 1 ( 𝐾 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝐛 𝜏 1 𝑛 − 1 𝑘 = 0 𝛼 𝑘 𝜆 𝑘 1 ⋮ 𝐛 𝜏 𝑖 𝑛 − 1 𝑘 = 0 𝛼 𝑘 𝜆 𝑘 𝑖 ⋮ 𝐛 𝜏 𝑙 𝑛 − 1 𝑘 = 0 𝛼 𝑘 𝜆 𝑘 𝑙 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝑠 1 . ( 5 4 )

Proof. See Appendix A.

Lemma 9. For the diagonal matrix 𝐽 𝑗 = 𝐽 𝑗 1 ⨁ 𝐽 𝑗 2 ⨁ ⋯ ⨁ 𝐽 𝑗 𝑑 𝑗 ∈ ℳ 𝜏 𝑗 , the following integral is given by (55): l i m 𝜎 → 0 𝑡 − ∞ 𝑒 − 𝐽 𝑗 𝑡 𝐛 𝜏 𝑗 𝑢 ( 𝜏 ) 𝑑 𝜏 = Φ − 1 ( 𝐾 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝐕 𝑗