Complete Controllability of Linear Fractional Differential Systems with Singularity

This paper is concerned with the controllability of a class of linear fractional differential systems with singularity.Themethodwhich is used to deal with the fast subsystem N⋅cDα 0,t x 2 (t) = x 2 (t) + B 2 u(t) and y 2 (t) = C 2 x 2 (t) is an improvement of the known ones. Based on the movement orbit of the state equation, we obtain several controllability criteria which are sufficient and necessary.


Introduction and Preliminaries
Singular systems are also commonly called descriptor systems, generalized state-space systems, differential-algebraic systems, or semistate systems whose behaviors are described by differential equations (or difference equations) and algebraic equations.In the past few decades, singular systems have attracted much attention for their extensive applications in robotics [1], power systems [2], networks, economic systems [3,4], highly interconnected large-scale systems [5], and so on.Many fundamental notions and conclusions based on regular systems have been extended to singular systems.For detail, see the monographs [6,7].
The concept of controllability plays an important role in the analysis and design of control systems.Recently, the controllability of fractional differential systems has been gaining much attention.For example, by applying Schauder's fixed point theorems, the authors of the paper [8] obtained a set of sufficient conditions for the controllability of nonlinear fractional differential systems.Using Sadovskii's fixed point theorem and Krasnoselskii's fixed point theorem, respectively, and properties of characteristic solution operators, the authors of the paper [9] established the complete controllability criteria for fractional evolution systems.Without involving the compactness of characteristic solution operators, the authors of the paper [10] considered the controllability of nonlinear dynamical systems with time varying multiple delays and distributed delays, respectively.In the paper [11], the controllability of fractional impulsive neutral integrodifferential systems was investigated.By using Krasnoselskii's fixed point theorem and the properties of resolvent operators, sufficient conditions for the controllability were established.By the representation of the state solution and construction of suitable control inputs, the authors established the controllability criteria for a class of linear neutral fractional timeinvariant differential systems.These criteria are sufficient and necessary [12].In 2013, the paper [13] was concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach space.With the help of two characteristic solution operators and their properties, the authors obtained the controllability criteria corresponding to two admissible control sets via the well-known Schauder's fixed point theorem.In 2014, the paper [14] dealt with the controllability of Sobolev type fractional evolutions and some controllability criteria were derived.
However, as far as we know, little attention has been paid to singular fractional differential systems.Motivated by this fact, this paper is devoted to the controllability for the singular fractional differential system where    0, is Caputo's derivative of order  with the lower limit 0, 0 <  < 1, ,  ∈ R × are  ×  constant matrices, the matrix couple (, ) is regular which will be defined later,  ∈ R × and  ∈ R × are the known constant matrices, rank  ≜  < ,  ∈ R  is the state variable,  ∈ R  is the control input, and  ∈ R  is the output.
Under the assumption that the matrix couple ( ) is regular, there exist two nonsingular matrices  and  satisfying where Then system (1) can be written as where , and  1 +  2 = .Subsystems (3)-( 4) and ( 5)-( 6) are called the slow subsystem and the fast subsystem, respectively.
Note that subsystem (3)-( 4) is a normal fractional differential linear system, whose controllability has been discussed in [15].
Hence, in order to investigate the controllability of system (1), it is sufficient to investigate the controllability of subsystem ( 5)-( 6), which is our main task in this paper.
Before giving our main results, we first recall some definitions and lemmas.For more details, please refer to [6,16,17].
Particularly, when 0 <  < 1, it holds that The Laplace transform of Caputo's fractional derivative where () is the Laplace transform of ().
Particularly, for 0 <  < 1, it holds that In addition, the Laplace transforms of the th derivative of () and the th derivative of the Dirac function () are Lemma 5 (see [17]).
Throughout this paper, "||" denotes the norm of the matrix "", "  " denotes the transpose of the matrix "", C denotes the complex plane, and "⇔" denotes equivalence.
Proof.An application of the Laplace transform on both sides of (5) yields where  2 () and () are the Laplace transform of  2 () and (), respectively.That is Since  ∈ R  2 × 2 is nilpotent whose index is ℎ and we get Now, we consider the following two cases.
Remark 7. Obviously, the output response  2 () is given by

Controllability
In this section, we will establish some controllability criteria for subsystem ( 5)-( 6) and system (1).We begin with the concept of the controllability of subsystem ( 5)-( 6).
The following two theorems and a corollary are our main results of this paper.