Reachability and Controllability of Fractional Singular Dynamical Systems with Control Delay

This paper is concerned with the reachability and controllability of fractional singular dynamical systems with control delay. The factors of such systems including the Caputo’s fractional derivative, control delay, and singular coefficient matrix are taken into account synchronously.The state structure of fractional singular dynamical systems with control delay is characterized by analysing the state response and reachable set. A set of sufficient and necessary conditions of controllability for such systems are established based on the algebraic approach. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed criteria.


Introduction
Singular systems play important roles in mathematical modeling of real-life problems arising in a wide range of applications.Depending on the area of application, these models are also called descriptor systems, semistate systems, differentialalgebraic systems, or generalized state-space systems.In the past few decades, singular systems with integer-order derivative have been extensively studied due to the fact that singular systems describe practical systems better than normal ones (see [1][2][3][4][5][6][7][8] and references therein).
Fractional calculus and its applications to the sciences and engineering is a recent focus of interest to many researchers.Fractional differential equations have been proved to be an excellent tool in the modelling of many phenomena in various fields of engineering, physics, and economics.Actually, fractional differential equations are considered as alternative model to integer differential equations.Many practical systems can be represented more accurately through fractional derivative formulation.For more details on fractional calculus theory, one can see the monographs of Miller and Ross [9], Podlubny [10], Diethelm [11], and Kilbas et al. [12].Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attention in [13][14][15][16][17][18] and references therein.
On the other hand, it is well-known that the issue of controllability plays an important role in engineering and control theory, which has close connections to pole assignment, structural decomposition, quadratic optimal control, and controller design, and so on.The problem of controllability for various kinds of dynamical systems with integer derivative has been extensively studied [1,[3][4][5][19][20][21][22].In particular, Dai [1] has established the theory of integer-order linear singular control systems.Yip and Sincovec [3] and Tang and Li [4] have investigated the controllability and observability of integer-order singular dynamical systems.It is worth mentioning that notable contributions have been made to fractional control systems [23][24][25][26][27][28][29][30][31][32][33][34][35].The different techniques have been applied to investigate the controllability of various fractional dynamical systems, such as algebraic method [23,24], geometrical analysis [25], fixed point theorems [26][27][28][29][30], and semigroup theory [31].Recently, Sakthivel et al. [32][33][34][35] have investigated the approximate controllability problem for various kinds of fractional dynamical systems by using fixed point techniques, which have further enriched and developed the control theory of fractional dynamical systems.The earlier studies concerning the controllability of fractional dynamical systems with control delay can be found in [25,27,29], which especially focused on fractional normal systems.However, it should be emphasized that the control theory of fractional singular systems is not yet sufficiently elaborated, compared to that of fractional normal systems.In this regard, it is necessary and important to study the controllability problems for fractional singular dynamical systems.To the best of our knowledge, there are no relevant reports on reachability and controllability of fractional singular dynamical systems as treated in the current literature.Motivated by this consideration, in this paper, we investigate the reachability and controllability of fractional singular dynamical systems with control delay.
In this paper, we consider the reachability and controllability of the following fractional singular dynamical systems with control delay: where    () denotes an  order Caputo fractional derivative of (), and 0 <  ≤ 1; , , , and  are the known constant matrices, ,  ∈ R × ,  ∈ R × ,  ∈ R × , and rank() < ;  ∈ R  is the state variable;  ∈ R  is the control input;  > 0 is the time control delay; and () is the initial control function.
The main purpose of this paper is to establish reachability and controllability criteria for system (1) based on the algebraic approach.The factors of fractional singular dynamical systems including the Caputo's fractional derivative, control delay, and singular coefficient matrices are taken into account synchronously.As discussed by Dai [1], we transform system (1) into two subsystems by the first equivalent form (FE1), which is more convenient to characterize the state reachable set.The state response, reachable set, and sufficient and necessary conditions for the controllability are obtained, respectively.The proposed criteria are applicable to a larger class of fractional dynamical systems.Therefore, we extend the known results [1,23,25] to a more general case.
This paper is organized as follows.In the next section, we recall some definitions and preliminary facts used in the paper.In Section 3, the state structure of system (1) is characterized by analysing the state response and reachable set.In Section 4, the necessary and sufficient conditions of controllability for two subsystems and system (1) are derived, respectively.In Section 5, an example is provided to illustrate the effectiveness and applicability of the proposed criteria.Finally, some concluding remarks are drawn in Section 6.

Preliminaries
In this section, we first recall some definitions of fractional calculus and preliminary facts.For more details, one can see [9][10][11][12].Next, the first equivalent form (FE1) of system (1) is given.
(a) The Riemann-Liouville's fractional integral of order  > 0 with the lower limit zero for a function  :  + →   is defined as provided that the right side is pointwise defined on [0, +∞), where Γ(⋅) is the Gamma function.
(b) The Caputo's fractional derivative of order  for a function  :  + →   is defined as For 0 <  ≤ 1, (c) The Mittag-Leffler function in two parameters is defined as where  > 0,  > 0, and  ∈ C, C denotes the complex plane.
In particular, for  = 1, has the interesting property (d) The Laplace transform of a function () is defined as where () is -dimensional vector-valued function.For  ∈ (0, 1], it follows from [10] that (e) For  ∈ N, the sequential fractional derivative for suitable function () is defined by where  = 1, . . ., , and D  is any fractional differential operator; here we still mention it as    .Now, we introduce the first equivalent form (FE1) of system (1) by the nonsingular transform, which is also called the standard decomposition of a singular system.
Assume that (, ) is regular throughout this paper.From [1,3], there exist two nonsingular matrices ,  ∈ R × , such that system (1) is equivalent to canonical system as follows: where and  is nilpotent whose nilpotent index is denoted by ]; that is,  ] = 0,  ]−1 ̸ = 0, and Applying the method in [3], we can obtain the precise form of the admissible initial state set () for system (11).
Thus, the set of admissible initial data is

State Response and State Reachability
In this section, we characterize the state structure of system (11) by analysing the state response and reachable set.
Theorem 1.For any admissible initial data ( 0 , ) ∈ A and the control function () ∈ C ℎ  ([0, +∞), R  ), the state response for system (11) can be represented in the following form: where  , (⋅) is the Mittag-Leffler function and    (⋅) is the sequential fractional derivative.
Proof.From [10], the Laplace transform of the  order Caputo fractional derivative of function () is Taking the Laplace transform with respect to  in both sides of the first equation of system (11), we obtain Hence, From [10], we have Then ( 19) is equivalent to The convolution theorem of the Laplace transform applied to (21) yields the form Applying the inverse Laplace transform, we obtain On the other hand, inserting ( 16) into both sides of the second equation in system (11) yields Then ( 16) is a solution of the second equation in system (11).
Next, we will show that any solution of the second equation in system (11) has the form of (16).Define linear operator  : Then the second equation in system (11) becomes Since  is nilpotent,  and  can exchange, and Thus, ( 16) is indeed a solution of the second equation in system (11).Therefore, the proof of Theorem 1 is completed.
For convenience sake, some notations are introduced as follows.
Given matrices  ∈ R × and ,  ∈ R × , denote Im() as the range of ; that is, Let Then the space ⟨ 1 |  1 ,  1 ⟩ is spanned by the columns of and the space ⟨| 2 ,  2 ⟩ is spanned by the columns of For any nonzero polynomial (), we define (, ) : where * denotes the matrix transpose.
The following lemma is required for the main result, which is a natural extension of Lemma 2-1.1 in [1].
Let R( 0 , ) be the reachable set from any admissible initial data ( 0 , ) ∈ A. Then Then Therefore, (60) is true.

Controllability Criteria
In this section, we proceed to investigate the controllability criteria of system (1) and system (11) based on the reachable set.A set of sufficient conditions and necessary conditions for the controllability are derived based on the algebraic approach.
Definition 5. System (1) (or (11)) is said to be controllable at   > 0 if one can reach any state at   from any admissible initial data ( 0 , ) ∈ A.
Theorem 6. Canonical system (11) is controllable if and only if Proof.We firstly prove the necessity of Theorem 6.If system (11) is controllable, for any  = [  1  2 ] ∈ R  , to the initial state  0 = 0 and initial control function  ≡ 0, there exists   > 0 and a control function (⋅) ∈ C ℎ  ([0, +∞) such that  1 ,  2 could be written in the form of ( 58) and (59).That is which implies that Obviously, Consequently, we have Thus, (66) and (67) are true.Next, we prove the sufficiency of Theorem 6.If (66) and (67) are true, then we know that (71) holds.For any  ∈ R  and any initial state  0 and initial control function (⋅), let (73) been investigated.The state structure of fractional singular dynamical systems with control delay has been characterized by analysing the state response and reachable set.A set of sufficient and necessary conditions of controllability criteria for such systems has been established based on the algebraic approach.An example is also presented to illustrate the effectiveness and applicability of the results obtained.Both the proposed criteria and the example show that the controllability property of fractional linear singular dynamical systems is dependent neither on the order of fractional derivative nor on control delay.Comparing with some existing results [1,23,25], we find that the algebraic approach has been extended to consider the controllability of more general fractional singular dynamical systems.Motivated and inspired by the works of Sakthivel et al. [32][33][34][35], the approximate controllability of linear (nonlinear) fractional singular dynamical systems will become our future investigative work.