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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 567089, 10 pages http://dx.doi.org/10.1155/2013/567089 Research Article Reachability and Controllability of Fractional Singular Dynamical Systems with Control Delay Hai Zhang, 1,2 Jinde Cao, 1,3 and Wei Jiang 4 1 Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China 2 Department of Mathematics, Anqing Normal University, Anqing, Anhui 246133, China 3 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia 4 School of Mathematical Sciences, Anhui University, Hefei, Anhui 230039, China Correspondence should be addressed to Jinde Cao; [email protected] Received 3 August 2013; Revised 25 September 2013; Accepted 25 September 2013 Academic Editor: Han H. Choi Copyright © 2013 Hai Zhang et al. is 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. is paper is concerned with the reachability and controllability of fractional singular dynamical systems with control delay. e factors of such systems including the Caputo’s fractional derivative, control delay, and singular coefficient matrix are taken into account synchronously. e 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. 1. Introduction Singular systems play important roles in mathematical mod- eling of real-life problems arising in a wide range of applica- tions. Depending on the area of application, these models are also called descriptor systems, semistate systems, differential- algebraic systems, or generalized state-space systems. In the past few decades, singular systems with integer-order deriva- tive have been extensively studied due to the fact that singular systems describe practical systems better than normal ones (see [18] 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 var- ious fields of engineering, physics, and economics. Actually, fractional differential equations are considered as alternative model to integer differential equations. Many practical sys- tems can be represented more accurately through fractional derivative formulation. For more details on fractional calcu- lus theory, one can see the monographs of Miller and Ross [9], Podlubny [10], Diethelm [11], and Kilbas et al. [12]. Frac- tional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attention in [1318] and references therein. On the other hand, it is well-known that the issue of con- trollability 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. e problem of controllability for various kinds of dynamical systems with integer derivative has been extensively studied [1, 35, 1922]. 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 [2335]. e 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 [2630], and semigroup theory [31]. Recently, Sakthivel et al. [3235] have investigated the approximate controllability problem for various kinds of fractional dynamical systems by using fixed point techniques, which have further enriched and

Research Article Reachability and Controllability of ...Reachability and Controllability of Fractional Singular Dynamical Systems with Control Delay HaiZhang, 1,2 JindeCao, 1,3 andWeiJiang

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  • Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 567089, 10 pageshttp://dx.doi.org/10.1155/2013/567089

    Research ArticleReachability and Controllability of Fractional SingularDynamical Systems with Control Delay

    Hai Zhang,1,2 Jinde Cao,1,3 and Wei Jiang4

    1 Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China2Department of Mathematics, Anqing Normal University, Anqing, Anhui 246133, China3Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia4 School of Mathematical Sciences, Anhui University, Hefei, Anhui 230039, China

    Correspondence should be addressed to Jinde Cao; [email protected]

    Received 3 August 2013; Revised 25 September 2013; Accepted 25 September 2013

    Academic Editor: Han H. Choi

    Copyright © 2013 Hai Zhang 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.

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

    1. Introduction

    Singular systems play important roles in mathematical mod-eling of real-life problems arising in a wide range of applica-tions. Depending on the area of application, these models arealso called descriptor systems, semistate systems, differential-algebraic systems, or generalized state-space systems. In thepast few decades, singular systems with integer-order deriva-tive have been extensively studied due to the fact that singularsystems describe practical systems better than normal ones(see [1–8] and references therein).

    Fractional calculus and its applications to the sciences andengineering is a recent focus of interest to many researchers.Fractional differential equations have been proved to be anexcellent tool in the modelling of many phenomena in var-ious fields of engineering, physics, and economics. Actually,fractional differential equations are considered as alternativemodel to integer differential equations. Many practical sys-tems can be represented more accurately through fractionalderivative formulation. For more details on fractional calcu-lus theory, one can see the monographs of Miller and Ross[9], Podlubny [10], Diethelm [11], and Kilbas et al. [12]. Frac-tional differential equations involving the Riemann-Liouville

    fractional derivative or the Caputo fractional derivative havebeen paid more and more attention in [13–18] and referencestherein.

    On the other hand, it is well-known that the issue of con-trollability plays an important role in engineering and controltheory, which has close connections to pole assignment,structural decomposition, quadratic optimal control, andcontroller design, and so on. The problem of controllabilityfor various kinds of dynamical systemswith integer derivativehas been extensively studied [1, 3–5, 19–22]. In particular,Dai [1] has established the theory of integer-order linearsingular control systems. Yip and Sincovec [3] and Tang andLi [4] have investigated the controllability and observabilityof integer-order singular dynamical systems. It is worthmentioning that notable contributions have been made tofractional control systems [23–35]. The different techniqueshave been applied to investigate the controllability of variousfractional dynamical systems, such as algebraic method [23,24], geometrical analysis [25], fixed point theorems [26–30],and semigroup theory [31]. Recently, Sakthivel et al. [32–35] have investigated the approximate controllability problemfor various kinds of fractional dynamical systems by usingfixed point techniques, which have further enriched and

  • 2 Journal of Applied Mathematics

    developed the control theory of fractional dynamical systems.The earlier studies concerning the controllability of fractionaldynamical 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 offractional 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 controllabilityproblems for fractional singular dynamical systems. To thebest of our knowledge, there are no relevant reports on reach-ability and controllability of fractional singular dynamicalsystems as treated in the current literature. Motivated by thisconsideration, in this paper, we investigate the reachabilityand controllability of fractional singular dynamical systemswith control delay.

    In this paper, we consider the reachability and controlla-bility of the following fractional singular dynamical systemswith control delay:

    𝐸𝑐

    𝐷𝛼

    𝑥 (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) + 𝐶𝑢 (𝑡 − 𝜏) , 𝑡 ≥ 0,

    𝑢 (𝑡) = 𝜓 (𝑡) , −𝜏 ≤ 𝑡 ≤ 0,

    𝑥 (0) = 𝑥0,

    (1)

    where 𝑐𝐷𝛼𝑥(𝑡) denotes an 𝛼 order Caputo fractional deriva-tive of 𝑥(𝑡), and 0 < 𝛼 ≤ 1; 𝐸, 𝐴, 𝐵, and 𝐶 are the knownconstant matrices, 𝐸,𝐴 ∈ R𝑛×𝑛, 𝐵 ∈ R𝑛×𝑚, 𝐶 ∈ R𝑛×𝑚, andrank(𝐸) < 𝑛;𝑥 ∈ R𝑛 is the state variable;𝑢 ∈ R𝑚 is the controlinput; 𝜏 > 0 is the time control delay; and 𝜓(𝑡) is the initialcontrol function.

    Themain purpose of this paper is to establish reachabilityand controllability criteria for system (1) based on the alge-braic approach. The factors of fractional singular dynamicalsystems including the Caputo’s fractional derivative, controldelay, and singular coefficientmatrices are taken into accountsynchronously. 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 reachableset. The state response, reachable set, and sufficient andnecessary conditions for the controllability are obtained,respectively. The proposed criteria are applicable to a largerclass of fractional dynamical systems. Therefore, we extendthe 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 inthe paper. In Section 3, the state structure of system (1) ischaracterized by analysing the state response and reachableset. In Section 4, the necessary and sufficient conditions ofcontrollability for two subsystems and system (1) are derived,respectively. In Section 5, an example is provided to illustratethe effectiveness and applicability of the proposed criteria.Finally, some concluding remarks are drawn in Section 6.

    2. Preliminaries

    In this section, we first recall some definitions of fractionalcalculus and preliminary facts. For more details, one can see[9–12]. Next, the first equivalent form (FE1) of system (1) isgiven.

    (a) The Riemann-Liouville’s fractional integral of order𝛼 > 0 with the lower limit zero for a function 𝑓 : 𝑅+ → 𝑅𝑛is defined as

    𝐷−𝛼

    𝑓 (𝑡) =1

    Γ (𝛼)∫

    𝑡

    0

    (𝑡 − 𝑠)𝛼−1

    𝑓 (𝑠) 𝑑𝑠, 𝑡 > 0, (2)

    provided that the right side is pointwise defined on [0, +∞),where Γ(⋅) is the Gamma function.

    (b) The Caputo’s fractional derivative of order 𝛼 for afunction 𝑓 : 𝑅+ → 𝑅𝑛 is defined as

    𝑐

    𝐷𝛼

    𝑓 (𝑡) =1

    Γ (𝑚 − 𝛼 + 1)∫

    𝑡

    0

    (𝑡 − 𝑠)𝑚−𝛼

    𝑓(𝑚+1)

    (𝑠) 𝑑𝑠,

    0 ≤ 𝑚 ≤ 𝛼 < 𝑚 + 1.

    (3)

    For 0 < 𝛼 ≤ 1,

    𝐷−𝛼

    {𝑐

    𝐷𝛼

    𝑓 (𝑡)} = 𝑓 (𝑡) − 𝑓 (0) . (4)

    (c) The Mittag-Leffler function in two parameters isdefined as

    𝐸𝛼,𝛽

    (𝑧) =

    +∞

    𝑘=0

    𝑧𝑘

    Γ (𝑘𝛼 + 𝛽), (5)

    where 𝛼 > 0, 𝛽 > 0, and 𝑧 ∈ C, C denotes the complex plane.In particular, for 𝛽 = 1,

    𝐸𝛼,1

    (𝑧) = 𝐸𝛼(𝑧) =

    +∞

    𝑘=0

    𝑧𝑘

    Γ (𝑘𝛼 + 1)(6)

    has the interesting property

    𝑐

    𝐷𝛼

    𝐸𝛼(𝜆𝑧𝛼

    ) = 𝜆𝐸𝛼(𝜆𝑧𝛼

    ) , 𝜆, 𝑧 ∈ C. (7)

    (d)The Laplace transform of a function 𝑓(𝑡) is defined as

    𝐹 (𝑠) = £ [𝑓 (𝑡)] = ∫+∞

    0

    𝑒−𝑠𝑡

    𝑓 (𝑡) 𝑑𝑡, 𝑠 ∈ C, (8)

    where 𝑓(𝑡) is 𝑛-dimensional vector-valued function. For 𝛼 ∈(0, 1], it follows from [10] that

    £ [(𝑐𝐷𝛼𝑓) (𝑡)] = 𝑠𝛼£ [𝑓 (𝑡)] − 𝑠𝛼−1𝑓 (0) . (9)

    (e) For 𝑛 ∈ N, the sequential fractional derivative forsuitable function 𝑦(𝑥) is defined by

    𝑦(𝑘)

    := (D𝑘𝛼𝑦) (𝑥) = (D𝛼D(𝑘−1)𝛼𝑦) (𝑥) , (10)

    where 𝑘 = 1, . . . , 𝑛, andD𝛼 is any fractional differential oper-ator; here we still mention it as 𝑐𝐷𝛼.

    Now, we introduce the first equivalent form (FE1) ofsystem (1) by the nonsingular transform, which is also calledthe standard decomposition of a singular system.

  • Journal of Applied Mathematics 3

    Assume that (𝐸, 𝐴) is regular throughout this paper. From[1, 3], there exist two nonsingular matrices 𝑃,𝑄 ∈ R𝑛×𝑛, suchthat system (1) is equivalent to canonical system as follows:

    𝑐

    𝐷𝛼

    𝑥1(𝑡) = 𝐴

    1𝑥1(𝑡) + 𝐵

    1𝑢 (𝑡) + 𝐶

    1𝑢 (𝑡 − 𝜏) , 𝑡 ≥ 0,

    𝑁𝑐

    𝐷𝛼

    𝑥2(𝑡) = 𝑥

    2(𝑡) + 𝐵

    2𝑢 (𝑡) + 𝐶

    2𝑢 (𝑡 − 𝜏) , 𝑡 ≥ 0,

    𝑢 (𝑡) = 𝜓 (𝑡) , −𝜏 ≤ 𝑡 ≤ 0,

    𝑥1(0) = 𝑥

    10, 𝑥

    2(0) = 𝑥

    20,

    (11)

    where 𝑥1∈ R𝑛1 , 𝑥

    2∈ R𝑛2 , 𝑛

    1+ 𝑛2= 𝑛, and 𝑁 is nilpotent

    whose nilpotent index is denoted by ]; that is, 𝑁] = 0,𝑁

    ]−1̸= 0, and

    𝑃𝐸𝑄 = [𝐼𝑛1

    0

    0 𝑁] , 𝑃𝐴𝑄 = [

    𝐴1

    0

    0 𝐼𝑛2

    ] ,

    𝑃𝐵 = [𝐵1

    𝐵2

    ] , 𝑃𝐶 = [𝐶1

    𝐶2

    ] , 𝑄−1

    𝑥 (𝑡) = [𝑥1(𝑡)

    𝑥2(𝑡)

    ] .

    (12)

    Let ℎ = [𝛼]] + 1, where [𝛼]] denotes the integral partof 𝛼]. Let Cℎ

    𝑝([0, +∞),R𝑚) be the set of ℎ times piecewise

    continuously differentiable functions mapping the interval[0, +∞) into R𝑚. Moreover, Cℎ([−𝜏, 0],R𝑚) denotes the setof ℎ times continuously differentiable functions mapping theinterval [−𝜏, 0] into R𝑚.

    Applying the method in [3], we can obtain the preciseform of the admissible initial state set 𝑆(𝜓) for system (11).For 𝜓 ∈ Cℎ([−𝜏, 0],R𝑚), we have

    𝑆 (𝜓)

    := {𝑥 ∈ R𝑛

    | 𝑥 = [𝑥1

    𝑥2

    ] , 𝑥1∈ R𝑛1 ,

    𝑥2= −

    ]−1

    𝑘=0

    [𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    𝜓 (0) + 𝑁𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    𝜓 (−𝜏)]} .

    (13)

    Thus, the set of admissible initial data is

    A := {(𝑥0, 𝜓) | 𝜓 ∈ Cℎ ([−𝜏, 0] ,R𝑚) , 𝑥

    0∈ 𝑆 (𝜓)} . (14)

    3. 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 (𝑥0, 𝜓) ∈ A and

    the control function 𝑢(𝑡) ∈ Cℎ𝑝([0, +∞),R𝑚), the state response

    for system (11) can be represented in the following form:𝑥1(𝑡) = 𝐸

    𝛼,1(𝐴1𝑡𝛼

    ) 𝑥1(0)

    + ∫

    𝑡

    𝑡−𝜏

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1𝑢 (𝑠) 𝑑𝑠

    + ∫

    𝑡−𝜏

    0

    {(𝑡− 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡− 𝑠)

    𝛼

    ] 𝐵1+ (𝑡− 𝜏 − 𝑠)

    𝛼−1

    × 𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1} 𝑢 (𝑠) 𝑑𝑠

    + ∫

    0

    −𝜏

    (𝑡 − 𝜏 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1𝜓 (𝑠) 𝑑𝑠,

    𝑡 ≥ 0,

    (15)

    𝑥2(𝑡) = −

    ]−1

    𝑘=0

    [𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡) + 𝑁𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡 − 𝜏)] , 𝑡 ≥ 0,

    (16)

    where𝐸𝛼,𝛽

    (⋅) is theMittag-Leffler function and 𝑐𝐷𝑘𝛼𝑢(⋅) is thesequential fractional derivative.

    Proof. From [10], the Laplace transformof the𝛼orderCaputofractional derivative of function 𝑥(𝑡) is

    £ [𝐷𝛼𝑥 (𝑡)] = 𝑠𝛼£ [𝑥 (𝑡)] − 𝑠𝛼−1𝑥 (0) . (17)Taking the Laplace transform with respect to 𝑡 in both sidesof the first equation of system (11), we obtain

    𝑠𝛼£ [𝑥1(𝑡)] − 𝑠

    𝛼−1

    𝑥1(0)

    = 𝐴1£ [𝑥1(𝑡)] + 𝐵

    1£ [𝑢 (𝑡)] + 𝐶

    1£ [𝑢 (𝑡 − 𝜏)] .

    (18)

    Hence,

    £ [𝑥1(𝑡)] = (𝑠

    𝛼

    𝐼 − 𝐴1)−1

    𝑠𝛼−1

    𝑥1(0)

    + (𝑠𝛼

    𝐼 − 𝐴1)−1£ [𝐵

    1𝑢 (𝑡) + 𝐶

    1𝑢 (𝑡 − 𝜏)] .

    (19)

    From [10], we have£ [𝑡𝛽−1𝐸

    𝛼,𝛽(𝐴1𝑡𝛼

    )] = 𝑠𝛼−𝛽

    (𝑠𝛼

    𝐼 − 𝐴1)−1

    . (20)

    Then (19) is equivalent to£ [𝑥1(𝑡)] = £ [𝐸

    𝛼,1(𝐴1𝑡𝛼

    )] 𝑥1(0)

    + £ [𝑡𝛼−1𝐸𝛼,𝛼

    (𝐴1𝑡𝛼

    )] £ [𝐵1𝑢 (𝑡) + 𝐶

    1𝑢 (𝑡 − 𝜏)] .

    (21)The convolution theorem of the Laplace transform applied to(21) yields the form

    £ [𝑥1(𝑡)] = £ [𝐸

    𝛼,1(𝐴1𝑡𝛼

    )] 𝑥1(0)

    + £{∫𝑡

    0

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ]

    × [𝐵1𝑢 (𝑠) + 𝐶

    1𝑢 (𝑠 − 𝜏)] 𝑑𝑠} .

    (22)

  • 4 Journal of Applied Mathematics

    Applying the inverse Laplace transform, we obtain

    𝑥1(𝑡) = 𝐸

    𝛼,1(𝐴1𝑡𝛼

    ) 𝑥1(0)

    + ∫

    𝑡

    0

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ]

    × [𝐵1𝑢 (𝑠) + 𝐶

    1𝑢 (𝑠 − 𝜏)] 𝑑𝑠

    = 𝐸𝛼,1

    (𝐴1𝑡𝛼

    ) 𝑥1(0)

    + ∫

    𝑡

    𝑡−𝜏

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1𝑢 (𝑠) 𝑑𝑠

    + ∫

    𝑡−𝜏

    0

    {(𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1+(𝑡 − 𝜏 − 𝑠)

    𝛼−1

    × 𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1} 𝑢 (𝑠) 𝑑𝑠

    + ∫

    0

    −𝜏

    (𝑡 − 𝜏 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1𝜓 (𝑠) 𝑑𝑠.

    (23)

    On the other hand, inserting (16) into both sides of the secondequation in system (11) yields

    𝑁𝑐

    𝐷𝛼

    𝑥2(𝑡) − 𝑥

    2(𝑡) − 𝐵

    2𝑢 (𝑡) − 𝐶

    2𝑢 (𝑡 − 𝜏)

    = −

    ]−1

    𝑘=0

    [𝑁𝑘+1

    𝐵2

    𝑐

    𝐷(𝑘+1)𝛼

    𝑢 (𝑡) + 𝑁𝑘+1

    𝐶2

    𝑐

    𝐷(𝑘+1)𝛼

    𝑢 (𝑡 − 𝜏)]

    − 𝐵2𝑢 (𝑡) − 𝐶

    2𝑢 (𝑡 − 𝜏)

    +

    ]−1

    𝑘=0

    [𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡) + 𝑁𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡 − 𝜏)]

    = 0.

    (24)

    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 linearoperator 𝑇 : C1([0, +∞),R𝑛2) → C([0, +∞),R𝑛2) by𝑇𝑥2(𝑡) =

    𝑐

    𝐷𝛼

    𝑥2(𝑡). Then the second equation in system (11)

    becomes

    (𝐼 − 𝑁𝑇) 𝑥2(𝑡) = −𝐵

    2𝑢 (𝑡) − 𝐶

    2𝑢 (𝑡 − 𝜏) . (25)

    Since 𝑁 is nilpotent, 𝑁 and 𝑇 can exchange, and (𝐼 −𝑁𝑇)(∑

    ]−1𝑘=0

    𝑁𝑘

    𝑇𝑘

    ) = 𝐼, then we have

    𝑥2(𝑡) = −(𝐼 − 𝑁𝑇)

    −1

    [𝐵2𝑢 (𝑡) + 𝐶

    2𝑢 (𝑡 − 𝜏)]

    = −(

    +∞

    𝑘=0

    𝑁𝑘

    𝑇𝑘

    ) [𝐵2𝑢 (𝑡) + 𝐶

    2𝑢 (𝑡 − 𝜏)]

    = −(

    ]−1

    𝑘=0

    𝑁𝑘

    𝑇𝑘

    ) [𝐵2𝑢 (𝑡) + 𝐶

    2𝑢 (𝑡 − 𝜏)]

    = −

    ]−1

    𝑘=0

    [𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡) + 𝑁𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡 − 𝜏)] .

    (26)

    Thus, (16) is indeed a solution of the second equation insystem (11).

    Therefore, the proof of Theorem 1 is completed.

    For convenience sake, some notations are introduced asfollows.

    Given matrices 𝐴 ∈ R𝑛×𝑛 and 𝐵, 𝐶 ∈ R𝑛×𝑚, denote Im(𝐵)as the range of 𝐵; that is,

    Im (𝐵) = {𝑦 | 𝑦 = 𝐵𝑥, ∀𝑥 ∈ R𝑚} . (27)

    Let

    ⟨𝐴 | 𝐵, 𝐶⟩ = Im (𝐵) + Im (𝐴𝐵) + ⋅ ⋅ ⋅ + Im (𝐴𝑛−1𝐵)

    + Im (𝐶) + Im (𝐴𝐶) + ⋅ ⋅ ⋅ + Im (𝐴𝑛−1𝐶) .(28)

    Then the space ⟨𝐴1| 𝐵1, 𝐶1⟩ is spanned by the columns of

    [𝐵1, 𝐴1𝐵1, . . . , 𝐴

    𝑛1−1

    1𝐵1, 𝐶1, 𝐴1𝐶1, . . . , 𝐴

    𝑛1−1

    1𝐶1] , (29)

    and the space ⟨𝑁|𝐵2, 𝐶2⟩ is spanned by the columns of

    [𝐵2, 𝑁𝐵2, . . . , 𝑁

    𝑛2−1

    𝐵2, 𝐶2, 𝑁𝐶2, . . . , 𝑁

    𝑛2−1

    𝐶2] . (30)

    For any nonzero polynomial 𝑓(𝑠), we define𝑊(𝑓, 𝑡) : R𝑛1 →R𝑛1 by

    𝑊(𝑓, 𝑡) 𝑧

    = ∫

    𝑡

    𝑡−𝜏

    𝑓 (𝑠) (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ]

    × 𝐵1𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴∗

    1(𝑡 − 𝑠)

    𝛼

    ] 𝑓 (𝑠) 𝑧 𝑑𝑠

    + ∫

    𝑡−𝜏

    0

    𝑓 (𝑠) {(𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1

    +(𝑡 − 𝜏 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1}

    × {𝐶∗

    1(𝑡 − 𝜏 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴∗

    1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ]

    +𝐵∗

    1(𝑡 −𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴∗

    1(𝑡− 𝑠)

    𝛼

    ] 𝑓 (𝑠)} 𝑧 𝑑𝑠,

    (31)

    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].

    Lemma 2. Given matrices 𝐴1∈ R𝑛1×𝑛1 and 𝐵

    1, 𝐶1∈ R𝑛1×𝑚1 ,

    then

    Im𝑊(𝑓, 𝑡) = ⟨𝐴1| 𝐵1, 𝐶1⟩ . (32)

    Proof. The formula (32) holds if and only if

    Ker𝑊(𝑓, 𝑡) =𝑛1−1

    𝑖=0

    Ker𝐵∗1(𝐴∗

    1)𝑖

    𝑛1−1

    𝑖=0

    Ker𝐶∗1(𝐴∗

    1)𝑖

    . (33)

  • Journal of Applied Mathematics 5

    If 𝑧0∈ Ker𝑊(𝑓, 𝑡) and 𝑧

    0̸= 0, 𝑧∗0𝐺(𝑡)𝑧0= 0, that is,

    𝑡

    𝑡−𝜏

    𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [(𝐴∗

    1(𝑡 − 𝑠)] 𝑓 (𝑠) 𝑧

    0

    2

    𝑑𝑠

    + ∫

    𝑡−𝜏

    0

    [𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝑠)) + 𝐶

    1(𝑡 − 𝜏 − 𝑠)

    𝛼−1

    ×𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝜏 − 𝑠)) ] 𝑓 (𝑠) 𝑧

    0

    2

    𝑑𝑠 = 0;

    (34)

    then we have

    𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴∗

    1(𝑡 − 𝑠)] 𝑓 (𝑠) 𝑧

    0= 0, 𝑠 ∈ [𝑡 − 𝜏, 𝑡] ,

    (35)

    [𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝑠)) + 𝐶

    1(𝑡 − 𝜏 − 𝑠)

    𝛼−1

    × 𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝜏 − 𝑠))] 𝑓 (𝑠) 𝑧

    0= 0, 𝑠 ∈ [0, 𝑡 − 𝜏] .

    (36)

    Since polynomial 𝑓(𝑠) only has a finite number of zero on𝑡 − 𝜏 ≤ 𝑠 ≤ 𝑡 and 0 ≤ 𝑠 ≤ 𝑡 − 𝜏, we immediately have

    𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴∗

    1(𝑡 − 𝑠)] 𝑧

    0= 0, 𝑠 ∈ [𝑡 − 𝜏, 𝑡] , (37)

    [𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝑠)) + 𝐶

    1(𝑡 − 𝜏 − 𝑠)

    𝛼−1

    × 𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝜏 − 𝑠))] 𝑧

    0= 0, 𝑠 ∈ [0, 𝑡 − 𝜏] .

    (38)

    Repeatedly taking the Caputo’s derivative on both sides of(37), we have

    𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴∗

    1(𝑡 − 𝑠)] (𝐴

    1)𝑖

    𝑧0= 0,

    𝑖 = 1, 2, . . . , 𝑛1− 1, 𝑠 ∈ [𝑡 − 𝜏, 𝑡] .

    (39)

    Substituting 𝑠 = 𝑡 in (39) yields

    𝐵∗

    (𝐴∗

    )𝑖

    𝑧0= 0, 𝑖 = 1, 2, . . . , 𝑛

    1− 1, (40)

    which implies that 𝑧0∈ Ker𝐵∗

    1(𝐴∗

    1)𝑖, 𝑖 = 1, 2, . . . , 𝑛

    1− 1.

    According to Cayley-Hamilton theorem [4], (𝑡 −𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] can be represented as

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] =

    +∞

    𝑘=0

    (𝑡 − 𝑠)𝑘𝛼+𝛼−1

    Γ (𝑘𝛼 + 𝛼)𝐴𝑘

    1

    =

    𝑛1−1

    𝑘=0

    𝛾𝑘(𝑡 − 𝑠) 𝐴

    𝑘

    1.

    (41)

    For 𝑠 ∈ [0, 𝑡 − 𝜏], it follows from (39) and (41) that

    𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝑠)) 𝑧

    0

    =

    𝑛1−1

    𝑘=0

    𝛾𝑘(𝑡 − 𝑠) 𝐵

    1𝐴∗

    1

    𝑘

    𝑧0= 0.

    (42)

    Inserting (42) into (38) yields

    𝐶∗

    1(𝑡 − 𝜏 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝜏 − 𝑠)) 𝑧

    0= 0. (43)

    Repeatedly taking the Caputo’s fractional derivative on bothsides of (43) and letting 𝑠 = 𝑡 − 𝜏, we have

    𝐶∗

    1𝐴∗

    1

    𝑘

    𝑧0= 0, 𝑘 = 1, 2, . . . , 𝑛

    1− 1, (44)

    which implies that 𝑧0∈ Ker𝐶∗

    1(𝐴∗

    1)𝑖, 𝑖 = 1, 2, . . . , 𝑛

    1− 1.

    Therefore, we have

    Ker𝑊(𝑓, 𝑡) ⊆𝑛1−1

    𝑖=0

    Ker𝐵∗1(𝐴∗

    1)𝑖

    𝑛1−1

    𝑖=0

    Ker𝐶∗1(𝐴∗

    1)𝑖

    . (45)

    Conversely, if 𝑧0∈ ⋂𝑛1−1

    𝑖=0Ker𝐵∗1(𝐴∗

    1)𝑖

    ⋂𝑛1−1

    𝑖=0Ker𝐶∗

    1(𝐴∗

    1)𝑖

    and 𝑧0

    ̸= 0, then 𝑧0∈ Ker𝐵∗

    1(𝐴∗

    1)𝑖, 𝑧0∈ Ker𝐶∗

    1(𝐴∗

    1)𝑖, 𝑖 =

    1, 2, . . . , 𝑛1− 1, and

    𝐵∗

    1(𝐴∗

    1)𝑖

    𝑧0= 𝐶∗

    1(𝐴∗

    1)𝑖

    𝑧0= 0, 𝑖 = 1, 2, . . . , 𝑛

    1− 1. (46)

    For 𝑠 ∈ [𝑡 − 𝜏, 𝑡], it follows from (41) that

    𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝑠)) 𝑧

    0

    =

    𝑛1−1

    𝑘=0

    𝛾𝑘(𝑡 − 𝑠) 𝐵

    1𝐴∗

    1

    𝑘

    𝑧0= 0.

    (47)

    For 𝑠 ∈ [0, 𝑡 − 𝜏], the same argument yields

    [𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝑠)) + 𝐶

    1(𝑡 − 𝜏 − 𝑠)

    𝛼−1

    ×𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝜏 − 𝑠)) ] 𝑧

    0

    =

    𝑛1−1

    𝑘=0

    𝛾𝑘(𝑡 − 𝑠) 𝐵

    1𝐴∗

    1

    𝑘

    𝑧0

    +

    𝑛1−1

    𝑘=0

    𝜆𝑘(𝑡 − 𝜏 − 𝑠) 𝐶

    1𝐴∗

    1

    𝑘

    𝑧0= 0.

    (48)

    Hence, 𝑧0∈ Ker𝑊(𝑓, 𝑡), which implies

    Ker𝑊(𝑓, 𝑡) ⊇𝑛1−1

    𝑖=0

    Ker𝐵∗1(𝐴∗

    1)𝑖

    𝑛1−1

    𝑖=0

    Ker𝐶∗1(𝐴∗

    1)𝑖

    . (49)

    From (45) and (49), we know that (32) is true. The proof istherefore completed.

    The reachable set for system (1) (or (11)) may be definedas follows.

    Definition 3. Any vector 𝜔 ∈ R𝑛 in 𝑛-dimensional vectorspace is said to reachable if there exists an admissibleinitial data (𝑥

    0, 𝜓) ∈ A, admissible control input 𝑢(𝑡) ∈

    Cℎ𝑝([0, +∞),R𝑚), and 𝑡

    𝑓> 0 such that the solution of system

    (1) (or (11)) satisfies 𝑥(𝑡𝑓, 𝑥0, 𝜓) = 𝜔.

    Let R(𝑥0, 𝜓) be the reachable set from any admissible

    initial data (𝑥0, 𝜓) ∈ A. Then

    R (𝑥0, 𝜓)

    = {𝜔 ∈ R𝑛

    | (𝑥0, 𝜓) ∈ A, 𝑢 (𝑡) ∈ Cℎ

    𝑝([0, +∞) ,R

    𝑚

    ) ,

    𝑡𝑓> 0, 𝑥 (𝑡

    𝑓, 𝑥0, 𝜓) = 𝜔} .

    (50)

  • 6 Journal of Applied Mathematics

    Considering the reachable set from the initial conditions𝑥0= 0 and 𝜓 ≡ 0, we have the following theorem.

    Theorem 4. For system (11), the reachable set R(0, 0) can berepresented as

    R (0, 0) = ⟨𝐴1| 𝐵1, 𝐶1⟩ ⊕ ⟨𝑁 | 𝐵

    2, 𝐶2⟩ , (51)

    where “⊕” is the direct sum in vector space.

    Proof. Let 𝜔 ∈ R(0, 0), from (15), (16), and (50); then thereexists 𝑡

    𝑓> 0 and 𝑢(𝑡) ∈ Cℎ

    𝑝([0, +∞),R𝑚) such that

    𝜔1= ∫

    𝑡𝑓

    0

    (𝑡𝑓− 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡𝑓− 𝑠)𝛼

    ] 𝐵1𝑢 (𝑠) 𝑑𝑠

    + ∫

    𝑡𝑓−𝜏

    0

    (𝑡𝑓− 𝜏 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡𝑓− 𝜏 − 𝑠)

    𝛼

    ]

    × 𝐶1𝑢 (𝑠) 𝑑𝑠.

    (52)

    Combining (41) and (52) yields

    𝜔1=

    𝑛1−1

    𝑘=0

    𝑡𝑓

    0

    𝜑𝑘(𝑡𝑓− 𝑠)𝐴

    𝑘

    1𝐵1𝑢 (𝑠) 𝑑𝑠

    +

    𝑛1−1

    𝑘=0

    𝑡𝑓−𝜏

    0

    𝜙𝑘(𝑡𝑓− 𝜏 − 𝑠)𝐴

    𝑘

    1𝐶1𝑢 (𝑠) 𝑑𝑠.

    (53)

    Therefore, 𝜔1∈ ⟨𝐴1| 𝐵1, 𝐶1⟩. Moreover, we also have

    𝜔2= −

    ]−1

    𝑘=0

    [𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡𝑓) + 𝑁

    𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡𝑓− 𝜏)] ,

    (54)

    which implies that 𝜔2∈ ⟨𝑁 | 𝐵

    2, 𝐶2⟩. Thus,

    R (0, 0) ⊆ ⟨𝐴1| 𝐵1, 𝐶1⟩ ⊕ ⟨𝑁 | 𝐵

    2, 𝐶2⟩ . (55)

    On the other hand, we need to prove

    R (0, 0) ⊇ ⟨𝐴1| 𝐵1, 𝐶1⟩ ⊕ ⟨𝑁 | 𝐵

    2, 𝐶2⟩ . (56)

    Suppose that

    𝜔 = [𝜔1

    𝜔2

    ] ∈ ⟨𝐴1| 𝐵1, 𝐶1⟩ ⊕ ⟨𝑁 | 𝐵

    2, 𝐶2⟩ , (57)

    where 𝜔1∈ ⟨𝐴1| 𝐵1, 𝐶1⟩, 𝜔2∈ ⟨𝑁 | 𝐵

    2, 𝐶2⟩, and 𝜔

    1̸= 0,

    𝜔2

    ̸= 0. For the initial conditions 𝑥0= 0 and 𝜓(𝑡) ≡ 0, 𝑡 ∈

    [−𝜏, 0], let 𝑢(𝑠) = 𝑢1(𝑠) + 𝑢

    2(𝑠); then we have

    𝑥1(𝑡) = ∫

    𝑡

    𝑡−𝜏

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1[𝑢1(𝑠) + 𝑢

    2(𝑠)] 𝑑𝑠

    + ∫

    𝑡−𝜏

    0

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ]

    × 𝐵1[𝑢1(𝑠) + 𝑢

    2(𝑠)] 𝑑𝑠

    + ∫

    𝑡−𝜏

    0

    (𝑡 − 𝜏 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ]

    × 𝐶1[𝑢1(𝑠) + 𝑢

    2(𝑠)] 𝑑𝑠,

    (58)

    𝑥2(𝑡) = −

    ]−1

    𝑘=0

    [𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    𝑢1(𝑡) + 𝑁

    𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    𝑢1(𝑡 − 𝜏)]

    ]−1

    𝑘=0

    [𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    𝑢2(𝑡) + 𝑁

    𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    𝑢2(𝑡 − 𝜏)] .

    (59)

    As discussed by Yip and Sincovec [3], we choose 𝑢1(𝑠) =

    𝑓(𝑠)𝑦(𝑠) to satisfy

    𝑡−𝜏

    0

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1𝑓 (𝑠) 𝑔 (𝑠) 𝑑𝑠

    + ∫

    𝑡

    𝜏

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐶1𝑓 (𝑠 − 𝜏) 𝑔 (𝑠 − 𝜏) 𝑑𝑠

    = 𝜔1− ∫

    𝑡−𝜏

    0

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1𝑢2(𝑠) 𝑑𝑠

    − ∫

    𝑡−𝜏

    0

    (𝑡 − 𝜏 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ]

    × 𝐶1𝑢2(𝑠) 𝑑𝑠 ≡ �̂�

    1,

    𝑐

    𝐷𝑘𝛼

    𝑢1(𝑡) =

    𝑐

    𝐷𝑘𝛼

    𝑢1(𝑡 − 𝜏) = 0, 𝑘 = 0, 1, 2, . . . , 𝜐 − 1.

    (60)

    Then 𝑢1(𝑠) contributes nothing to 𝑥

    2(𝑡) at 𝑠 = 𝑡 and 𝑠 = 𝑡 − 𝜏.

    Now, we prove that (60) is true.In fact, for �̂�

    1∈ ⟨𝐴1| 𝐵1, 𝐶1⟩, fromLemma 2, there exists

    𝑧 ∈ R𝑛 such that𝑊(𝑓, 𝑡)𝑧 = �̂�1.

    Let

    𝑦 (𝑠) =

    {{{{{{{{{

    {{{{{{{{{

    {

    𝑓 (𝑠) [𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝑠))

    +𝐶∗

    1(𝑡 − 𝜏 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝜏 − 𝑠))] 𝑧,

    0 ≤ 𝑠 ≤ 𝑡 − 𝜏,

    𝑓 (𝑠) 𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    (𝐴∗

    1(𝑡 − 𝑠)) 𝑧,

    𝑡 − 𝜏 ≤ 𝑠 ≤ 𝑡, −𝜏 ≤ 𝑠 ≤ 0.

    (61)

  • Journal of Applied Mathematics 7

    Then

    𝑡−𝜏

    0

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1𝑢1(𝑠) 𝑑𝑠

    + ∫

    𝑡

    𝜏

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐶1𝑢1(𝑠 − 𝜏) 𝑑𝑠

    = ∫

    𝑡

    𝑡−𝜏

    𝑓 (𝑠) (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ]

    × 𝐵1𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴∗

    1(𝑡 − 𝑠)

    𝛼

    ] 𝑓 (𝑠) 𝑧 𝑑𝑠

    + ∫

    𝑡−𝜏

    0

    𝑓 (𝑠) {(𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1

    +(𝑡 − 𝜏 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1}

    × {𝐶∗

    1(𝑡 − 𝜏 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴∗

    1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ]

    +𝐵∗

    1(𝑡 − 𝑠)

    𝛼−1

    𝐸𝛼,𝛼

    [𝐴∗

    1(𝑡 − 𝑠)

    𝛼

    ] 𝑓 (𝑠)} 𝑧 𝑑𝑠

    = 𝑊(𝑓, 𝑡) 𝑧 = �̂�1.

    (62)

    Therefore, (60) is true.For 𝜔2∈ ⟨𝑁 | 𝐵

    2, 𝐶2⟩, there exist 𝑦

    𝑘, 𝑧𝑘such that

    𝜔2= −

    ]−1

    𝑘=0

    𝑁𝑘

    𝐵2𝑦𝑘−

    ]−1

    𝑘=0

    𝑁𝑘

    𝐶2𝑧𝑘. (63)

    There exists a function 𝑝(𝑠) such that 𝑐𝐷𝑘𝛼𝑝(0) = 0,𝑐

    𝐷𝑘𝛼

    𝑝(𝑡) = 𝑦𝑘, and 𝑐𝐷𝑘𝛼𝑝(𝑡 − 𝜏) = 𝑧

    𝑘. Let

    𝑢2(𝑠) = {

    𝑝 (𝑠) , 0 ≤ 𝑠 ≤ 𝑡,

    0, −𝜏 ≤ 𝑠 ≤ 0;(64)

    then we have

    𝑥2(𝑡) = −

    ]−1

    𝑘=0

    𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    𝑢2(𝑡)

    ]−1

    𝑘=0

    𝑁𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    𝑢2(𝑡 − 𝜏) = 𝜔

    2,

    (65)

    and 𝑥2(0) = 0. It follows from (60) and (65) that 𝜔 ∈ R(0, 0).

    Thus, (56) holds. Combining (55) and (56), we know that (51)is true. Then the proof of Theorem 4 is completed.

    4. Controllability Criteria

    In this section, we proceed to investigate the controllabilitycriteria of system (1) and system (11) based on the reachableset. A set of sufficient conditions and necessary conditionsfor the controllability are derived based on the algebraicapproach.

    Definition 5. System (1) (or (11)) is said to be controllable at𝑡𝑓> 0 if one can reach any state at 𝑡

    𝑓from any admissible

    initial data (𝑥0, 𝜓) ∈ A.

    Theorem 6. Canonical system (11) is controllable if and only if

    rank [𝐵1, 𝐴1𝐵1, . . . , 𝐴

    𝑛1−1

    1𝐵1, 𝐶1, 𝐴1𝐶1, . . . , 𝐴

    𝑛1−1

    1𝐶1] = 𝑛1,

    (66)

    rank [𝐵2, 𝑁𝐵2, . . . , 𝑁

    𝑛2−1

    𝐵2, 𝐶2, 𝑁𝐶2, . . . , 𝑁

    𝑛2−1

    𝐶2] = 𝑛2.

    (67)

    Proof. We firstly prove the necessity of Theorem 6. If system(11) is controllable, for any 𝜔 = [ 𝜔1𝜔

    2] ∈ R𝑛, to the initial state

    𝑥0= 0 and initial control function 𝜓 ≡ 0, there exists 𝑡

    𝑓> 0

    and a control function 𝑢(⋅) ∈ Cℎ𝑝([0, +∞) such that 𝜔

    1, 𝜔2

    could be written in the form of (58) and (59). That is𝜔 ∈ ⟨𝐴

    1| 𝐵1, 𝐶1⟩ ⊕ ⟨𝑁 | 𝐵

    2, 𝐶2⟩ , (68)

    which implies thatR𝑛

    ⊆ ⟨𝐴1| 𝐵1, 𝐶1⟩ ⊕ ⟨𝑁 | 𝐵

    2, 𝐶2⟩ . (69)

    Obviously,R𝑛

    ⊇ ⟨𝐴1| 𝐵1, 𝐶1⟩ ⊕ ⟨𝑁 | 𝐵

    2, 𝐶2⟩ . (70)

    Consequently, we have

    R𝑛

    = ⟨𝐴1| 𝐵1, 𝐶1⟩ ⊕ ⟨𝑁 | 𝐵

    2, 𝐶2⟩ ,

    ⟨𝐴1| 𝐵1, 𝐶1⟩ = R

    𝑛1 , ⟨𝑁 | 𝐵

    2, 𝐶2⟩ = R

    𝑛2 .

    (71)

    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 𝜔 ∈ R𝑛and any initial state 𝑥

    0and initial control function 𝜓(⋅), let

    𝑘1= 𝜔1− 𝐸𝛼,1

    (𝐴1𝑡𝛼

    ) 𝑥1(0)

    − ∫

    𝑡

    𝑡−𝜏

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1𝜓 (0) 𝑑𝑠

    − ∫

    𝑡−𝜏

    0

    {(𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1+ (𝑡 − 𝜏 − 𝑠)

    𝛼−1

    × 𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1} 𝜓 (0) 𝑑𝑠

    − ∫

    0

    −𝜏

    (𝑡 − 𝜏 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1𝜓 (𝑠) 𝑑𝑠,

    𝑘2= 𝜔2+ 𝐵2𝜓 (0) + 𝐶

    2𝜓 (0) .

    (72)

    For 𝑘 = [ 𝑘1𝑘2

    ] ∈ R𝑛 = ⟨𝐴1| 𝐵1, 𝐶1⟩ ⊕ ⟨𝑁 | 𝐵

    2, 𝐶2⟩, we have

    𝑘 ∈ R(0, 0). Then there exists a control �̃�(𝑠) such that

    𝑘1= ∫

    𝑡

    𝑡−𝜏

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1�̃� (𝑠) 𝑑𝑠

    + ∫

    𝑡−𝜏

    0

    [(𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1+ (𝑡 − 𝜏 − 𝑠)

    𝛼−1

    × 𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1] �̃� (𝑠) 𝑑𝑠,

    𝑘2= −

    ]−1

    𝑘=0

    [𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    �̃� (𝑡) + 𝑁𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    �̃� (𝑡 − 𝜏)] .

    (73)

  • 8 Journal of Applied Mathematics

    Let 𝑢(𝑠) = �̃�(𝑠) + 𝜓(0), then we have

    𝜔1= 𝐸𝛼,1

    (𝐴1𝑡𝛼

    ) 𝑥1(0)

    + ∫

    𝑡

    𝑡−𝜏

    (𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1𝑢 (𝑠) 𝑑𝑠

    + ∫

    𝑡−𝜏

    0

    {(𝑡 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝑠)

    𝛼

    ] 𝐵1

    +(𝑡− 𝜏 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡− 𝜏 −𝑠)

    𝛼

    ] 𝐶1} 𝑢 (𝑠) 𝑑𝑠

    + ∫

    0

    −𝜏

    (𝑡 − 𝜏 − 𝑠)𝛼−1

    𝐸𝛼,𝛼

    [𝐴1(𝑡 − 𝜏 − 𝑠)

    𝛼

    ] 𝐶1𝜓 (𝑠) 𝑑𝑠,

    (74)

    𝜔2= −

    ]−1

    𝑘=0

    [𝑁𝑘

    𝐵2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡) + 𝑁𝑘

    𝐶2

    𝑐

    𝐷𝑘𝛼

    𝑢 (𝑡 − 𝜏)] . (75)

    Thus, by Definition 5, system (11) is controllable. Therefore,the proof is completed.

    Applying the results of Yip and Sincovec [3], Tang and Li[4], andTheorem 6 to the pair of matrices, we can obtain thefollowing results.

    Theorem 7. Canonical system (11) is controllable if and only if

    rank [𝑠𝐼 − 𝐴1, 𝐵1, 𝐶1] = 𝑛1, ∀𝑠 ∈ C, 𝑠 finite,

    rank [𝑁, 𝐵2, 𝐶2] = 𝑛2.

    (76)

    Theorem 8. System (1) is controllable if and only if

    rank [𝑠𝐸 − 𝐴, 𝐵, 𝐶] = 𝑛, ∀𝑠 ∈ C, 𝑠 finite,

    rank [𝐸, 𝐵, 𝐶] = 𝑛.(77)

    Remark 9. If we choose 𝛼 = 1 and 𝐶 = 0, then system (1)reduces to an integer-order singular system without controldelay. FromTheorem 8, system (1) is controllable if and onlyif

    rank [𝑠𝐸 − 𝐴, 𝐵] = 𝑛, ∀𝑠 ∈ C, 𝑠 finite,

    rank [𝐸, 𝐵] = 𝑛,(78)

    which is just the result in [1].

    If we choose 𝐸 = 𝐼, 𝐶 = 0, then system (1) reducesto a fractional normal system without control delay. FromTheorem 8, system (1) is controllable if and only if

    rank [𝐵, 𝐴𝐵, . . . , 𝐴𝑛−1𝐵] = 𝑛, (79)

    which is just Corollary 3.4 in [23].If we choose 𝐸 = 𝐼, then system (1) reduces to a fractional

    normal system with control delay. From Theorem 8, system(1) is controllable if and only if

    rank [𝐵, 𝐴𝐵, . . . , 𝐴𝑛−1𝐵, 𝐶, 𝐴𝐶, . . . , 𝐴𝑛−1𝐶] = 𝑛, (80)

    which is justTheorem 3 in [25].Therefore, our results extendthe existing results [1, 23, 25].

    Remark 10. Theorem 8 actually offers an algebraic criterion ofthe exact controllability for linear fractional singular dynam-ical systems, which is concise and convenient to check thecontrollability of such systems. Motivated and inspired bythe works of Sakthivel et al. [32–35], the approximate con-trollability of linear (nonlinear) fractional singular dynamicalsystems with the control delay will become our future inves-tigative work.

    5. An Illustrative Example

    In this section, we give an example to illustrate the applica-tions of Theorem 8.

    Example 11. Consider the controllability of fractional singu-lar dynamical systems as follows:

    [

    [

    1 0 0

    0 0 1

    0 0 0

    ]

    ]

    𝐷1/2

    𝑥 (𝑡) = [

    [

    1 0 0

    0 1 1

    1 0 1

    ]

    ]

    𝑥 (𝑡) + [

    [

    1 0

    1 1

    0 1

    ]

    ]

    𝑢 (𝑡)

    + [

    [

    1 0

    0 1

    1 1

    ]

    ]

    𝑢(𝑡 −𝜋

    4) , 𝑡 ∈ [0, +∞) ,

    𝑢 (𝑡) = 𝜓 (𝑡) , 𝑡 ∈ [−𝜋

    4, 0] ,

    𝑥 (0) = 𝑥0,

    (81)

    where the admissible initial data (𝑥0, 𝜓) ∈ A. Now, we apply

    Theorem 8 to prove that system (81) is controllable. Let ustake

    𝛼 =1

    2, 𝐸 = [

    [

    1 0 0

    0 0 1

    0 0 0

    ]

    ]

    , 𝐴 = [

    [

    1 0 0

    0 1 1

    1 0 1

    ]

    ]

    ,

    𝐵 = [

    [

    1 0

    1 1

    0 1

    ]

    ]

    , 𝐶 = [

    [

    1 0

    0 1

    1 1

    ]

    ]

    .

    (82)

    If we choose 𝜆 = 1 such that det(𝜆𝐸 + 𝐴) = 2 ̸= 0, then (𝐸, 𝐴)is regular. By using the elementary transformation of matrix,one can obtain

    rank [𝑠𝐸 − 𝐴, 𝐵, 𝐶] = rank[

    [

    1 0 1 ⋆ ⋆ ⋆ ⋆

    0 1 −1 ⋆ ⋆ ⋆ ⋆

    0 0 2 ⋆ ⋆ ⋆ ⋆

    ]

    ]

    = 3,

    rank [𝐸, 𝐵, 𝐶] = rank[

    [

    1 0 1 ⋆ ⋆ ⋆ ⋆

    0 1 0 ⋆ ⋆ ⋆ ⋆

    0 0 1 ⋆ ⋆ ⋆ ⋆

    ]

    ]

    = 3.

    (83)

    Thus, byTheorem 8, system (81) is controllable.

    6. Conclusions

    In this paper, the reachability and controllability of frac-tional singular dynamical systems with control delay have

  • Journal of Applied Mathematics 9

    been investigated. The state structure of fractional singulardynamical systems with control delay has been characterizedby analysing the state response and reachable set. A set ofsufficient and necessary conditions of controllability criteriafor such systems has been established based on the algebraicapproach. An example is also presented to illustrate the effec-tiveness and applicability of the results obtained. Both theproposed criteria and the example show that the controllabil-ity property of fractional linear singular dynamical systemsis dependent neither on the order of fractional derivative noron control delay. Comparing with some existing results [1, 23,25], we find that the algebraic approach has been extended toconsider the controllability of more general fractional singu-lar dynamical systems. Motivated and inspired by the worksof Sakthivel et al. [32–35], the approximate controllability oflinear (nonlinear) fractional singular dynamical systems willbecome our future investigative work.

    Conflicts of Interests

    The authors declare that there is no conflict of interestsregarding the publication of this paper.

    Acknowledgments

    The authors are very grateful to the Associate Editor, Profes-sorHanH.Choi, and the anonymous reviewers for their help-ful and valuable comments and suggestions which improvedthe quality of the paper. This work is jointly supported by theNational Natural Science Foundation of China under Grantnos. 11071001, 11072059, and 61272530, the Natural ScienceFoundation of Jiangsu Province of China under Grant no.BK2012741, and the Programs of Educational Commissionof Anhui Province of China under Grant nos. KJ2011A197and KJ2013Z186. This work is also funded by the Deanshipof Scientific Research, King Abdulaziz University (KAU),under Grant no. 3-130/1434/HiCi. Therefore, the authorsacknowledge technical and financial support of KAU.

    References

    [1] L. Dai, Singular Control Systems, vol. 118 of Lecture Notes in Con-trol and Information Sciences, Springer, Berlin, Germany, 1989.

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