Controllability of nonlinear fractional delay dynamical systems with prescribed controls

https://doi.org/10.15388/NA.2018.1.1 Nonlinear Analysis: Modelling and Control, Vol. 23, No. 1, 1–18 ISSN 1392-5113 Controllability of nonlinear fractional delay dynamical systems with prescribed controls Xiao-Li Dinga,1,2, Juan J. Nietob,3 aDepartment of Mathematics, Xi’an Polytechnic University, Xi’an, Shaanxi 710048, China [email protected] bDepartamento de Análise Matemática, Estatística e Optimización, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain Received: December 13, 2016 / Revised: July 7, 2017 / Published online: December 14, 2017 Abstract. In this paper, we consider controllability of nonlinear fractional delay dynamical systems with prescribed controls. We firstly give the solution representation of the fractional delay dynamical systems using Laplace transform and Mittag–Leffler functions. Then we give necessary and sufficient conditions for the controllability criteria of linear fractional delay dynamical systems with prescribed controls. Further, we use a fixed point theorem to establish the sufficient condition for the controllability of nonlinear fractional delay dynamical systems with prescribed controls. In particular, we determine several sufficient conditions on the nonlinear function term so that if the linear system is controllable, then the nonlinear system is controllable. Finally, we give two examples to demonstrate the applicability of our obtained results. Keywords: fractional delay dynamical systems, prescribed controls, controllability, Mittag–Leffler function, fixed point theorem. 1 Introduction Fractional calculus is a generalization of integer order calculus. Unlike the integer order calculus, the fractional calculus is defined by nonlocal operators. Due to this fact, the fractional calculus has proved to be useful tools in the investigation of many phenomena with memory in engineering [32], physics [28], medicine [3], electrical circuits [1], electrodynamics of complex medium, and other fields; see, for instance, [12, 21]. In recent 1Corresponding author. 2The author was supported by the Natural Science Foundation of China (NSFC) under grant 11501436 and Young Talent fund of University Association for Science and Technology in Shaanxi, China, under grant 20170701. 3The author was partially supported by the Ministerio de Economa y Competitividad of Spain under grant MTM2013-43014-P and XUNTA de Galicia under grant GRC2015-004. c Vilnius University, 2018 2X.-L. Ding, J.J. Nieto years, there has been a growing interest in investigating fractional mathematical models to improve the quality of modeling towards real world applications. On the other hand, some authors have generalized integer order controllers to noninteger order controllers. As earlier as 1961, Manabe [27] has been devoted to fractional order systems in the area of automatic control. In 2009, Chen et al. [11] gave a clear discussion on fractional calculus as well as several known fractional order controllers and the discretisation techniques. After that, some authors began to discuss the applications of fractional calculus in control. In 2016, Ammar Soukkou et al. wrote a paper about review, design, optimization, and stability analysis of fractional-order PID controller [35]. Some pioneering works on fractional calculus in dynamic systems and controls are found in the literature (for example, see [7,16,18, 22, 37–39, 42]). As one of the important topics in mathematical control theory, controllability plays an important role in the analysis and design of control systems. Controllability for various kinds of fractional differential equations has been extensively studied by many researchers [5, 6, 10, 14, 15, 17, 20, 30, 40, 41]. The above most works resort in using tools for unconstrained dynamical systems, patched with a collection of heuristic rules. However, in real world problems, many control systems have an associated set of constraints. So handling constraints in control system design is an important issue. One possible strategy for dealing with constraints is to modify the design so that limits are never violated. In the literature, the investigations of constrained controllability for both linear and nonlinear systems are numerous [2,24,25]. It is worth mentioning that Krishnan and Jayskumar [25] discussed the controllability of fractional dynamical systems with prescribed controls using Schauder’s fixed point theorem. Fractional delay dynamical systems are an important kind of fractional order systems in real life. In these years, some authors pay attention to the study about the fractional delay dynamical systems (for example, see [4,9,26, 34,36]). Motivated by this literature, we propose to study the controllability for the linear fractional delay dynamical systems of the form Dα 0+x(t) = Ax(t) + Bx(t−h) + Cu(t),0< α ⩽1, t ∈[0, T ], x(t) = φ(t), t ∈[−h, 0], x(T) = xT, u(0) = u0, u(T) = uT (1) and the nonlinear fractional delay dynamical system Dα 0+x(t) = Ax(t) + Bx(t−h) + Cu(t) + ft, x(t), x(t−h), u(t), 0< α ⩽1, t ∈[0, T ], x(t) = φ(t), t ∈[−h, 0], x(T) = xT, u(0) = u0, u(T) = uT, (2) where Dα 0+is the Caputo fractional derivative (the definition of the Caputo fractional derivative will be given in Section 2), A,B ∈Rn×n,C∈Rn×m, the nonlinear function f https://www.mii.vu.lt/NA Controllability of nonlinear fractional delay dynamical systems 3 is continuous on Rn,x(t)∈Rn, and u(t)∈Rmare state vector and control input of the systems. The above systems have initial and final conditions such that x(0) = x0, x(T) = xTand u(0) = u0,u(T) = uT. We need to find some conditions on A,B, C, and f, which ensure that, for any given x0, xT∈Rn, there exists a control u∈Rm with u(0) = u0,u(T) = uT, which produces a response x(t;u)satisfying the boundary conditions x(0; u) = x0and x(T;u) = xT. Here we will restrict ourselves to considering the case of controllability using continuous control functions. Hence, we will assume that the above two systems are continuous. This assumption simplifies our arguments somewhat. This article is organized as follows. In Section 2, we briefly review some basic definitions and properties, which will be used in this paper. In Section 3, we give the solution representation of fractional delay dynamical systems using Laplace transforms and Mittag–Leffler functions. In Section 4, we give necessary and sufficient conditions for the controllability criteria of linear fractional delay dynamical systems with prescribed controls. In Section 5, we use the fixed point theorem to establish the sufficient condition for the controllability of nonlinear fractional delay dynamical systems with prescribed controls. In Section 6, the applications of the presented theory are demonstrated with two examples. 2 Preliminaries In this section, we give some basic definitions and results that are used throughout this paper. For more details, please see [23,31]. Definition 1. Let [a, b]be a finite interval on the real axis R. The fractional integral of order α > 0with the lower limit afor the function xis defined as Iα a+x(t) = 1 Γ(α) t Za (t−τ)α−1x(τ) dτ, a < t ⩽b, provided the right-hand side is pointwise defined on [a, b], where Γ(·)is the gamma function. Definition 2. Let [a, b]be a finite interval on the real axis R,n−1⩽α < n,n∈N+, and let the function x(t)have continuous derivatives up to order nsuch that x(n)(t)is absolutely continuous on [a, b]. The Caputo fractional derivative (Dα a+x)(t)of order αis defined as Dα a+x(t) = 1 Γ(n−α) t Za (t−τ)n−α−1x(n)(τ) dτ, a < t ⩽b. The Laplace transform of the Caputo’s fractional derivative (Dα 0+x)(t)is LDα 0+x(t); s=sαLx(t); s− n−1 X i=0 sα−i−1x(i)0+, t > 0. Nonlinear Anal. Model. Control, 23(1):1–18 4X.-L. Ding, J.J. Nieto Definition 3. Three-parameter Mittag–Leffler function is defined as Eρ α,β(z) = ∞ X i=0 (ρ)k Γ(αk +β) zk k!, α, β, ρ > 0, z ∈R,(3) where (ρ)kis the Pochhammer symbol, which is defined as (ρ)k=ρ(ρ+1) · · · (ρ+k−1). The Laplace transform of the three-parameter Mittag–Leffler function is Lzβ−1Eρ α,β±azα;s=sαρ−β (sα∓a)ρ, provided that |as−α|<1. An important function occurring in electrical systems is the delayed unit step function ua(t) = (1, t ⩾a, 0, t < a, and its Laplace transformation is given by Lua(t); s=e−as s,Re(s)>0. If F(s)is the Laplace transformation of the function f(t), i.e., F(s) = L{f(t); s}, then Leatf(t); s=F(s−a) and Lua(t)f(t−a); s= e−asF(s), a ⩾0, and also we have L−1e−asF(s); t=ua(t)f(t−a), a ⩾0.(4) 3 Solution representation In this section, we give the solution representation of fractional delay dynamical systems. Consider a fractional delay differential equation of the following form: Dα 0+x(t) = Ax(t) + Bx(t−h) + f(t),0< α ⩽1, t ∈[0, T], x(t) = φ(t), t ∈[−h, 0],(5) where A, B ∈Rn×n,φ(t) : [−h, 0] →Rnand f: [0, T]→Rnare two real-value continuous functions, and x∈Rnis to be solved. https://www.mii.vu.lt/NA Controllability of nonlinear fractional delay dynamical systems 5 Following the idea in [29], we take the Laplace transform on both sides of (5) to get sαX(s)−sα−1φ(0) = AX(s) + B ∞ Z0 e−stx(t−h) dt+F(s), and by simple calculations we have sαX(s)−sα−1φ(0) = AX(s) + Be−hs 0 Z −h e−sτ x(τ) dτ+Be−hsX(s) + F(s), where X(s) = ∞ Z0 e−stx(t) dt, F(s) = ∞ Z0 e−stf(t) dt. It follows that X(s) = sα−1 sαI−A−Be−hs φ(0) + Be−hs sαI−A−Be−hs 0 Z −h e−sτ φ(τ) dτ +F(s) sαI−A−Be−hs . Using Laplace inverse transform and property of the convolution, we get x(t) = L−1sα−1 sαI−A−Be−hs ;tφ(0) +L−1B sαI−A−Be−hs ;t∗ L−1(e−hs 0 Z −h e−sτ φ(τ) dτ;t) +L−11 sαI−A−Be−hs ;t∗f(t). For brevity, we denote Qα(t) := L−1sα−1 sαI−A−Be−hs ;t, Qα,α(t) = L−11 sαI−A−Be−hs ;t. (6) Define a new staircase function p(t)on [−h, ∞)such that p(t) = (0, t ⩾0, 1,−h⩽t < 0,(7) Nonlinear Anal. Model. Control, 23(1):1–18 6X.-L. Ding, J.J. Nieto and extend the function φ(t)to [−h, ∞)such that φ(t) = φ(0) for t⩾0. Based on the extension, it has e−hs 0 Z −h e−sτ φ(τ) dτ= ∞ Z0 e−stφ(−h+t)p(−h+t) dt =Lφ(−h+t)p(−h+t); s. Hence, the solution of system (5) is given as x(t) = Qα(t)φ(0) + t Z0 Qα,α(t−τ)Bφ(τ−h)p(τ−h) dτ + t Z0 Qα,α(t−τ)f(τ) dτ, and so x(t) = Qα(t)φ(0) + t−h Z −h Qα,α(t−τ−h)Bφ(τ)p(τ) dτ + t Z0 Qα,α(t−τ)f(τ) dτ. Furthermore, according the definition of p(t)in (7), the solution xof (5) can be written compactly as x(t) = x(t;φ) + t Z0 Qα,α(t−τ)f(τ) dτ, (8) where x(t;φ)is expressed as x(t;φ) = (Qα(t)φ(0) + Rt−h −hQα,α(t−τ−h)Bφ(τ) dτ, 0⩽t < h, Qα(t)φ(0) + R0 −hQα,α(t−τ−h)Bφ(τ) dτ, h ⩽t⩽T. (9) In particular, if A=a∈R,B=b∈R, then using (3) and (4), we can obtain Qα(t) := L−1sα−1 sα−a−be−hs ;t) =L−1sα−1 (sα−a)(1 −(sα−a)−1be−hs;t =L−1sα−1 sα−a ∞ X n=0 bne−nhs (sα−a)n;t https://www.mii.vu.lt/NA Controllability of nonlinear fractional delay dynamical systems 7 = ∞ X n=0 bnL−1sα−1e−nhs (sα−a)n+1 ;t = ∞ X n=0 bn(t−nh)αnEn+1 α,αn+1a(t−nh)αunh(t) = [t/h] X n=0 bn(t−nh)αnEn+1 α,αn+1a(t−nh)α and Qα,α(t) := L−11 sα−a−be−hs ;t =L−11 (sα−a)(1 −(sα−a)−1be−hs);t =L−11 sα−a ∞ X n=0 bne−nhs (sα−a)n;t = ∞ X n=0 bnL−1(e−nhs (sα−a)n+1 ;t) = ∞ X n=0 bn(t−nh)α(n+1)−1En+1 α,α(n+1)a(t−nh)αunh(t) = [t/h] X n=0 bn(t−nh)α(n+1)−1En+1 α,α(n+1)a(t−nh)α. Therefore, in this case, the solution xof system (5) is given explicitly by x(t) = x(t;φ) + [t/h] X n=0 bn t−nh Z0 (t−τ−nh)α(n+1)−1 ×En+1 α,α(n+1)a(t−τ−nh)αf(τ) dτ, (10) where x(t;φ)is expressed as x(t;φ) = [t/h] X n=0 bn (t−nh)αnEn+1 α,αn+1a(t−nh)αφ(0) +b t−(n+1)h Z −ht−τ−(n+ 1)hα(n+1)−1 ×En+1 α,α(n+1)at−τ−(n+ 1)hαφ(τ) dτ!(11a) if 0⩽t < (n+ 1)h, Nonlinear Anal. Model. Control, 23(1):1–18 8X.-L. Ding, J.J. Nieto x(t;φ) = [t/h] X n=0 bn (t−nh)αnEn+1 α,αn+1a(t−nh)αφ(0) + [t/h] X n=0 b 0 Z −ht−τ−(n+ 1)hα(n+1)−1 ×En+1 α,α(n+1)at−τ−(n+ 1)hαφ(τ) dτ!(11b) if (n+ 1)h⩽t⩽T. With respect to representations of solutions of functional differential equations, one can refer to [8,19, 29, 33]. 4 Controllability for linear systems Definition 4. System (1) (or (2)) is said to be controllable on [0, T]if, for every given initial state φand xT, there exists a control u∈Rmwith u(0) = u0, u(T) = uTsuch that the solution of system (1) (or (2)) satisfies the boundary conditions x(0; u) = x0and x(T;u) = xT. According to (8), the solution xof system (1) can be expressed as x(t) = x(t;φ) + t Z0 Qα,α(t−τ)Cu(τ) dτ, where Qα(t),Qα,α(t), and x(t;φ)are defined as (6) and (9), respectively. For brevity, let us denote χ(t) = t Z0 Qα,α(τ)Cdτ, (12) Θ(t;T) = T Z T−t χ>(τ) dτ−t T T Z0 χ>(τ) dτ, (13) Υ(t;T) = t Z0 Qα,α(t−τ)CΘ(τ;T) dτ, (14) WT= T Z0 χ(τ)χ>(τ) dτ−1 T T Z0 χ(τ) dτ T Z0 χ>(τ) dτ, (15) where “>” denotes the matrix transpose. https://www.mii.vu.lt/NA Controllability of nonlinear fractional delay dynamical systems 9 Define the control function of system (1) as u(t) = 1−t Tu0+t TuT+Θ(t;T)y(T),(16) where y(T) = W−1 T"xT−x(T;φ)−χ(T)u0−1 T T Z0 χ(τ) dτ!(uT−u0)#. Lemma 1. Let u∈Rmbe defined as (16). Then we have t Z0 Qα,α(t−τ)Cu(τ) dτ=χ(t)u0+1 T t Z0 χ(τ) dτ!(uT−u0) +Υ(t;T)y(T) and Υ(T;T) = WT. Proof. The idea of the proof of this lemma is exactly parallel to the proof of Lemma 4.1 in [25]. So we omit the proof. Theorem 1. Assume that the matrix WTdefined in (15) is nonsingular. Then for an arbitrary xT∈Rn, the control udefined as (16) transfers system (1) from φ(0) ∈Rnto xT∈Rnat time Twith boundary conditions u(0) = u0and u(T) = uT. Proof. Since WTis nonsingular, the control uis well defined, and it satisfies the conditions u(0) = u0and u(T) = uT. Furthermore, according to Lemma 1, we can deduce that x(t) = x(t;φ) + χ(t)u0+1 T t Z0 χ(τ) dτ!(uT−u0) + Υ(t;T)y(T). It is trivial to verify that x(0) = φ(0) and x(T) = xT. Thus, the control udefined as (16) transfers system (1) from φ(0) ∈Rnto xT∈Rnat time T. That is to say, system (1) is controllable on [0, T]. The proof is completed. In fact, using the controllability of system (1), we can establish the following statement. Theorem 2. The system is controllable on [0, T ]if and only if WTis positive definite. Proof. Sufficiency. Since WTis positive definite, WTis nonsingular. Then we can construct a control usuch that it steers system (1) from the initial state φ(t)to xTwith boundary conditions u(0) = u0and u(T) = uT. Thus, the system is controllable on [0, T]. Nonlinear Anal. Model. Control, 23(1):1–18 16 X.-L. Ding, J.J. Nieto Therefore, the control uis defined as u(t) = 1−t 2u0+t 2uT+Θ(t; 2)y(2), t ∈[0,2], where Θ(t; 2) = 2 Z 2−t χ>(ω) dω−t 2 2 Z0 χ>(ω) dω, t ∈[0,2], y(2) = W−1 T"xT−χ(2)u0−1 2 2 Z0 χ(τ) dτ!(uT−u0)#. If we choose φ(t)=0for t∈[−1,0],u(0) = 0,u(2) = 1, and x(0) = 0,x(2) = 6, then the state x(t)for system (22) is shown in Fig. 1(b). Example 2. Consider the following nonlinear fractional delay system with delay h= 1: D1/2 0+x(t) = x(t) + x(t−1) + u(t) + tsin x(t), t ∈[0,2], x(t)=1, t ∈[−1,0], x(T) = xT, u(0) = u0, u(T) = uT. According to the analysis in Example 1, one knows that the linear system D1/2 0+x(t) = x(t) + x(t−1) + u(t), t ∈[0,2], x(t)=1, t ∈[−1,0], x(T) = xT, u(0) = u0, u(T) = uT (23) is controllable on [0,2]. Also, the nonlinear continuous function f=tsin x(t)satisfies condition (18) in Theorem 3, hence, by Theorem 3, system (23) is controllable on [0,2]. References 1. A. Alsaedi, J.J. Nieto, V. Venktesh, Fractional electrical circuits, Adv. Mech. Eng.,7(12), 2015. 2. G. Anichini, Global controllability of nonlinear control processes with prescribed controls, J. Optim. Theory Appl.,32(2):183–199, 1980. 3. I. Area, H. Batarfi, J. Losada, J.J. Nieto, W. Shammakh, A. Torres, On a fractional order Ebola epidemic model, Adv. Difference Equ.,2015(1):1–12, 2015. 4. G.M. Bahaa, Fractional optimal control problem for differential system with delay argument, Adv. Difference Equ.,2017:69, 2017. https://www.mii.vu.lt/NA Controllability of nonlinear fractional delay dynamical systems 17 5. K. Balachandran, V. Govindaraj, L. Rodríguez-Germá, J.J. Trujillo, Controllability of nonlinear higher order fractional dynamical systems, Nonlinear Dyn.,71(4):605–612, 2013. 6. K. Balachandran, Y. Zhou, J. Kokila, Relative controllability of fractional dynamical systems with distributed delays in control, Comput. Math. Appl.,64(10):3201–3209, 2012. 7. D. Baleanu, G.-C. Wu, Y.-R. Bai, F.-L. Chen, Stability analysis of Caputo-like discrete fractional systems, Commun. Nonlinear Sci. Numer. Simul.,48:520–530, 2017. 8. H.T. Banks, Representations for solutions of linear functional differential equations, J. Differ. Equations,5(2):399–409, 1969. 9. J. ˇ Cermák, Z. Došlá, T. Kisela, Fractional differential equations with a constant delay: Stability and asymptotics of solutions, Appl. Math. Comput.,298:336–350, 2017. 10. Y.Q. Chen, H.-S. Ahn, D. Xue, Robust controllability of interval fractional order linear time invariant systems, Signal Process.,86(10):2794–2802, 2006. 11. Y.Q. Chen, Petras I., D. Xue, Fractional order control: A tutorial, in Proceedings of the 2009 American Control Conference (ACC’09), June 10–12, 2009, St. Louis, IEEE, Piscataway, NJ, 2009, pp. 1397–1411. 12. J.H. Cushman, T.R. Ginn, Nonlocal dispersion in media with continuously evolving scales of heterogeneity, Transp. Porous Media,13(1):123–138, 1993. 13. J.P. Dauer, Nonlinear perturbations of quasi-linear control systems, J. Math. Anal. Appl., 54(3):717–725, 1976. 14. A. Debbouche, D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl.,62(3):1442–1450, 2011. 15. X. Ding, J.J. Nieto, Controllability and optimality of linear time-invariant neutral control systems with different fractional orders, Acta Math. Sci.,35(5):1003–1013, 2015. 16. A.M.A. El-Sayed, H.M. Nour, A. Elsaid, A.E. Matouk, A. Elsonbaty, Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system, Appl. Math. Modelling,40(5):3516–3534, 2016. 17. Z. Fan, Q. Dong, G. Li, Approximate controllability for semilinear composite fractional relaxation equations, Fract. Calc. Appl. Anal.,19(1):267–284, 2016. 18. G. Fernández-Anaya, G. Nava-Antonio, J. Jamous-Galante, R. Muñoz-Vega, E.G. Hernández- Martínez, Asymptotic stability of distributed order nonlinear dynamical systems, Commun. Nonlinear Sci. Numer. Simul.,48:541–549, 2017. 19. J. Hale, Theory of Functional Differential Equations, Appl. Math. Sci., Vol. 3, Springer, New York, 1977. 20. B.-B. He, H.-C. Zhou, C.-H. Kou, The controllability of fractional damped dynamical systems with control delay, Commun. Nonlinear Sci. Numer. Simul.,32:190–198, 2016. 21. R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. 22. S. Huang, R. Zhang, D. Chen, Stability of nonlinear fractional-order time varying systems, J. Comput. Nonlinear Dyn.,11(3):031007–1, 2016. 23. A. Kilbas, H. Srivastave, J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., Vol. 204, Elsevier Science, Amsterdam, 2006. Nonlinear Anal. Model. Control, 23(1):1–18 18 X.-L. Ding, J.J. Nieto 24. J. Klamka, Constrained controllability of semilinear systems, Nonlinear Anal., Theory Methods Appl.,47(5):2939–2949, 2001. 25. B. Krishnan, K. Jayakumar, Controllability of fractional dynamical systems with prescribed controls, IET Control Theory Appl.,7(9):1242–1248, 2013. 26. M. Li, J.R. Wang, Finite time stability of fractional delay differential equations, Appl. Math. Lett.,64:170–176, 2017. 27. S. Manabe, The non-integer integral and its application to control systems, Electr. Eng. Jpn., 6(3–4):83–87, 1961. 28. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep.,339(1):1–77, 2000. 29. S. Murakami, Representation of solutions of linear functional difference equations in phase space, Nonlinear Anal., Theory Methods Appl.,30(2):1153–1164, 1997. 30. R.J. Nirmala, K. Balachandran, L. Rodríguez-Germá, J.J. Trujillo, Controllability of nonlinear fractional delay dynamical systems, Rep. Math. Phys.,77(1):87–104, 2016. 31. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, CA, 1999. 32. S.D. Purohit, Solutions of fractional partial differential equations of quantum mechanics, Adv. Appl. Math. Mech.,5(5):639–651, 2013. 33. F.C. Shu, On explicit representations of solutions of linear delay systems, Appl. Math. E-Notes, 13:120–135, 2013. 34. S. Song, X. Song, I.T. Balsera, Adaptive projective synchronization for fractional-order TS fuzzy neural networks with time-delay and uncertain parameters, Optik,129:140–152, 2017. 35. A. Soukkou, M.C. Belhour, S. Leulmi, Review, design, optimization and stability analysis of fractional-order PID controller, Int. J. Intell. Syst. Appl.,8(7):73–96, 2016. 36. Y. Tang, N. Li, M. Liu, Y. Lu, W. Wang, Identification of fractional-order systems with time delays using block pulse functions, Mech. Syst. Signal Process.,91:382–394, 2017. 37. D. Valério, J.S. da Costa, Introduction to single-input, single-output fractional control, IET Control Theory Appl.,5(8):1033–1057, 2011. 38. B.M. Vinagre, C.A. Monje, A.J. Calderon, Fractional order systems and fractional order control actions, in Proceedings of the 41st IEEE Conference on Decision and Control, December 10– 13, 2002, Las Vegas, NV, IEEE, Piscataway, NJ, 2002, pp. 2550–2554. 39. X.-J. Wen, Z.-M. Wu, J.-G. Lu, Stability analysis of a class of nonlinear fractional-order systems, IEEE Trans. Circuits Syst. II: Express Briefs,55(11):1178–1182, 2008. 40. Sun Yi, Patrick W Nelson, A Galip Ulsoy, Controllability and observability of systems of linear delay differential equations via the matrix Lambert Wfunction, IEEE Trans. Autom. Control, 53(3):854–860, 2008. 41. X.-F. Zhou, J. Wei, L.-G. Hu, Controllability of a fractional linear time-invariant neutral dynamical system, Appl. Math. Lett.,26(4):418–424, 2013. 42. Z. Zhou, W. Gong, Finite element approximation of optimal control problems governed by time fractional diffusion equation, Comput. Math. Appl.,71(1):301–318, 2016. https://www.mii.vu.lt/NA