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December 2017, 7(4): 537-562. doi: 10.3934/mcrf.2017020

## Controllability of fractional dynamical systems: A functional analytic approach

 Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram-695 547, India

* Corresponding author: govindaraj.maths@gmail.com

Received  July 2016 Revised  January 2017 Published  September 2017

In this paper, we investigate controllability of fractional dynamical systems involving monotone nonlinearities of both Lipchitzian and non-Lipchitzian types. We invoke tools of nonlinear analysis like fixed point theorem and monotone operator theory to obtain controllability results for the nonlinear system. Examples are provided to illustrate the results. Controllability results of fractional dynamical systems with monotone nonlinearity is new.

Citation: Venkatesan Govindaraj, Raju K. George. Controllability of fractional dynamical systems: A functional analytic approach. Mathematical Control & Related Fields, 2017, 7 (4) : 537-562. doi: 10.3934/mcrf.2017020
##### References:
 [1] M. Axtell and M. E. Bise, Fractional calculus applications in control systems, Proceedings of the IEEE 1990 National Aerospace and Electronics conference, New York (1990), 563–566. doi: 10.1109/NAECON.1990.112826. [2] R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, Journal of Applied Mechanics, 51 (1984), 294-298. doi: 10.1115/1.3167615. [3] R. L. Bagley and R. A. Calico, Fractional order state equations for the control of viscoelastically damped structure, Journal of Guidance, Control and Dynamics, 14 (1991), 304-311. doi: 10.2514/6.1989-1213. [4] K. Balachandran and J. P. Dauer, Controllability of nonlinear systems via fixed-point theorem, Journal of Optimization Theory and Applications, 53 (1987), 345-352. doi: 10.1007/BF00938943. [5] K. Balachandran, J. Y. Park and J. J. Trujillo, Controllability of nonlinear fractional dynamical systems, Nonlinear Analysis, 75 (2012), 1919-1926. doi: 10.1016/j.na.2011.09.042. [6] K. Balachandran, J. Kokila and J. J. Trujillo, Relative controllability of fractional dynamical systems with multiple delays in control, Computer and Mathematics with Applications, 64 (2012), 3037-3045. doi: 10.1016/j.camwa.2012.01.071. [7] K. Balachandran, V. Govindaraj, L. Rodriguez-Germá and J. J. Trujillo, Controllability of nonlinear higher order fractional dynamical systems, Nonlinear Dynamics, 71 (2013), 605-612. doi: 10.1007/s11071-012-0612-y. [8] K. Balachandran, V. Govindaraj, L. Rodriguez-Germá and J. J. Trujillo, Controllability results for nonlinear fractional-order dynamical systems, Journal of Optimization Theory and Applications, 156 (2013), 33-44. doi: 10.1007/s10957-012-0212-5. [9] K. Balachandran and V. Govindaraj, Numerical controllability of fractional dynamical systems, Optimization, 63 (2014), 1267-1279. doi: 10.1080/02331934.2014.906416. [10] K. Balachandran, V. Govindaraj, M. Rivero and J. J. Trujillo, Controllability of fractional damped dynamical systems, Applied Mathematics and Computation, 257 (2015), 66-73. doi: 10.1016/j.amc.2014.12.059. [11] M. Bettayeb and S. Djennoune, New results on the controllability and observability of fractional dynamical systems, Journal of Vibration and Control, 14 (2008), 1531-1541. doi: 10.1177/1077546307087432. [12] Y. Chen, H. S. Ahn and D. Xue, Robust controllability of interval fractional order linear time invariant systems, Signal Processing, 86 (2006), 2794-2802. doi: 10.1115/DETC2005-84744. [13] V. Dolezal, Monotone Operators and Applications in Control and Network Theory Elsevier Scientific Publishing Company, Amsterdam, 1979. [14] S. Guermah, S. Djennoune and M. Bettayeb, Controllability and observability of linear discrete-time fractional-order systems, International Journal of Applied Mathematics and Computer, 18 (2008), 213-222. doi: 10.2478/v10006-008-0019-6. [15] P. Hess, On nonlinear equations of Hammerstein type in Banach spaces, Proceedings of the American Mathematical Society, 30 (1971), 308-312. doi: 10.1090/S0002-9939-1971-0282268-X. [16] M. C. Joshi, On the existence of an optimal control in Banach spaces, Bulletin of the Australian Mathematical Society, 27 (1983), 395-401. doi: 10.1017/S0004972700025892. [17] M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis Wiley Eastern Limited, New Delhi, 1985. [18] M. C. Joshi and T. E. Govindan, Stochastic global stability of random feed-back system, Stochastic Analysis and Applications, 7 (1989), 169-186. doi: 10.1080/07362998908809175. [19] M. C. Joshi and R. K. George, Controllability of nonlinear systems, Numerical Functional Analysis and Optimization, 10 (1989), 139-166. doi: 10.1080/01630568908816296. [20] T. Kaczorek, Selected Problems of Fractional Systems Theory Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-20502-6. [21] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier, Amsterdam, 2006. [22] J. Klamka, Controllability of dynamical systems -a survey, Arch. Control Sci., 2 (1993), 283-310. [23] J. A. T. Machado, Analysis and design of fractional order digital control systems, Systems Analysis Modelling Simulation, 27 (1997), 107-122. [24] R. Magin, Fractional Calculus in Bioengineering Begell House Inc. , Redding, 2006. [25] D. Matignon and B. d'Andréa-Novel, Some results on controllability and observability of finite dimensional fractional differential systems, Proceedings of the IAMCS, IEEE Conference on Systems, Man and Cybernetics Lille, France, (1996), 952–956. [26] C. A. Monje, Y. Q. Chen, B. M. Vinagre, X. Xue and V. Feliu, Fractional-order Systems and Controls: Fundamentals and Applications Springer, London, 2010. doi: 10.1007/978-1-84996-335-0. [27] D. Mozyrska and D. F. M. Torres, Minimal modified energy control for fractional linear control systems with the Caputo derivative, Carpathian Journal of Mathematics, 26 (2010), 210-221. [28] D. Mozyrska and D. F. M. Torres, Modified optimal energy and initial memory of fractional continuous-time linear systems, Signal Processing, 91 (2011), 379-385. doi: 10.1016/j.sigpro.2010.07.016. [29] I. Petráŝ, Control of fractional order Chua's system, Journal of Electrical Engineering, 53 (2002), 219-222. [30] I. Podlubny, Fractional Differential Equations Academic Press, New York, 1999. [31] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & sons, Canada, 1980. [32] B. M. Vinagre, C. A. Monje and A. J. Calderon, Fractional order systems and fractional order control actions, Lecture 3 of the IEEE CDCO2 TW# 2: Fractional Calculus Applications in Automatic Control and Robotics, Las Vegas, 2002.

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##### References:
 [1] M. Axtell and M. E. Bise, Fractional calculus applications in control systems, Proceedings of the IEEE 1990 National Aerospace and Electronics conference, New York (1990), 563–566. doi: 10.1109/NAECON.1990.112826. [2] R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, Journal of Applied Mechanics, 51 (1984), 294-298. doi: 10.1115/1.3167615. [3] R. L. Bagley and R. A. Calico, Fractional order state equations for the control of viscoelastically damped structure, Journal of Guidance, Control and Dynamics, 14 (1991), 304-311. doi: 10.2514/6.1989-1213. [4] K. Balachandran and J. P. Dauer, Controllability of nonlinear systems via fixed-point theorem, Journal of Optimization Theory and Applications, 53 (1987), 345-352. doi: 10.1007/BF00938943. [5] K. Balachandran, J. Y. Park and J. J. Trujillo, Controllability of nonlinear fractional dynamical systems, Nonlinear Analysis, 75 (2012), 1919-1926. doi: 10.1016/j.na.2011.09.042. [6] K. Balachandran, J. Kokila and J. J. Trujillo, Relative controllability of fractional dynamical systems with multiple delays in control, Computer and Mathematics with Applications, 64 (2012), 3037-3045. doi: 10.1016/j.camwa.2012.01.071. [7] K. Balachandran, V. Govindaraj, L. Rodriguez-Germá and J. J. Trujillo, Controllability of nonlinear higher order fractional dynamical systems, Nonlinear Dynamics, 71 (2013), 605-612. doi: 10.1007/s11071-012-0612-y. [8] K. Balachandran, V. Govindaraj, L. Rodriguez-Germá and J. J. Trujillo, Controllability results for nonlinear fractional-order dynamical systems, Journal of Optimization Theory and Applications, 156 (2013), 33-44. doi: 10.1007/s10957-012-0212-5. [9] K. Balachandran and V. Govindaraj, Numerical controllability of fractional dynamical systems, Optimization, 63 (2014), 1267-1279. doi: 10.1080/02331934.2014.906416. [10] K. Balachandran, V. Govindaraj, M. Rivero and J. J. Trujillo, Controllability of fractional damped dynamical systems, Applied Mathematics and Computation, 257 (2015), 66-73. doi: 10.1016/j.amc.2014.12.059. [11] M. Bettayeb and S. Djennoune, New results on the controllability and observability of fractional dynamical systems, Journal of Vibration and Control, 14 (2008), 1531-1541. doi: 10.1177/1077546307087432. [12] Y. Chen, H. S. Ahn and D. Xue, Robust controllability of interval fractional order linear time invariant systems, Signal Processing, 86 (2006), 2794-2802. doi: 10.1115/DETC2005-84744. [13] V. Dolezal, Monotone Operators and Applications in Control and Network Theory Elsevier Scientific Publishing Company, Amsterdam, 1979. [14] S. Guermah, S. Djennoune and M. Bettayeb, Controllability and observability of linear discrete-time fractional-order systems, International Journal of Applied Mathematics and Computer, 18 (2008), 213-222. doi: 10.2478/v10006-008-0019-6. [15] P. Hess, On nonlinear equations of Hammerstein type in Banach spaces, Proceedings of the American Mathematical Society, 30 (1971), 308-312. doi: 10.1090/S0002-9939-1971-0282268-X. [16] M. C. Joshi, On the existence of an optimal control in Banach spaces, Bulletin of the Australian Mathematical Society, 27 (1983), 395-401. doi: 10.1017/S0004972700025892. [17] M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis Wiley Eastern Limited, New Delhi, 1985. [18] M. C. Joshi and T. E. Govindan, Stochastic global stability of random feed-back system, Stochastic Analysis and Applications, 7 (1989), 169-186. doi: 10.1080/07362998908809175. [19] M. C. Joshi and R. K. George, Controllability of nonlinear systems, Numerical Functional Analysis and Optimization, 10 (1989), 139-166. doi: 10.1080/01630568908816296. [20] T. Kaczorek, Selected Problems of Fractional Systems Theory Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-20502-6. [21] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier, Amsterdam, 2006. [22] J. Klamka, Controllability of dynamical systems -a survey, Arch. Control Sci., 2 (1993), 283-310. [23] J. A. T. Machado, Analysis and design of fractional order digital control systems, Systems Analysis Modelling Simulation, 27 (1997), 107-122. [24] R. Magin, Fractional Calculus in Bioengineering Begell House Inc. , Redding, 2006. [25] D. Matignon and B. d'Andréa-Novel, Some results on controllability and observability of finite dimensional fractional differential systems, Proceedings of the IAMCS, IEEE Conference on Systems, Man and Cybernetics Lille, France, (1996), 952–956. [26] C. A. Monje, Y. Q. Chen, B. M. Vinagre, X. Xue and V. Feliu, Fractional-order Systems and Controls: Fundamentals and Applications Springer, London, 2010. doi: 10.1007/978-1-84996-335-0. [27] D. Mozyrska and D. F. M. Torres, Minimal modified energy control for fractional linear control systems with the Caputo derivative, Carpathian Journal of Mathematics, 26 (2010), 210-221. [28] D. Mozyrska and D. F. M. Torres, Modified optimal energy and initial memory of fractional continuous-time linear systems, Signal Processing, 91 (2011), 379-385. doi: 10.1016/j.sigpro.2010.07.016. [29] I. Petráŝ, Control of fractional order Chua's system, Journal of Electrical Engineering, 53 (2002), 219-222. [30] I. Podlubny, Fractional Differential Equations Academic Press, New York, 1999. [31] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & sons, Canada, 1980. [32] B. M. Vinagre, C. A. Monje and A. J. Calderon, Fractional order systems and fractional order control actions, Lecture 3 of the IEEE CDCO2 TW# 2: Fractional Calculus Applications in Automatic Control and Robotics, Las Vegas, 2002.
The trajectory of the system (29) steers from the initial state $\left[\begin{array}{r}0\\ 0\end{array}\right]$ to the finial state $\left[\begin{array}{r}1\\ -1\end{array}\right]$ during the interval $[0, 1]$
The steering control $u(t)$ of the system (29) during the interval $[0, 1]$
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