The present paper studies a chaotic system with one cubic nonlinear term and deals with optimal control of this system. The problem analysis technique of this paper, which is a major issue in oscillators, robotics, lasers, etc., has not been proposed in previous studies. Modal Series technique was used to solve the problem of optimal control with infinite time horizon for chaotic system. Nonlinear boundary value obtained in this technique is converted to a sequence of time invariable linear boundary value using Pontryagin's minimum principle. By resolving this sequence, state trajectory and optimal control law are obtained in the form of series with uniform convergence. Moreover, this technique allows for selection of suitable number of answers to reach an appropriate approximation of the main answer. In addition, the number of series terms is not limited. A reverse algorithm for drawing approximate state trajectory and sub-optimal control law. The results of simulations confirmed efficiency and accuracy of the proposed algorithm.
optimal control, chaos, four-scroll chaotic system, pontryagin's minimum principle, modal series method
 Lorenz EN. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 20(2): 130-141. https://doi.org/10.1175/15200469(1963)020<0130:DNF>2.0.CO;2
 Rössler OE. (1976). An equation for continuous chaos. Physics Letters A 57(5): 397-398. https://doi.org/10.1016/0375-9601(76)90101-8
 Rabinovich MI, Fabrikant AL. (1979). Stochastic self-modulation of waves in nonequilibrium media. Journal of Experimental and Theoretical Physics 77: 617-629.
 Arneodo A, Coullet P, Tresser C. (1981). Possible new strange attractors with spiral structure. Communications in Mathematical Physics 79(4): 573-579. https://doi.org/10.1007/BF01209312
 Sprott JC. (1994). Some simple chaotic flows. Physical Review E 50(2): 647-650. https://doi.org/10.1103/PhysRevE.50.R647
 Chen G, Ueta T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and Chaos 9(07): 1465-1466. https://doi.org/10.1142/S0218127499001024
 Lü J, Chen G. (2002). A new chaotic attractor coined. International Journal of Bifurcation and Chaos 12(03): 659-661. https://doi.org/10.1142/S0218127402004620
 Shaw R. (1981). Strange attractors, chaotic behavior, and information flow. Zeitschrift für Naturforschung A 36(1): 80-112. https://doi.org/10.1515/zna-1981-0115
 Cai G, Tan Z. (2007). Chaos synchronization of a new chaotic system via nonlinear control. Journal of Uncertain Systems 1(3): 235-240. https://doi.org/10.1.1.509.4486
 Tigan G, Opriş D. (2008). Analysis of a 3D chaotic system. Chaos, Solitons & Fractals 36(5): 1315-1319. https://doi.org/10.1016/j.chaos.2006.07.052
 Kennedy MP. (1994). Chaos in the colpitts oscillator. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 41(11): 771-774. https://doi.org/10.1109/81.331536
 Zhou W, Xu Y, Lu H, Pan L. (2008). On dynamics analysis of a new chaotic attractor. Physics Letters A, 372(36): 5773-5777. https://doi.org/10.1016/j.physleta.2008.07.032
 Li D. (2008). A three-scroll chaotic attractor. Physics Letters A 372(4): 387-393. https://doi.org/10.1016/j.physleta.2007.07.045
 Sundarapandian V. (2013). Analysis and anti-synchronization of a novel chaotic system via active and adaptive controllers. Journal of Engineering Science and Technology Review 6(4): 45-52.
 Sundarapandian V, Pehlivan I. (2012). Analysis, control, synchronization, and circuit design of a novel chaotic system. Mathematical and Computer Modelling 55(7-8): 1904-1915. https://doi.org/10.1016/j.mcm.2011.11.048
 Zhu CX, Liu YH, Guo Y. (2010). Theoretic and numerical study of a new chaotic system. Intelligent Information Management 2(2): 104. https://doi.org/10.4236/iim.2010.22013
 Pan L, Zhou W, Fang JA, Li D. (2010). A new three-scroll unified chaotic system coined. International Journal of Nonlinear Science 10(4): 462-474.
 Liu X, Shen XS, Zhang H. (2012). Multi-scroll chaotic and hyperchaotic attractors generated from chen system. International Journal of Bifurcation and Chaos 22(02): 1250033. https://doi.org/10.1142/S0218127412500332
 Kengne J, Chedjou JC, Kenne G, Kyamakya K. (2012). Dynamical properties and chaos synchronization of improved colpitts oscillators. Communications in Nonlinear Science and Numerical Simulation 17(7): 2914-2923. https://doi.org/10.1016/j.cnsns.2011.10.038
 Li N, Pan W, Yan L, Luo B, Zou X. (2014). Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers. Communications in Nonlinear Science and Numerical Simulation 19(6): 1874-1883. https://doi.org/10.1016/j.cnsns.2013.09.036
 Iqbal S, Zang X, Zhu Y, Zhao J. (2014). Bifurcations and chaos in passive dynamic walking: A review. Robotics and Autonomous Systems 62(6): 889-909. https://doi.org/10.1016/j.robot.2014.01.006
 Volos CK, Kyprianidis IM, Stouboulos IN. (2013). Experimental investigation on coverage performance of a chaotic autonomous mobile robot. Robotics and Autonomous Systems 61(12): 1314-1322. https://doi.org/10.1016/j.robot.2013.08.004
 Naas TT, Lasbet Y, Benzaoui A, Loubar K. (2016). Characterization of pressure drops and heat transfer of non-newtonian power-law fluid flow flowing in chaotic geometry. International Journal of Heat and Technology 34(2): 251-260. DOI: 10.18280/ijht.340214
 Boukhalkhal AL, Lasbet Y, Makhlouf M, Loubar K. (2017). Numerical study of the chaotic flow in three-dimensional open geometry and its effect on the both fluid mixing and heat performances. International Journal of Heat and Technology 35(1). https://doi.org/10.18280/ijht.350101
 Kyprianidis IM, Makri AT. (2013). Complex dynamics of FitzHugh-Nagumo type neurons coupled with gap junction under external voltage stimulation. Journal of Engineering Science and Technology Review 6(4): 104-114.
 Abdullah A. (2013). Synchronization and secure communication of uncertain chaotic systems based on full-order and reduced-order output-affine observers. Applied Mathematics and Computation 219(19): 10000-10011. https://doi.org/10.1016/j.amc.2013.03.133
 Paul S, Dasgupta P, Naskar PK, Chaudhuri A. (2017). Secured image encryption scheme based on DNA encoding and chaotic map. International Information and Engigneering Technology Assocation 1: 0-75. https://doi.org/10.18280/rces.040206
 Huber AW. (1989). Adaptive control of chaotic system. Helv Acta 62: 343-346
 Li N, Yuan H, Sun H, Zhang Q. (2013). An impulsive multi-delayed feedback control method for stabilizing discrete chaotic systems. Nonlinear Dynamics 73(3): 1187-1199. https://doi.org/10.1016/j.chaos.2003.12.071
 Sun K, Liu X, Zhu C, Sprott JC. (2012). Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system. Nonlinear Dynamics 69(3): 1383-1391. https://doi.org/10.1007/s11071-012-0354-x
 Zhang X, Zhu H, Yao H. (2012). Analysis of a new three-dimensional chaotic system. Nonlinear Dynamics 67(1): 335-343. https://doi.org/10.1007/s11071-011-9981-x
 Tsai JSH, Fang JS, Yan JJ, Dai MC, Guo SM, Shieh LS. (2018). Hybrid robust discrete sliding mode control for generalized continuous chaotic systems subject to external disturbances. Nonlinear Analysis: Hybrid Systems 29: 74-84. https://doi.org/10.1016/j.nahs.2018.01.001
 Zhang R, Zeng D, Park JH, Liu Y, Zhong S. (2018). A new approach to stabilization of chaotic systems with nonfragile fuzzy proportional retarded sampled-data control. IEEE Transactions on Cybernetics. https://doi.org/10.1109/TCYB.2018.2831782
 Bouzeriba A, Boulkroune A, Bouden T. (2015, May). Fuzzy adaptive synchronization of a class of fractional-order chaotic systems. In Control, Engineering & Information Technology (CEIT), 2015 3rd International Conference on 1-6. IEEE. https://doi.org/10.1109/CEIT.2015.7233073
 Vaidyanathan S. (2017). A new 3-D jerk chaotic system with two cubic nonlinearities and its adaptive backstepping control. Archives of Control Sciences 27(3): 409-439. https://doi.org/10.1515/acsc-2017-0026
 Lien CH, Vaidyanathan S, Sambas A, Mamat M, Sanjaya WSM. (2018). A new two-scroll chaotic attractor with three quadratic nonlinearities, its adaptive control and circuit design. In IOP Conference Series: Materials Science and Engineering 332(1): 012010. https://doi.org/10.1088/1757-899X/332/1/012010
 Mamat M, Vaidyanathan S, Sambas A, Sanjaya WSM. (2018). A novel double-convection chaotic attractor, its adaptive control and circuit simulation. In IOP Conference Series: Materials Science and Engineering 332(1): https://doi.org/10.1088/1757-899X/332/1/012033
 Vaidyanathan S. (2015). Adaptive control design for the anti-synchronization of novel 3-D chemical chaotic reactor systems. Parameters 8: 654–668.
 Vaidyanathan S, Volos C. (2015). Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system. Archives of Control Sciences 25(3): 333-353.
 Vaidyanathan S. (2015). Dynamics and control of Tokamak system with symmetric and magnetically confined plasma. International Journal of ChemTech Research 8(6): 795-803.
 Tang GY. (2005). Suboptimal control for nonlinear systems: a successive approximation approach. Systems & Control Letters 54(5): 429-434. https://doi.org/10.1016/j.sysconle.2004.09.012
 Tang GY, Qu HP, Gao YM. (2002). Sensitivity approach of suboptimal control for a class of nonlinear systems. J. Ocean Univ. Qingdao 32(4): 615-620.
 Pariz N. (2001). Analysis of nonlinear system behavior: the case of stressed power systems. Ph. D. dissertation, Department of Electrical Engineering, Ferdowsi University, Mashhad, Iran.
 Jajarmi A, Pariz N, Effati S, Kamyad AV. (2012). Infinite horizon optimal control for nonlinear interconnected large‐scale dynamical systems with an application to optimal attitude control. Asian Journal of Control 14(5): 1239-1250. https://doi.org/10.1002/asjc.452
 Jajarmi A, Pariz N, Vahidian Kamyad A, Effati S. (2011). A novel modal series representation approach to solve a class of nonlinear optimal control problems. International Journal of Innovative, Computing, Information and Control 7: 1413–1425.
 Jajarmi A, Pariz N, Kamyad AV, Effati S. (2012). A highly computational efficient method to solve nonlinear optimal control problems. Scientia Iranica 19(3): 759-766. https://doi.org/10.1016/j.scient.2011.08.029
 Jajarmi A, Pariz N, Effati S, Kamyad AV. (2011). Solving infinite horizon nonlinear optimal control problems using an extended modal series method. Journal of Zhejiang University SCIENCE C 12(8): 667. https://doi.org/10.1631/jzus.C1000325
 Sampath S, Vaidyanathan S, Volos CK, Pham VT. (2015). An eight-term novel four-scroll chaotic system with cubic nonlinearity and its circuit simulation. Journal of Engineering Science and Technology Review 8(2): 1-6.