Rafikov, Marat; Balthazar, José Manoel On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. (English) Zbl 1221.93230 Commun. Nonlinear Sci. Numer. Simul. 13, No. 7, 1246-1255 (2008). Summary: We present the control and synchronization of chaos by designing linear feedback controllers. The linear feedback control problem for nonlinear systems is formulated under optimal control theory viewpoint. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation thus guaranteeing both stability and optimality. The formulated theorem expresses explicitly the form of minimized functional and gives the sufficient conditions that allow using the linear feedback control for nonlinear system. The numerical simulations were provided in order to show the effectiveness of this method for the control of the chaotic Rössler system and synchronization of the hyperchaotic Rössler system. Cited in 81 Documents MSC: 93D15 Stabilization of systems by feedback 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 34H10 Chaos control for problems involving ordinary differential equations 49N10 Linear-quadratic optimal control problems Keywords:chaos control; synchronization; linear feedback control; chaotic and hyperchaotic Rössler systems PDFBibTeX XMLCite \textit{M. Rafikov} and \textit{J. M. Balthazar}, Commun. Nonlinear Sci. Numer. Simul. 13, No. 7, 1246--1255 (2008; Zbl 1221.93230) Full Text: DOI References: [1] Anderson, B. D.O.; Moor, J. B., Optimal control: linear quadratic methods (1990), Prentice-Hall: Prentice-Hall NY [2] Bryson, A. E.; Ho, Y., Applied optimal control (1975), Hemisphere Publ. Corp: Hemisphere Publ. Corp Washington, DC [3] Bewley, T. R., Linear control and estimation of nonlinear chaotic convection: harnessing the butterfly effect, Phys Fluids, 11, 5, 1169-1186 (1999) · Zbl 1147.76326 [4] Jiang, G. P.; Chen, G.; Tang, W. K.S., A new criterion for chaos synchronization using linear state feedback control, Int J Bifurcat Chaos, 13, 8, 2343-2351 (2003) · Zbl 1064.37515 [5] Kapitaniak, T., Controlling chaos (1996), Academic Press: Academic Press San Diego, CA · Zbl 0958.70504 [6] Nijmeijer, H., A dynamical control view on synchronization, Physica D, 154, 219-228 (2001) · Zbl 0981.34053 [7] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys Rev Lett, 64, 1196-1199 (1990) · Zbl 0964.37501 [8] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys Rev Lett, 64, 821-824 (1990) · Zbl 0938.37019 [9] Rafikov, M.; Balthazar, J. M., On a optimal control design for Rössler system, Phys Lett A, 333, 241-245 (2004) · Zbl 1123.49300 [10] Rafikov M, Balthazar JM. Optimal linear and nonlinear control design for chaotic systems. In: Proceedings of ASME international design engineering technical conferences and computers and information in engineering conference, Long Beach, CA, USA, 24-28 September 2005.; Rafikov M, Balthazar JM. Optimal linear and nonlinear control design for chaotic systems. In: Proceedings of ASME international design engineering technical conferences and computers and information in engineering conference, Long Beach, CA, USA, 24-28 September 2005. [11] Rössler, O. E., An equation for continuous chaos, Phys Lett A, 57, 397-398 (1976) · Zbl 1371.37062 [12] Rössler, O. E., An equation for hyperchaos, Phys Lett A, 71, 155-157 (1979) · Zbl 0996.37502 [13] Sinha, S. C.; Henrichs, J. T.; Ravindra, B. A., A general approach in the design of active controllers for nonlinear systems exhibiting chaos, Int J Bifurcat Chaos, 10, 1, 165-178 (2000) · Zbl 1090.37528 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.