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Optimal control of Lorenz system during different time intervals. (English) Zbl 1036.49028

Summary: The problem of optimal control of the equilibrium states of the Lorenz system in both finite and infinite time intervals has been studied. The optimal control functions ensuring asymptotic stability of desired states in both cases are obtained as functions of the phase state and time. The squared Euclidean norm of the perturbed state of the Lorenz system in both cases is obtained as transcendental function of time. As an application, it is shown that the equilibrium states of the Lorenz system are asymptotically stable. Graphical and numerical simulation studies for the obtained results are presented.

MSC:

49K20 Optimality conditions for problems involving partial differential equations
34D20 Stability of solutions to ordinary differential equations
76D05 Navier-Stokes equations for incompressible viscous fluids
93D20 Asymptotic stability in control theory
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] Yu, X., Controlling Lorenz chaos, Int. J. Syst. Sci., 27, 4, 355-359 (1996) · Zbl 0853.93028
[2] Sparrow, C., The Lorenz Equations: Bifurcation, Chaos and Strange Attractors (1982), Springer: Springer New York · Zbl 0504.58001
[3] Stoten, D. P.; Bernardo, Di, Application of the minimal control synthesis algorithm to the control synchronization of chaotic systems, Int. J. Contr., 65, 6, 925-938 (1996) · Zbl 0867.93073
[4] Bai, Er-wei; Lonngen, K. E., Synchronization of two Lorenz systems using active control, Chaos Soliton Fract., 8, 1, 51-56 (1997) · Zbl 1079.37515
[5] E.A. El-Rifai, A generalization of Lotka-Volerra, GLV, systems with some dynamical and topological properties, Chaos Soliton Fract. 11 (2000) 1747-1751; E.A. El-Rifai, A generalization of Lotka-Volerra, GLV, systems with some dynamical and topological properties, Chaos Soliton Fract. 11 (2000) 1747-1751 · Zbl 0955.92035
[6] Malescio, G., Synchronization of the Lorenz system through continuous feedback control, Phys. Rev. E, 53, 6, 6566-6568 (1996)
[7] Glendinning, P., Stability, Unstability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations (1994), Cambridge University Press: Cambridge University Press Cambridge
[8] Krasovskii, N., Problems in the stabilization of controlled motion, (Malkin, I. G., The Stability of Motion (1966), Nauka Press: Nauka Press Moscow) · Zbl 0969.49022
[9] Francis, C., Chaotic Vaibrationa, An Introduction for Applied Scientists and Engineers (1987), Ithaca: Ithaca New York · Zbl 0745.58003
[10] Xie, Q.; Chen, G., Synchronization stability analysis of the chaotic Rossler system, Int. J. Bifur. Chaos, 6, 11, 2153-2161 (1996) · Zbl 1298.34096
[11] Bellman, R.; Dreufos, S., Applied Problems in the Dynamic Programming (1965), Nauka: Nauka Moscow
[12] Foy, W. H., Fuel minimization in flight vehicle attitude control, IEEE Trans. Autom. Contr., Ac-8, 2, 84-88 (1963)
[13] Cook, P. A., Nonlinear Dynamical System (1986), Prentice Hall International: Prentice Hall International Englewood Cliffs, NJ · Zbl 0588.93001
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