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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:
49K20Optimal control problems with PDE (optimality conditions)
34D20Stability of ODE
76D05Navier-Stokes equations (fluid dynamics)
93D20Asymptotic stability of control systems
37D45Strange attractors, chaotic dynamics
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References:
[1] Yu, X.: Controlling Lorenz chaos. Int. J. Syst. sci. 27, No. 4, 355-359 (1996) · Zbl 0853.93028
[2] Sparrow, C.: The Lorenz equations: bifurcation, chaos and strange attractors. (1982) · 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, No. 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, No. 1, 51-56 (1997)
[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 · Zbl 0955.92035
[6] Malescio, G.: Synchronization of the Lorenz system through continuous feedback control. Phys. rev. E 53, No. 6, 6566-6568 (1996)
[7] Glendinning, P.: Stability, unstability and chaos: an introduction to the theory of nonlinear differential equations. (1994) · Zbl 0808.34001
[8] Krasovskii, N.: Problems in the stabilization of controlled motion. The stability of motion (1966)
[9] Francis, C.: Chaotic vaibrationa, an introduction for applied scientists and engineers. (1987) · Zbl 0745.58003
[10] Xie, Q.; Chen, G.: Synchronization stability analysis of the chaotic Rössler system. Int. J. Bifur. chaos 6, No. 11, 2153-2161 (1996) · Zbl 1298.34096
[11] Bellman, R.; Dreufos, S.: Applied problems in the dynamic programming. (1965)
[12] Foy, W. H.: Fuel minimization in flight vehicle attitude control. IEEE trans. Autom. contr. 8, No. 2, 84-88 (1963)
[13] Cook, P. A.: Nonlinear dynamical system. (1986) · Zbl 0588.93001