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Control of a chaotic system. (English) Zbl 0747.93071
Given a Lorenz system subject to control \[ \dot x_ 1=-sx_ 1+sx_ 2, \dot x_ 2=rx_ 1-x_ 2-x_ 1 x_ 3+u, \dot x_ 3=x_ 1 x_ 2- bx_ 3, \] the authors suggest two different controllers to stabilize unstable equilibrium point of the uncontrolled system. They analyze a particular case \(s=10\), \(r=28\), \(b=8/3\), which has three unstable equilibrium points.
The first controller is given by \(u=-k(x_ 1-x_{10})\). Then a sufficiently large \(k>0\) guarantees the stability, but the motion may contain chaotic transients of different time lengths depending on the magnitude of \(k\).
In the second case, the controllability minimum principle produces a stabilizing bang-bang control \(u=-10\text{ sgn}(x^ 2_ 1-(8/3)x_ 3)\).
Reviewer: J.Kucera (Pullman)

MSC:
93D15 Stabilization of systems by feedback
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
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[1] J. Brindley and I. M. Motoz, ”Lorenz attractor behavior in a continuously stratified baroclinic fluid,”Phys. Lett., vol. 77A, pp. 441–444, 1980.
[2] P. Ehrhard and U. Müller, ”Dynamical behavior of natural convection in a single-phase loop,”J. Fluid Mech. (in press).
[3] J.E. Gayek and T.L. Vincent, ”On the asymptotic stability of boundary trajectories,”Int. J. Control, vol. 41, pp. 1077–1086, 1985. · Zbl 0562.93078
[4] J.D. Gibbon and M.J. McGuinness, ”A derivation of the Lorenz equation for some unstable dispersive physical systems,”Phys. Lett., vol. 77A, pp 295–299, 1980.
[5] W.J. Grantham and T.L. Vincent, ”A controllability minimum principle.”J. Optimization Theory Applications, vol. 17, pp. 93–114, 1975. · Zbl 0292.93002
[6] H. Haken, ”Analogy between higher instabilities in fluids and lasers,”Phys. Lett., vol. 53A, pp. 77–78, 1975.
[7] J.L. Kaplan and J.A. Yorke, ”Preturbulence: a regime observed in a fluid flow model of Lorenz,”Commun. Math. Phys., vol. 67, pp. 93–108, 1979. · Zbl 0443.76059
[8] E. Knobloch, ”Chaos in a segmented disc dynamo,”Phys Lett., vol. 82A, pp. 439–440, 1981.
[9] E.N. Lorenz, ”Deterministic non-periodic flow,”J. Atmos. Sci., vol. 20, pp. 130–141, 1963. · Zbl 1417.37129
[10] M.V.R. Malkus, ”Non-peroidic convection at high and low Prandtl number,”Mémoires Société Royale des Sciences de Liége, Series 6, vol. 4, pp. 125–128, 1972.
[11] J. Pedlosky, ”Limit cycles and unstable baroclinic waves,”J. Atmos. Sci., vol. 29, p. 53, 1972.
[12] J. Pedlosky and C. Frenzen, ”Chaotic and periodic behavior of finite amplitude baroclinic waves,”J. Atmos. Sci., vol. 37, pp. 1177–1196, 1980.
[13] C. Sparrow, ”The Lorenz equations: bifurcations, chaos, and strange attractor,”Appl. Math. Sci., vol. 41, 1982. · Zbl 0504.58001
[14] E.D. Yorke and J.A. Yorke, ”Metastable chaos: transition to sustained chaotic behavior in the Lorenz model,”J. Stat. Phys., vol. 21, pp. 263–277, 1979.
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