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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)

93D15 Stabilization of systems by feedback
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
Full Text: DOI
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