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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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##### References:
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