Control of a chaotic system (Q1179524)

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 \(u=-10\text{ sgn}(x^ 2_ 1-(8/3)x_ 3)\).