Chaos control for the Lorenz system

A scalar additive control action with inequality constraints is injected into the chaotic Lorenz system to stabilize the unstable periodic orbits embedded within chaotic attractors. The control problem is synthesized using the minimum principle, resulting in a bang-bang control law, in which the control variable switches once or more times from one point to another point of the boundary of the feasible control region. Two cases are considered, that is, the control action is respectively imposed on the first and second states of the Lorenz system. Surprisingly, the simple bang-bang control is shown to be effective for chaos control, even if a small control action is applied.