Optimal control in a noisy system

We describe a simple method to control a known unstable periodic orbit (UPO) in the presence of noise. The strategy is based on regarding the control method as an optimization problem, which allows us to calculate a control matrix $A$⁠. We illustrate the idea with the Rossler system, the Lorenz system, and a hyperchaotic system that has two exponents with positive real parts. Initially, a UPO and the corresponding control matrix are found in the absence of noise in these systems. It is shown that the strategy is useful even if noise is added as control is applied. For low noise, it is enough to find a control matrix such that the maximum Lyapunov exponent $λmax<0$⁠, and with a single non-null entry. If noise is increased, however, this is not the case, and the full control matrix $A$ may be required to keep the UPO under control. Besides the Lyapunov spectrum, a characterization of the control strategies is given in terms of the average distance to the UPO and the control effort required to keep the orbit under control. Finally, particular attention is given to the problem of handling noise, which can affect considerably the estimation of the UPO itself and its exponents, and a cleaning strategy based on singular value decomposition was developed. This strategy gives a consistent manner to approach noisy systems, and may be easily adapted as a parametric control strategy, and to experimental situations, where noise is unavoidable.