Optimal control of chaotic Hamiltonian dynamics

Abstract

Two general methods are developed to optimally control chaotic dynamics and, for each method, control approximations are presented. The first method consists of seeking a path, defined by an average trajectory, to meet the objective at a target time. The second method has the goal of achieving control by drawing in the irregular trajectories around an imposed regular fiducial trajectory. These two methods are complementary in their approaches to achieving control over chaotic behavior. A Lagrange-multiplier technique and a penalty method are used in the process of finding the optimal solution that minimizes a cost functional. Approximations are introduced for the Lagrange multipliers, based on preserving and controlling only the average trajectory. Numerical results are presented for the two methods. The optimal external perturbation ε(t) influences the chaotic trajectories by redirecting them to a region in phase space where it is possible to maintain a tight bundle of trajectories with a minimum external interaction. Moreover, the results show that the interaction energy is small in comparison with the total energy of the system, indicating an efficient control strategy.