Targeting in Hamiltonian systems that have mixed regular/chaotic phase spaces

Area of Science:

• Dynamical Systems and Chaos Theory
• Celestial Mechanics
• Control Theory

Background:

• Directing trajectories of chaotic dynamical systems to a target is a known challenge.
• Previous research demonstrated that chaos enables targeting with minimal control inputs.
• Hamiltonian systems exhibit complex phase space structures with both regular and chaotic regions.

Purpose of the Study:

• To investigate trajectory targeting within Hamiltonian systems.
• To develop and evaluate an efficient and robust targeting method.
• To leverage the inherent phase space structure for optimal control.

Main Methods:

• A multistep forward-backward targeting strategy using strategic intermediate points.
• Exploitation of the mixed regular and chaotic phase space structure.
• Testing the method on the standard map and the restricted circular three-body problem.

Main Results:

• The multistep method is found to be efficient and robust for targeting.
• The approach effectively utilizes phase space structure, potentially yielding optimal transport times.
• The method demonstrates resilience to small noise, modeling errors, and temporary control loss.

Conclusions:

• The developed multistep forward-backward method offers an effective solution for trajectory control in Hamiltonian systems.
• The strategy's robustness makes it suitable for real-world applications, such as space probe navigation.
• This approach provides a significant advancement in controlling chaotic dynamical systems.