In this contribution, the optimal stabilization problem of periodic orbits is studied via invariant manifold theory and symplectic geometry. The stable manifold theory for the optimal point stabilization case is generalized to the case of periodic orbit stabilization, where a normally hyperbolic invariant manifold (NHIM) plays the role of a hyperbolic equilibrium. A sufficient condition for the existence of an NHIM of an associated Hamiltonian system is derived in terms of a periodic Riccati differential equation. It is shown that the problem of optimal orbit stabilization has a solution if a linearized periodic system satisfies stabilizability and detectability. A moving orthogonal coordinate system is employed along the periodic orbit which is a natural framework for orbital stabilization and linearization argument. Examples illustrated include an optimal control problem for a spring-mass oscillator system, which should be stabilized at a certain energy level, and an orbit transfer problem for a satellite, which constitutes a typical control problem of orbital mechanics.

Submitted for a journal on November 29 2024