Optimal Stabilization of Periodic Orbits: A Symplectic Geometry Approach

TL;DR

This paper develops a symplectic geometry-based method for the optimal stabilization of periodic orbits, extending invariant manifold theory and providing conditions for the existence of stabilizing controllers with applications in mechanics.

Contribution

It generalizes stable manifold theory to periodic orbits using symplectic geometry and derives conditions for stabilizing periodic orbits via a periodic Riccati equation.

Findings

The method successfully stabilizes a spring-mass oscillator at a target energy.

The approach outperforms traditional linear control in satellite orbit transfer.

Conditions for the existence of stabilizing controllers are established for Hamiltonian systems.

Abstract

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 plays the role of a hyperbolic equilibrium point. A sufficient condition for the existence of an NHIM of an extended 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 is stabilizable and detectable. A moving orthogonal coordinate system is employed along the periodic orbit, which is a natural framework for orbital stabilization and linearization along the orbit. Two illustrative examples are presented: the first involves stabilizing a spring-mass oscillator at a target energy level, and the second addresses an orbit transfer problem for a satellite-a classic scenario in orbital mechanics. In both cases, we show that the proposed nonlinear feedback controller outperforms traditional linear control.