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Optimal Stabilization of Periodic Orbits

Fabian Beck, Noboru Sakamoto

TL;DR

The paper extends optimal stabilization from equilibria to periodic orbits by combining invariant-manifold theory, moving transverse coordinates, and symplectic/Hamiltonian methods. It constructs a normally hyperbolic invariant manifold (NHIM) via a continuation of the trivial solution and shows a 2D NHIM structure with a stable/unstable splitting, under a period-1 stabilizing periodic Riccati solution. A local, feedback-optimal control is obtained by solving a Hamilton–Jacobi equation, and existence is tied to stabilizability and detectability along the orbit. Two orbital-engineering examples demonstrate the approach: energy-controlled mass-spring dynamics and satellite orbit transfer, illustrating practical applicability in mechanical and aerospace contexts. The work connects NHIM theory with periodic Riccati equations to yield a structured, local optimal control framework for orbital stabilization, while also outlining computational and boundary-handling considerations for implementation.

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 (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.

Optimal Stabilization of Periodic Orbits

TL;DR

The paper extends optimal stabilization from equilibria to periodic orbits by combining invariant-manifold theory, moving transverse coordinates, and symplectic/Hamiltonian methods. It constructs a normally hyperbolic invariant manifold (NHIM) via a continuation of the trivial solution and shows a 2D NHIM structure with a stable/unstable splitting, under a period-1 stabilizing periodic Riccati solution. A local, feedback-optimal control is obtained by solving a Hamilton–Jacobi equation, and existence is tied to stabilizability and detectability along the orbit. Two orbital-engineering examples demonstrate the approach: energy-controlled mass-spring dynamics and satellite orbit transfer, illustrating practical applicability in mechanical and aerospace contexts. The work connects NHIM theory with periodic Riccati equations to yield a structured, local optimal control framework for orbital stabilization, while also outlining computational and boundary-handling considerations for implementation.

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 (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.
Paper Structure (16 sections, 10 theorems, 96 equations, 4 figures)

This paper contains 16 sections, 10 theorems, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Base control system and its periodic orbit
  4. Change to a moving orthonormal system
  5. Continuation of trivial solution and a 2-dimensional invariant manifold
  6. Proof that $\mathscr{M}$ is an NHIM
  7. Existence of Optimal control
  8. Application examples
  9. Energy control for a mass-spring system
  10. Satellite orbit transfer
  11. Conclusion and outlook
  12. Continuation
  13. Theory of linear periodic systems and differential Riccati equations
  14. Normally hyperbolic invariant manifolds
  15. Modification of boundaries
  16. ...and 1 more sections

Key Result

Proposition III.1

Suppose that the periodic Riccati equation (eqn:p1Riccati) has a period-1 stabilizing solution $P_s(t)$. Then, the monodromy matrix of $\mathrm{Ham}(t)$ has $n-1$ eigenvalues inside the unit circle and $n-1$ eigenvalues outside of the unit circle.

Figures (4)

  • Figure 1: The invariant manifold $\mathscr{M}(\epsilon)$ and the orbit given by $\Gamma_0$ (black line).
  • Figure 2: Trajectories corresponding to the Hamiltonian flow along the stable manifold projected onto $x_0$-$x_1$ plane.
  • Figure 3: Plot of the control input for selected initial conditions.
  • Figure 4: Function $\Psi$ for the boundary modification.

Theorems & Definitions (22)

  • Remark II.1
  • Proposition III.1
  • proof
  • Proposition III.2
  • proof
  • Remark III.1
  • Theorem IV.1
  • proof
  • Theorem IV.2
  • Remark IV.1
  • ...and 12 more