The Closed Orbit Controllability Criterium

2010/03/05 by Valeri Marenitch, Marenitch, Valeri
Engineering · Mathematics · #20M20 #93B05 #93B29 #Aerospace Engineering and Control Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Space Satellite Systems and Control #Spacecraft Dynamics and Control #math.DS #msc:20M20 #msc:93B05 #msc:93B29

paper · pdf · doi:10.48550/arxiv.1003.1246

16 page

arxiv created 2010/03/05 · arxiv updated 2010/03/05 · openalex publication_date 2010/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that every closed "general" trajectory of the control system Σ_M has an open neighborhood on which Σ_M is controllable if 1) this orbit contains some point where the Lie algebra rank condition (LARC) is satisfied, and 2) the set of control vectors is "involved" at q. In particular, for the control systems Σ_M on the compact connected manifold Mⁿ with an open control set this gives the following "Closed Orbit Controllability Criterium": The dynamical system Σ_M of the considered type is controllable on Mⁿ if and only if for an arbitrary point q of Mⁿ there exists a closed trajectory of the control system going through this point. We also present examples which show that our conditions are necessary.

Related

• A necessary and sufficient condition for local controllability around closed orbits
• Local controllability does imply global controllability
• On the structural properties of the bounded control set of a linear control system
• Distributions and controllability problems (I)
• Ad-rank condition for controllability of linear control system on solvable Lie groups