Geometric Methods in System Theory pp 165-173 | Cite as
On Necessary and Sufficient Conditions for Local Controllability Along a Reference Trajectory
Conference paper
Abstract
Consider an n — dimensional control system modelled by the differential equations
where f is smooth and an admissible control u is a piecewise continuous function taking values in a given set U having nonempty interior in Rm . Denote by a(t,q) the set of all points attainable at time t by solutions of (1) corresponding to admissible controls and initiating from q at time 0 . Let u* be a given control which generates a reference trajectory φ with φ(0) = p . The system (1) is locally controllable along φ at p if for all ε > 0 ,φ(ε)is an interior point of a(ε,p) . Loosely speaking, this implies the ability to control the system to a full neighborhood of the reference trajectory over an arbitrarily small interval of time.
$$
\dot x(t) = f(x(t) ,u(t)), (\dot x(t) = dx/dt)
$$
(1)
Keywords
Vector Field Local Controllability Admissible Control Reference Trajectory Nonempty Interior
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
- 1.Sussmann, H.J. and Jurdjevic, V.; Controllability of Nonlinear Systems, J. Diff. Eqs. 12(1972), 95–116.MathSciNetMATHCrossRefGoogle Scholar
- 2.Lobry, C.; Contrôlabilité des systèmes non linèaires, SIAM J. Control, 8(1970), 573–605.MathSciNetMATHCrossRefGoogle Scholar
- 3.Hermes, H., LaSalle, J.P.; Functional Analysis and Time Optimal Control, Academic Press, N.Y. (1969).MATHGoogle Scholar
- 4.Krener, A.J.; A Generalization of Chow’s Theorem and the Bang-Bang Theorem to Nonlinear Control Systems (to appear), SIAM J. Control.Google Scholar
- 5.Hermes, H.; On Local and Global Controllability,Google Scholar
- 6.Hermes, H.; Controllability and the Singular Problem, SIAM J. Control, 2(1965), 241–260.MathSciNetGoogle Scholar
- 7.Hermes, H. and Haynes, G.W.; On the Nonlinear Control Problem with Control Appearing Linearly, J. SIAM Control, 1(1963), 85–108.MathSciNetMATHGoogle Scholar
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© D. Reidel Publishing Company, Dordrecht 1973