Abstract
A wide variety of systems in e.g. engineering or biological applications can be described or modelled by control systems (or controlled differential equations) of the general form \(\dot x = F\left( {x,u} \right)\) where the state a; is a point on a smooth manifold, the control u takes values in a compact interval or another manifold and is integrable as a function of time, and for each fixed value of u the vector field F(·,u) is smooth. One of the main topics in the investigation of such systems is the problem of understanding the local behaviour near an equilibrium point x0, or more generally near a reference trajectory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Aeyels, “Stabilization of a Class of Nonlinear Systems by a Smooth Feedback Control”, Syst. amp Control Letters, v. 5, 1985, pp. 289–294.
R. M. BIANCHINI and G. STEFANI, “Local Controllability about a Reference Trajectory”, Analysis and Optimization of Systems, A. Bensoussand & J. L. Lions eds., Lect. Notes Control, v. 83, 1986, Springer.
G. BIRKHOFF and G.-C. ROTA, Ordinary Differential Equations, BlaisdeU Publ., 1962.
R. W. BROCKETT, “Asymptotic Stability and Feedback Stabilization”, Differential Geometric Control Theory, R. W. Brockett, R. S. Millmann, and H. J. Sussmann eds., Progress in Mathematics, v. 27, 1983, pp. 181–191.
C. I. Byrnes and A. ISIDORI, “The Analysis and Design of Nonlinear Feedback Systems”, to appear in IEEE Transactions Automatic Control.
P. E. Crouch and C. I. Byrnes, “Local Accessiblity, Local Reachability, and Representations of Compact Groups”, Math. Systems Theory, v. 19, 1986, pp. 43–65.
Crouch,PE “Solvable Approximations to Control Systems”, SIAM J. Control & Opt, v. 22, 1984, pp. 40–54.
W. HAHN, “Uber Differentialgleichungen erster Ordnung mit Homogenen Rechten Seiten”, ZAMM, v. 46, 1966, pp. 357–361, (erratum ZAMM, v. 47, 1967, p. 284). (Russian and English summary.).
P. HARTMANN, Ordinary Differential Equations, Birkhäuser, 1982.
H.G. Hermes, “Controlled Stability”, Annali di Matematica pura ed applicata IV, v. 114, 1977, pp. 103–119.
M. KAWSK , “A Necessary Condition for Local.Controllability”, Differential Geometry: The Interface between Pure and Applied Mathematics, M. Luksic, C. F. Martin and W. Shadwick eds., Contemporary Mathematics, v. 68, 1987, pp. 143–155.
M. Kawski, “Stabilization of Nonlinear Systems in the Plane”, to appear in Systems & Control Letters, v. 12, 1989.
G. STEFANI, “On the Local Controllability of a Scalar-Input System”, Theory and Applications of Nonlinear Control Theory, C. Byrnes and A. Lindquist eds., 1986, pp. 267–279.
H. J. Sussmann, “Subanalytic Sets and Feedback Control”, J. Diff. Equations, v. 31, 1979, pp. 31–52.
H. J. Sussmann, “Lie Brackets and Local Controllability: A Sufficient Condition for Scalar-Input Systems”, SIAM J. Control & Opt., v. 21, 1983, pp. 686–713.
H. J. Sussmann, “A General Theorem on Local Controllability”, SIAM J. Control & Opt, v. 25, 1987, 158–194.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1989 Birkhäuser Boston
About this chapter
Cite this chapter
Kawski, M. (1989). Controllability, Approximations and Stabilization. In: Computation and Control. Progress in Systems and Control Theory, vol 1. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3704-4_11
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3704-4_11
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3438-4
Online ISBN: 978-1-4612-3704-4
eBook Packages: Springer Book Archive