Extensive research in nonlinear system theory in recent years has shown that a number of system theoretic questions are closely related to the behaviour of the (controllability) orbits of the system. Thus orbit minimality, namely the property of the system having its state space identical to one orbit, arises naturally when dealing with controllability questions. This paper is concerned with the more general case where orbit minimality is not available, and attempts to explore various ways the orbits are patched together on the state space. Concepts like stability, attraction and asymptotic stability are defined and studied with the aid of certain sets naturally associated with the system like the limit set and the prolongational set. Since the related questions are topological in nature, the problems are set up using only the topological dynamics of the system.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
H. Sussmann. Existence and Uniqueness of Minimal Realizations of Nonlinear Systems,Math. Systems Theory. vol. 10 (1977), pp. 263–284.
R. Hermann and A. J. Krener. Nonlinear Controllability and Observability,IEEE Trans on A. C., vol. AC-22, no. 5. Oct. 1977
C. Lobry. Controllabilite des Systemes non Lineaires,SIAM J. Cont. vol. 8. (1970) pp 573–605
H. Sussman and V. Jurdjevic. Controllability of Nonlinear Systems,J. Diff. Equations, vol. 12 (1972). pp 95–116
H. Sussmann. Orbits of Families of Vector Fields and Integrability of Distributions,Trans. Amer. Math. Soc., vol. 180 (1973). pp 171–188
A. Krener. A Generalization of Chow's Theorem and the Bang-Bang Theorem to Nonlinear Control Problems.SIAM J. Cont., vol. 12 (1974)
D. Elliott. A Consequence of Controllability.J. Diff. Equations vol. 10 (1971), pp 364–370
W. Boothby. A Transitivity Problem from Control Theory,J. Diff. Equations, vol. 17, no. 2, 1975
G. W. Haynes and H. Hermes. Non-linear Controllability via Lie Theory.SIAM J. Contr. vol. 8, pp 450–460, 1970
R. M. Hirschorn. Global Controntrability of Nonlinear Systems,SIAM J. Cont., Vol. 14, 1976, pp 700–711
N. Bhatia and G. Szegö.Stability Theory of Dynamical Systems, Springer-Verlag. Berlin (1970)
V. Nemytscii and V. Stepanov.Qualitative Theory of Differential Equations (English Translation) Princeton Univ. Press (1960)
N. Kalouptsidis. Accessibility and Stability Theory of Nonlinear Control Systems, D. Sc. Dissertation. Washington Univ. (1976)
N. Kalouptsidis and D. Elliott. Accessibility Properties of Smooth Nonlinear Control Systems. in Proceedings of the 1976 Ames Research Center Conf. on Geometric Control Theory. (Martin and Brockett. Eds.)
W. Wonham.Linear Multicariable Control. Geometric Approach, Lecture Notes in Economics and Mathematical Systems. vol. 101. Springer-Verlag. 1974
R. N. Brockett. System Theory on Group Manifolds and Coset Spaces,SIAM J. Com., vol. 10, (1972), pp 265–284
V. Jurdjevic and H. Sussman. Control Systems on Lie Groups,J. Diff. Equations, vol. 16 (1972) pp. 313–329
J. Auslander, P. Seibert. Prolongations and Stability in Dynamical Systems, Ann. Inst. Fourier (Grenoble), vol. 14. pp. 237–267 (1964)
N. Kalouptsidis. Prolongations and Lyapunov Functions in Control Systems,Math. Systems Theory, to appear.
This work was supported in part by the National Science Foundation under Grants ENG 76-16812 and ENG 78-22166.
About this article
Cite this article
Kalouptsidis, N., Elliott, D.L. Stability analysis of the orbits of control systems. Math. Systems Theory 15, 323–342 (1981). https://doi.org/10.1007/BF01786989
- Control System
- State Space
- Nonlinear System
- Stability Analysis
- Computational Mathematic