Abstract
It is shown that no resonance and small time local controllability imply many standard necessary conditions for the existence of a continuous, asymptotically stabilizing (state) feedback control (ASFC) for ann-dimensional, real analytic, single input affine control system. Furthermore, no resonance means there exist local coordinates in which the drift vector field is linear, yielding a canonical form for the study of sufficient conditions and construction of an ASFC. The roles of resonance and the Kawski necessary condition are examined in detail, together with their implications on regularity of viscosity solutions of Hamilton-Jacobi-Bellman equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Ancona, Ph.D. dissertation, University of Colorado, 1994.
M. Bardi, A boundary value problem for the minimum time function,SIAM J. Control Optim.,27 (1989), 776–785.
R. W. Brockett, Asymptotic stability and feedback stabilization, inDifferential Geometric Control Theory, Vol. 27 (R. W. Brockett, R. S. Millman, H. J. Sussmann, eds.), BirkhÄuser, Boston, 1983, pp. 181–191.
J. M. Coron, A necessary condition for feedback stabilization,Systems Control Lett.,14 (1990), 227–232.
J. M. Coron, Stabilization of controllable systems, inNonholonomic Geometry (A. Bellaiche and J. J. Risler, eds.), Progress in Mathematics, BirkhÄuser, Boston, to appear.
M. G. Crandall, L. C. Evans, and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations,Trans. Amer. Math. Soc.,282 (1984), 487–502.
L. C. Evans and M. R. James, The Hamilton-Jacobi-Bellman equation for time optimal control,SIAM J. Control Optim.,27 (1989), 1477–1489.
H. Hermes, Control systems which generate decomposable Lie algebras,J. Differential Equations,44 (1982), 166–187.
H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, inDifferential Equations, Stability and Control (S. Elaydi, ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 127, Marcel Dekker, New York, 1991, pp. 249–260.
H. Hermes, Nilpotent and high-order approximations of vector field systems,SIAM Rev. 33 (1991), 238–264.
H. Hermes, Asymptotically stabilizing feedback controls and the nonlinear regulator problem,SIAM J. Control Optim.,29 (1991), 185–196.
H. Hermes, Large time local controllability via homogeneous approximations,SIAM J. Control Optim., to appear.
H. Hermes and M. Kawski,Local Controllability of a Single-Input Affine System (V. Lakshmikantham, ed.), Lectures Notes in Pure and Applied Mathematics, Vol. 109, Marcel Dekker, New York 1987, pp. 235–248.
M. Kawski, Stabilization of nonlinear systems in the plane,Systems Control Lett.,12 (1989), 169–175.
M. Kawski, Asymptotic feedback stabilization in dimension three,Proceedings of the IEEE Conference on Decision and Control, Tampa, FL, Vol. II, 1989, pp. 1370–1375.
M. Kawski,Homogeneous Stabilizing Feedback Laws, Control Theory and Advanced Technology, Vol. 6, Mita Press, Tokyo, pp. 497–516.
E. B. Lee and L. Markus,Foundations of Optimal Control Theory, Wiley, New York, 1967.
R. Rosier, Homogeneous Liapunov function for continuous vector field,Systems Control Lett.,19 (1992), 467–473.
H. J. Sussmann, A general theorem on local controllability,SIAM J. Control Optim.,25 (1987), 158–194.
Author information
Authors and Affiliations
Additional information
This research was supported by NSF Grant DMS 93-01039. Some of the results, herein, were announced at NOCLOS '95.
Rights and permissions
About this article
Cite this article
Hermes, H. Resonance, stabilizing feedback controls, and regularity of viscosity solutions of Hamilton-Jacobi-Bellman equations. Math. Control Signal Systems 9, 59–72 (1996). https://doi.org/10.1007/BF01211518
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01211518