Abstract
For a closed target setS⊏ℝn and a control system (formulated as a differential inclusion and defined nearS), the present paper considers a sufficient condition for the property that every point nearS can be steered toS in finite time by some trajectory of the system. Estimates are obtained revealing how fast some such trajectory is nearing the target. A strong form of this condition is shown to imply that every trajectory of the system hits the target. With a further assumption on the target setS, we also consider conditions that guarantee that some trajectories enter the interior ofS.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Petrov, N. N.,Controllability of Autonomous Systems, Differential Equations, Vol. 4, pp. 311–317, 1968.
Bardi, M., andFalcone, M.,An Approximation Scheme for the Minimum Time Function, SIAM Journal of Control and Optimization, Vol. 28, pp. 950–965, 1990.
Cannarsa, P., andSinestrari, C.,Convexity Properties of the Minimum Time Function, Preprint.
Clarke, F. H., andLedyaev, Yu. S.,Mean-Value Inequalities, Proceedings of the American Mathematical Society, Vol. 122, pp. 1075–1083, 1994.
Clarke, F. H., Ledyaev, Yu. S., andWolenski, P. R.,Proximal Analysis and Minimization Principles, Journal of Mathematical Analysis and Applications (to appear).
Clarke, F. H.,Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, Vol. 57, 1989.
Loewen, P. D.,Optimal Control via Nonsmooth Analysis, CRM Proceedings and Lecture Notes, American Mathematical Society, Providence, Rhode Island, 1993.
Clarke, F. H., andLedyaev, Yu. S.,Mean-Value Inequalities in Hilbert Space, Transactions of the American Mathematical Society, Vol. 344, pp. 307–324, 1994.
Clarke, F. H., Stern, R. J., andWolenski, P. R.,Proximal Smoothness and the Lower C 2 Property, Journal of Convex Analysis (to appear).
Clarke, F. H.,Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, New York, 1983. Second Edition: Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, Vol. 5, 1990.
Deimling, K.,Multivalued Differential Equations, De Gruyter, Berlin, Germany, 1992.
Clarke, F. H., Ledyaev, Yu. S., Stern, R. J., andWolenski, P. R.,Qualitative Properties of Differential Inclusions: A Survey, Journal of Dynamical and Control Systems, Vol. 1, pp. 1–48, 1995.
Haddad, G.,Monotone Trajectories of Differential Inclusions and Functional Differential Inclusions with Memory, Israel Journal of Mathematics, Vol. 39, pp. 83–100, 1981.
Aubin, J. P.,Viability Theory, Birkhauser, Boston, Massachusetts, 1991.
Wolenski, P. R.,A Uniqueness Theorem for Differential Inclusions, Journal of Differential Equations, Vol. 84, pp. 165–182, 1990.
Rockafellar, R. T.,Clarke's Tangent Cones and the Boundaries of Closed Sets in ℝn, Nonlinear Analysis, Vol. 3, pp. 145–154, 1979.
Veliov, V. M.,Sufficient Conditions for Viability Under Imperfect Measurement, Set-Valued Analysis, Vol. 1, pp. 305–317, 1993.
Yue, R.,On the Properties of Bellman's Function in Time Optimal Control Problems, Preprint.
Author information
Authors and Affiliations
Additional information
Communicated by L. D. Berkovitz
This work was supported by the Natural Sciences and Engineering Research Council of Canada and by FCAR, Québec, Canada.
Rights and permissions
About this article
Cite this article
Clarke, F.H., Wolenski, P.R. Control of systems to sets and their interiors. J Optim Theory Appl 88, 3–23 (1996). https://doi.org/10.1007/BF02192020
Issue Date:
DOI: https://doi.org/10.1007/BF02192020