Abstract
A controllability minimum principle and two associated transversality conditions are presented, dealing with the controllability of nonlinear systems. The theorems represent necessary conditions for a control function to generate a system path which lies in the boundary of the set of points that are controllable to a target.
The theorems presented here are controllability counterparts to Pontryagin's maximum principle, and undoubtedly these results will seem familiar or may have occurred to other researchers in the area of optimal control. The purpose of this paper is to make the distinction explicit and to establish the validity of these controllability theorems on their own merits.
The theorems are demonstrated using a simple example and the principal result (a controllability minimum principle) is shown to be equivalent to the Kalman controllability criterion for linear systems.
Similar content being viewed by others
References
Kalman, R. E., Ho, Y. C., andNarendra, K. S.,Controllability of Linear Dynamical Systems, Contributions to Differential Equations, Vol. 1, No. 2, 1962.
Blaquière, A., andLeitmann, G.,On the Geometry of Optimal Processes, Topics in Optimization, Edited by G. Leitmann, Academic Press, New York, New York, 1967.
Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., andMishchenko, E. F.,The Mathematical Theory of Optimal Processes, John Wiley and Sons, New York, New York, 1962.
Leitmann, G., Professor, Department of Mechanical Engineering, University of California, Berkeley, Personal Correspondence, 1972.
Berkovitz, L. D.,On Control Problems with Bounded State Variables, Journal of Mathematical Analysis and Applications, Vol. 5, No. 3, 1962.
Bryson, A. E., Denham, W. F., and Dreyfus, S. E.,Optimal Programming Problems with Inequality Constraints, I, Necessary Conditions for External Solutions, AIAA Journal, Vol. 1, No. 11, 1963.
Leitmann, G.,An Introduction to Optimal Control, McGraw-Hill Book Company, New York, New York, 1966.
Litt, F. X.,Some Remarks on State-Constrained Optimal Control Problems, International Journal on Control, Vol. 17, No. 1, 1973.
Vincent, T. L., Cliff, E. M., Grantham, W. J., andPeng, W. Y.,A Problem of Collision Avoidance, University of Arizona, Engineering Experiment Station, Report No. 39, 1972.
Coddington, E. A., andLevinson, N.,Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York, New York, 1955.
Halkin, H.,Mathematical Foundations of Systems Optimization, Topics in Optimization, Edited by G. Leitmann, Academic Press, New York, New York, 1967.
Lee, E. B., andMarkus, L.,Foundations of Optimal Control Theory, John Wiley and Sons, New York, New York, 1967.
Grantham, W. J.,A Controllability Minimum Principle, Ph.D. Dissertation, University of Arizona, Tucson, Arizona, 1973.
Kuhn, H. W., andTucker, A. W.,Nonlinear Programming, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California, Berkeley, California, 1951.
Buck, R. C.,Advanced Calculus, McGraw-Hill Book Company, New York, New York, 1965.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
Rights and permissions
About this article
Cite this article
Grantham, W.J., Vincent, T.L. A controllability minimum principle. J Optim Theory Appl 17, 93–114 (1975). https://doi.org/10.1007/BF00933917
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00933917