Abstract
In this paper, we study intersections of extremals in a linear-quadratic Bolza problem of optimal control. The structure of the inter-sections is described. We show that this structure implies the semipositive definiteness of the quadratic cost functional. In addition, we derive necessary and sufficient conditions for the existence of minimizers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bolt'anski, V. G.,Sufficient Conditions of Optimality and the Justification of the Method of Dynamic Programming, SIAM Journal on Control, Vol. 4, pp. 326–361, 1966.
Young, L. C.,Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders Company, Philadelphia, Pennsylvania, 1969.
Brunovsky, P.,Existence of Regular Synthesis for the Linear-Quadratic Optimal Control with Linear Control Constraints, Journal of Differential Equations, Vol. 38, pp. 344–360, 1980.
Lojasiewicz, S., Jr.,The Structure of Optimal Solutions in Nonlinear Control Systems, Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 32, pp. 485–499, 1984.
Lojasiewicz, S., Jr.,The Existence of Optimal Feedback in Nonlinear Control Systems, Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 32, pp. 500–514, 1984.
Nowakowski, A.,Sufficient Conditions for a Strong Relative Minimum in an Optimal Control Problem, Journal of Optimization Theory and Applications, Vol. 50, pp. 129–147, 1986.
Bryson, A. E., andHo, Y. C.,Applied Optimal Control, Blaisdell, Waltham, Massachusetts, 1969.
Johnson, C. D.,Limits of Propriety for Linear-Quadratic Regulator Problems, International Journal of Control, Vol. 45, pp. 1835–1846, 1987.
Molinari, B. P.,Nonnegativity of a Quadratic Functional, SIAM Journal on Control, Vol. 13, pp. 792–806, 1975.
Coppel, W. A.,Linear-Quadratic Optimal Control, Proceedings of the Royal Society of Edinburgh, Vol. 73A, pp. 271–289, 1974/75.
Hestenes, M. R.,Application of the Theory of Quadratic Forms in Hilbert Space to the Calculus of Variations, Pacific Journal of Mathematics, Vol. 1, pp. 525–581, 1951.
Kogan, J.,Intersections of Extremals in Linear Quadratic Control Problems, Journal of Optimization Theory and Applications, Vol. 56, pp. 407–432, 1988.
Gelfand, I. M., andFomin, S. V.,Calculus of Variations, Prentice-Hall, Englewood Cliffs, New Jersey, 1963.
Jurdjevic, V., andKogan, J.,Optimality of Extremals for Linear Systems with Quadratic Costs, Technical Report No. 66, Center for Applied Mathematics, Purdue University, Lafayette, Indiana, 1987.
Kogan, J.,Structure of Extremals in Optimal Problems, SIAM Journal on Control, Vol. 25, pp. 951–966, 1987.
Caratheodory, C.,Calculus of Variations and Partial Differential Equations of the First Order, Vol. 2, Holden-Day, San Francisco, California, 1967.
Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, Krieger Publishing Company, New York, New York, 1980.
Berkovitz, L. D.,Optimal Control Theory, Springer-Verlag, Berlin, Germany, 1974.
Bliss, G. A.,Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Illinois, 1946.
Author information
Authors and Affiliations
Additional information
Communicated by L. D. Berkovitz
Rights and permissions
About this article
Cite this article
Kogan, J. Structure of minimizers in linear-quadratic Bolza problems of optimal control. J Optim Theory Appl 63, 225–260 (1989). https://doi.org/10.1007/BF00939576
Issue Date:
DOI: https://doi.org/10.1007/BF00939576