Abstract
The present paper is concerned with the study of quadratic control problems on linear spaces. In particular, we are concerned with the conditions under which a quadratic criterion function is positive on certain linear spaces. This involves the elementary theory of conjugate and focal points, the existence of a conjugate system with a nonvanishing determinant, and the existence of extremal fields. The results given are in part a translation into control language of known theory for the problem of Bolza. The method used is based on the Hilbert space techniques developed earlier by the author.
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References
Hestenes, M. R.,Applications of the Theory of Quadratic Forms in Hilbert Space to the Calculus of Variations, Pacific Journal of Mathematics, Vol. 1, pp. 525–582, 1951.
Hestenes, M. R.,Quadratic Variational Theory and Linear Elliptic Partial Differential Equations, Transactions of the American Mathematical Society, Vol. 101, pp. 306–350, 1961.
Mikami, E.,Quadratic Optimal Control Problems, Ph.D. Thesis, University of California, Los Angeles, California, 1968.
Hestenes, M. R.,Quadratic Variational Theory, Control Theory and the Calculus of Variations, Edited by A. V. Balakrishnan, Academic Press, New York, New York, 1969.
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The preparation of this paper was sponsored in part by the U.S. Army Research Office under Grant DA-31-124-ARO(D)-355 and in part by the Office of Naval Research under Contract NONR-233(76).
Paper presented to the All-Union Conference on Optimal Control and Minimal Surfaces, Tbilisi, USSR, 1969.
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Hestenes, M.R. Quadratic control problems. J Optim Theory Appl 17, 1–42 (1975). https://doi.org/10.1007/BF00933915
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DOI: https://doi.org/10.1007/BF00933915