Abstract
A class of singular control problems involving amplitude constraints on the controls is examined. IfL ∞ is the space of control functionsU, the control constraint setS can be identified with the unit ball inL ∞. Now, for anyn ∈ (1, ∞), an analogous problem may be set up withL n forU and the unit ball inL n forS. This modified problem is necessarily nonsingular for controllable systems. It is shown that, by takingn sufficiently large, the solution to the modified problem also solves the original problem arbitrarily closely (in a sense made precise). Behavior asn → ∞ is investigated.
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References
Taylor, A. E.,Introduction to Functional Analysis, John Wiley and Sons, New York, New York, 1958.
Apostol, T. M.,Mathematical Analysis, Addison-Wesley Publishing Company, Reading, Massachusetts, 1965.
Kreindler, E., andSarachik, P. E.,On the Concepts of Controllability and Observability of Linear Systems, IEEE Transactions on Automatic Control, Vol. 9, No. 2, 1964.
Dunford, N., andSchwartz, J. T.,Linear Operators, Vol. 1, John Wiley and Sons, Interscience Publishers, New York, New York, 1958.
Porter, W. A.,A Minimization Problem and its Application to Optimal Control and System Sensitivity, SIAM Journal on Control, Vol. 6, No. 2, 1968.
Athans, M., andFalb, P. L.,Optimal Control, McGraw-Hill Book Company, New York, New York, 1966.
Johnson, C. D.,Singular Solutions in Problems of Optimal Control, Advances in Control Systems, Vol. 2, Edited by C. T. Leondes, Academic Press, New York, New York, 1965.
Wonham, W. H., andJohnson, C. D.,Optimum Bang-Bang Control with Quadratic Performance Index, Journal of Basic Engineering, Vol. 86, No. 1, 1964.
Kopp, R. E., andMoyer, H. G.,Necessary Conditions for Singular Extremals, AIAA Journal, Vol. 3, No. 1, 1965.
Jacobson, D. H.,A New Necessary Condition of Optimality for Singular Control Problems, SIAM Journal on Control, Vol. 7, No. 4, 1969.
Gilbert, E. G.,An Iterative Procedure for Computing the Minimum of a Quadratic Form on a Convex Set, SIAM Journal on Control, Vol. 4, No. 1, 1966.
Ho, Y. C.,A Successive Approximation Technique for Optimal Control Systems Subject to Input Saturation, Journal of Basic Engineering, Vol. 84, No. 1, 1962.
Jacobson, D. H., Gershwin, S. B., andLele, M. M.,Computation of Optimal Singular Controls, IEEE Transactions on Automatic Control, Vol. 15, No. 1, 1970.
Jacobson, D. H., andSpeyer, J. L.,Necessary and Sufficient Conditions for Optimality for Singular Control Problems: A Limit Approach, Journal of Mathematical Analysis and Applications, Vol. 34, No. 2, 1971.
Neustadt, L. W., andPaiewonsky, B. H.,On Synthesizing Optimal Controls, Paper presented at 2nd IFAC Congress, Basle, Switzerland, 1963.
Fletcher, R., andPowell, M. J. D.,A Rapidly Convergent Descent Method for Minimization, Computer Journal, Vol. 6, No. 1, 1963.
Hille, E., andPhillips, R. S.,Functional Analysis and Semi-Groups, Colloquium Publications of the American Mathematical Society, Providence, Rhode Island, 1957.
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Communicated by S. E. Dreyfus
This research was supported by the Science Research Council of Great Britain and the Commonwealth Fund (Harkness Fellowship).
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Vinter, R.B. Approximate solution of a class of singular control problems. J Optim Theory Appl 13, 461–483 (1974). https://doi.org/10.1007/BF00934941
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DOI: https://doi.org/10.1007/BF00934941