Research ArticleOpen AccessGoogle Scholar indexed
A New Approach for a Class of Optimal Control Problems of Volterra Integral Equations
- 1
- 2
- 3
Intelligent Control and Automation·Volume 02 (2011)·Pages 121–125·Published 3 June 2011·DOI10.4236/ica.2011.22014
Copy link · social · email
Abstract
In this paper, we propose a new approach for a class of optimal control problems governed by Volterra integral equations which is based on linear combination property of intervals. We convert the nonlinear terms in constraints of problem to the corresponding linear terms. Discretization method is also applied to convert the new problems to the discrete-time problem. In addition, some numerical examples are presented to illustrate the effectiveness of the proposed approach.
KeywordsVolterra Integral EquationsOptimal ControlLinear Programming
- N. Hritonenko and Y. Yatsenko, “Mathematical Modeling in Economics, Ecology, and the Environment,” Kluwer Academic Publishers, Dordrecht, 1999.
- M. I. Kamien and N. L. Schwartz, “Dynamic Optimization. The Calculus of Variations and Optimal Control in Economics and Management,” North-Holland Publishing Co., Amsterdam, 1991.
- L. W. Neustadt, “Optimization,” Princeton University Press, Princeton, 1976.
- L. W. Neustadt, “Optimization: A Theory of Necessary Conditions,” Princeton University Press, Princeton, New Jersey, 1976.
- L. W. Neustadt and J. Warga, “Comments on the Paper ‘Optimal Control of Processes Described by Integral Equations, I’ by V. R. Vinokurov,” SIAM Journal on Control and Optimization, Vol. 8, 1970, p. 572. doi:10.1137/0308041
- V. L. Bakke, “A Maximum Principle for an Optimal Control Problem with Integral Constraints,” Journal of Optimization Theory and Applications, Vol. 13, No. 1, 1974, pp. 32- 55. doi:10.1007/BF00935608
- D. A. Carlson, “An Elementary Proof of the Maximum Principle for Optimal Control Problems Governed by a Volterra Integral Equation,” Journal of Optimization Theory and Applications, Vol. 54, No. 1, 1987, pp. 43-61. doi:10.1007/BF00940404
- V. R. Vinokurov, “Optimal Control of Processes Described by Integral Equations, Parts I, II, and III,” SIAM Journal of Control, Vol. 7, 1996, pp. 324-355. doi:10.1137/0307022
- N. G. Medhin, “Optimal Processes Governed by Integral Equations,” Journal of Mathematical Analysis and Applications, Vol. 120, No. 1, 1986, pp. 1-12. doi:10.1016/0022-247X(86)90199-X
- W. H. Schmidt, “Notwendige Optimalitaetsbedingungen fuer Prozesse mit Zeitvariablen Integralgleichungen in Banach-Raeumen,” Z. Angew. Math. Mech., Vol. 60, 1980, pp. 595-608. doi:10.1002/zamm.19800601107
- W. H. Schmidt, "Durch Integralgleichungen beschriebene optimale Prozesse mit Nebenbedingungen in Banachraeumen notwendige Optimalitaetsbendingungen," Zeschrift fur angewandte Mathematik und Mechanik, Vol. 62, 1982, pp. 66-75.
- W. H. Schmidt, “Volterra Integral Processes with State Constraints,” Scottish Association for Marine Science, Vol. 9, 1992, pp. 213-224.
- W. H. Schmidt, “Durch Integralgleichungen beschrienbene optimale Prozesse mit Nebenbedingungen in Banachraumen—Notwendige Optimalita tsbedingungen,” Zeschrift fur angewandte Mathematik und Mechanik, Vol. 62, 1982, pp. 65-75.