A New Approach for a Class of Optimal Control Problems of Volterra Integral Equations

Mohammad Hadi Noori Skandari, Hamid Reza Erfanian, Ali Vahidian Kamyad

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DOI: 10.4236/ica.2011.22014                

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.

Keywords

Volterra Integral Equations, Optimal Control, Linear Programming

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	N. Hritonenko and Y. Yatsenko, “Mathematical Modeling in Economics, Ecology, and the Environment,” Kluwer Academic Publishers, Dordrecht, 1999.
[2]	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.
[3]	L. W. Neustadt, “Optimization,” Princeton University Press, Princeton, 1976.
[4]	L. W. Neustadt, “Optimization: A Theory of Necessary Conditions,” Princeton University Press, Princeton, New Jersey, 1976.
[5]	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
[6]	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
[7]	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
[8]	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
[9]	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
[10]	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
[11]	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.
[12]	W. H. Schmidt, “Volterra Integral Processes with State Constraints,” Scottish Association for Marine Science, Vol. 9, 1992, pp. 213-224.
[13]	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.
[14]	L. Wolfersdorf, “Optimale Steuerung bei Hammerstein schen Integralgleichungen mit Schwach Singulaeren Kernen,” Math. Oper. Statist., Vol. 6, 1975, pp. 609-626.
[15]	G. N. Elnagar, M. A. Kazemi and Hoonjoo Kim, “Necessary and Sufficient Optimality Condittions for Control Systems Described by Integral Equations with Delay,” Journal of the Korean Mathematical Society, Vol. 37, No. 4, 2000, pp. 625-643.
[16]	L. P. Pan and K. L. Teo, “Near-Optimal Controls of a Class of Volterra Integral Systems1,” Journal of Optimization Theory and Applications, Vol. 101, No. 2, 1999, pp. 355-373. doi:10.1023/A:1021741627449
[17]	T. S. Angell, “Existence of Optimal Control without Convexity and a Bangbang Theorem for Linear Volterra Equations,” Journal of Optimization Theory and Applications, Vol. 19, No. 1, 1976, pp. 63-79. doi:10.1007/BF00934052
[18]	T. S. Angell, “On the Optimal Control of Systems Governed by Nonlinear Volterra Equations,” Journal of Optimization Theory and Applications, Vol. 19, No. 1, 1976, pp. 29-45. doi:10.1007/BF00934050
[19]	S. A. Belbas, “Iterative Schemes for Optimal Control of Volterra Integral Equations,” Nonlinear Analysis, Vol. 37, No. 1, 1999, pp. 57-79. doi:10.1016/S0362-546X(98)00144-8
[20]	S. A. Belbas, “A New Method for Optimal Control of Volterra Integral Equations,” Applied Mathematics and Computation, Vol. 189, No. 1, 2007, pp. 1902-1915. doi:10.1016/j.amc.2006.12.077
[21]	G. Carlier and R. Tahraoui, “On Some Optimal Control Problems Governed by a State Equation with Memory,” ESAIM: Control, Optimisation and Calculus of Variations, Vol. 14, No. 4, 2008, pp. 725-743. doi:10.1051/cocv:2008005
[22]	C. Burnap and M. Kazemi, “Optimal Control of a System Governed by Nonlinear Volterra Integral Equations with Delay,” IMA Journal of Mathematical Control and Information, Vol. 16, No. 1, 1999, pp. 73-89. doi:10.1093/imamci/16.1.73
[23]	C. de la Vega, “Necessary Conditions for Optimal Terminal Time Control Problems Governed by a Volterra Integral Equation,” Journal of Optimization Theory and Applications, Vol. 130, No. 1, 2006, pp. 79-93. doi:10.1007/s10957-006-9087-7
[24]	J. F. Bonnens and C. de la Vega, “Optimal Control of Stated Constrained Integral Equations,” Institut national de Recherche en Informatiqeue et en Automatique, 2010.
[25]	S. A. Belbas, “A Reduction Method for Optimal Control of Volterra Integral Equations,” Applied Mathematics and Computation, Vol. 197, No. 2, 2008, pp. 880-890. doi:10.1016/j.amc.2007.08.093
[26]	D. Luenberger, “Linear and Nonlinear Programming,” Addison-Wesley, New York, 1984.