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October  2011, 7(4): 967-975. doi: 10.3934/jimo.2011.7.967

A variational problem and optimal control

Hang-Chin Lai 1, , Jin-Chirng Lee 1, and  Shuh-Jye Chern 2,

1. 

Chung Yuan Christian University, Chung Li, Taiwan, Taiwan
2. 

National Tsing Hua University, Hsinchu, Taiwan

Received  January 2011 Revised  June 2011 Published  August 2011

A variational problem involving two variables, the state and the control variables, is reduced to another variational problem in which the objective has no control variable, but the constrained identity has one. We then establish that the two problems are equivalent with the same optimal (state) solution under some conditions.
Keywords: dynamic multimapping, bounded compactness condition, separable Banach space, Measurable selection, Radon Nikodym property..
Mathematics Subject Classification: Primary: 46G12, 28B20; Secondary: 54C6.
Citation: Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967
References:
[1]

G. P. Akilov and L. V. Kantorovich, "Functional Analysis," 2nd edition, Translated from the Russian by Howard L. Silcock, Pergamon Press, Oxford-Elmsford, NY, 1982.  Google Scholar

[2]

H. C. Lai, Duality of Banach function spaces and Radon Nikodym property, Acta Mathematics Hungarica, 47 (1986), 45-52. doi: 10.1007/BF01949123.  Google Scholar

[3]

H. C. Lai and J. W. Chen, On the generalized Euler-Lagrange equations, Journal of Mathematical Analysis and Applications, 213 (1997), 681-697.  Google Scholar

[4]

H. C. Lai and J. C. Lee, Convergent Theorems and $L$$p$-selections for Banach-valued multifunctions, Nihonkai Math. Journal, 4 (1993), 163-180.  Google Scholar

[5]

H. C. Lai and J. C. Lee, Integration Theory for Banach-valued multifunctions, Indian Journal of Mathematics, 34 (1992), 265-284.  Google Scholar

[6]

H. C. Lai and J. C. Lee, Integral representations and convergences for Banach-valued multifunctions, Fixed Point Theory and Applications (Halifax, NS, 1991), World Scientific Publ., River Edge, NJ, (1992), 169-188.  Google Scholar

[7]

H. C. Lai and L. J. Lin, The Fenchel-Moreau theorem for set functions, Proceedings of the American Mathematics Society, 103 (1988), 85-90. doi: 10.1090/S0002-9939-1988-0938649-4.  Google Scholar

[8]

C. Olech, Existence theorem in optimal control problems involving multiple integrals, Journal of Differential Equations, 6 (1969), 512-526. doi: 10.1016/0022-0396(69)90007-2.  Google Scholar

[9]

J. P. Raymond, Existence theorems in optimal control problems without convexity assumptions, Journal of Optimization Theory and Applications, 67 (1990), 109-132. doi: 10.1007/BF00939738.  Google Scholar

[10]

R. T. Rockafellar, Existence theorems for general control problems of Bolza and Lagrange, Advances in Mathematics, 15 (1975), 312-333. doi: 10.1016/0001-8708(75)90140-1.  Google Scholar

show all references

References:
[1]

G. P. Akilov and L. V. Kantorovich, "Functional Analysis," 2nd edition, Translated from the Russian by Howard L. Silcock, Pergamon Press, Oxford-Elmsford, NY, 1982.  Google Scholar

[2]

H. C. Lai, Duality of Banach function spaces and Radon Nikodym property, Acta Mathematics Hungarica, 47 (1986), 45-52. doi: 10.1007/BF01949123.  Google Scholar

[3]

H. C. Lai and J. W. Chen, On the generalized Euler-Lagrange equations, Journal of Mathematical Analysis and Applications, 213 (1997), 681-697.  Google Scholar

[4]

H. C. Lai and J. C. Lee, Convergent Theorems and $L$$p$-selections for Banach-valued multifunctions, Nihonkai Math. Journal, 4 (1993), 163-180.  Google Scholar

[5]

H. C. Lai and J. C. Lee, Integration Theory for Banach-valued multifunctions, Indian Journal of Mathematics, 34 (1992), 265-284.  Google Scholar

[6]

H. C. Lai and J. C. Lee, Integral representations and convergences for Banach-valued multifunctions, Fixed Point Theory and Applications (Halifax, NS, 1991), World Scientific Publ., River Edge, NJ, (1992), 169-188.  Google Scholar

[7]

H. C. Lai and L. J. Lin, The Fenchel-Moreau theorem for set functions, Proceedings of the American Mathematics Society, 103 (1988), 85-90. doi: 10.1090/S0002-9939-1988-0938649-4.  Google Scholar

[8]

C. Olech, Existence theorem in optimal control problems involving multiple integrals, Journal of Differential Equations, 6 (1969), 512-526. doi: 10.1016/0022-0396(69)90007-2.  Google Scholar

[9]

J. P. Raymond, Existence theorems in optimal control problems without convexity assumptions, Journal of Optimization Theory and Applications, 67 (1990), 109-132. doi: 10.1007/BF00939738.  Google Scholar

[10]

R. T. Rockafellar, Existence theorems for general control problems of Bolza and Lagrange, Advances in Mathematics, 15 (1975), 312-333. doi: 10.1016/0001-8708(75)90140-1.  Google Scholar

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