American Institute of Mathematical Sciences

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March  2016, 6(1): 53-94. doi: 10.3934/mcrf.2016.6.53

Optimal sampled-data control, and generalizations on time scales

Loïc Bourdin 1, and  Emmanuel Trélat 2,

1. 

Université de Limoges, Institut de recherche XLIM, Département de Mathématiques et d'Informatique, CNRS UMR 7252, Limoges, France
2. 

Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris

Received  January 2015 Revised  October 2015 Published  January 2016

In this paper, we derive a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. Our framework is actually much more general, and we treat optimal control problems for which the state variable evolves on a given time scale (arbitrary non-empty closed subset of $\mathbb{R}$), and the control variable evolves on a smaller time scale. Sampled-data systems are then a particular case. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle.
Keywords: Optimal control, sampled-data, Pontryagin maximum principle, time scale..
Mathematics Subject Classification: 49J15, 93C57, 34N99, 39A1.
Citation: Loïc Bourdin, Emmanuel Trélat. Optimal sampled-data control, and generalizations on time scales. Mathematical Control & Related Fields, 2016, 6 (1) : 53-94. doi: 10.3934/mcrf.2016.6.53
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show all references

References:
[1]

Results Math., 35 (1999), 3-22. doi: 10.1007/BF03322019.  Google Scholar

[2]

Math. Inequal. Appl., 4 (2001), 535-557. doi: 10.7153/mia-04-48.  Google Scholar

[3]

Adv. Difference Equ., (2006), Art. ID 38121, 14pp.  Google Scholar

[4]

Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[5]

In Proceedings of the Sixth International Conference on Difference Equations, 239-252, CRC, Boca Raton, FL, 2004.  Google Scholar

[6]

Math. Comput. Modelling, 43 (2006), 718-726. doi: 10.1016/j.mcm.2005.08.014.  Google Scholar

[7]

J. Math. Anal. Appl., 342 (2008), 1220-1226. doi: 10.1016/j.jmaa.2008.01.018.  Google Scholar

[8]

Dynam. Systems Appl., 13 (2004), 339-349.  Google Scholar

[9]

Comput. Math. Appl., 54 (2007), 45-57. doi: 10.1016/j.camwa.2006.10.032.  Google Scholar

[10]

Birkhäuser Boston Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[11]

Birkhäuser Boston Inc., Boston, MA, 2003. doi: 10.1007/978-0-8176-8230-9.  Google Scholar

[12]

John Wiley & Sons, New York-Toronto, Ont., 1978.  Google Scholar

[13]

Springer Verlag, 2003.  Google Scholar

[14]

Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37640-2.  Google Scholar

[15]

J. Difference Equ. Appl., 20 (2014), 526-547. doi: 10.1080/10236198.2013.862358.  Google Scholar

[16]

SIAM J. Control Optim., 51 (2013), 3781-3813. doi: 10.1137/130912219.  Google Scholar

[17]

J. Math. Anal. Appl., 411 (2014), 543-554. doi: 10.1016/j.jmaa.2013.10.013.  Google Scholar

[18]

Springfield, MO, 2007.  Google Scholar

[19]

Springer, New York, 2011.  Google Scholar

[20]

Hemisphere Publishing Corp. Washington, D. C., 1975. Optimization, estimation, and control, Revised printing.  Google Scholar

[21]

Texts in Applied Mathematics, 49, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[22]

J. Difference Equ. Appl., 11 (2005), 1013-1028. doi: 10.1080/10236190500272830.  Google Scholar

[23]

Math. Comput. Modelling, 43 (2006), 194-207. doi: 10.1016/j.mcm.2005.09.028.  Google Scholar

[24]

McGraw-Hill Book Co., New York, 1970.  Google Scholar

[25]

Applications of Mathematics, 17, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[26]

J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[27]

John Wiley & Sons, New York, 1964. Google Scholar

[28]

In Mathematical control theory and finance, pages 149-159. Springer, Berlin, 2008. doi: 10.1007/978-3-540-69532-5_9.  Google Scholar

[29]

Bull. Math. Biol., 64 (2002), 611-620. doi: 10.1006/bulm.2002.0286.  Google Scholar

[30]

In Mathematical events of the twentieth century, pages 85-99. Springer, Berlin, 2006. doi: 10.1007/3-540-29462-7_5.  Google Scholar

[31]

J. Math. Anal. Appl., 285 (2003), 107-127. doi: 10.1016/S0022-247X(03)00361-5.  Google Scholar

[32]

SIAM J. Control, 4 (1966), 90-111. doi: 10.1137/0304009.  Google Scholar

[33]

Robert E. Krieger Publishing Co. Inc., Huntington, N.Y., 1980. Corrected reprint of the 1966 original.  Google Scholar

[34]

PhD thesis, Universität Würzburg, 1988. Google Scholar

[35]

J. Math. Anal. Appl., 289 (2004), 143-166. doi: 10.1016/j.jmaa.2003.09.031.  Google Scholar

[36]

Comput. Math. Appl., 62 (2011), 3490-3503. doi: 10.1016/j.camwa.2011.08.065.  Google Scholar

[37]

Nonlinear Anal., 70 (2009), 3209-3226. doi: 10.1016/j.na.2008.04.025.  Google Scholar

[38]

Analysis (Munich), 28 (2008), 1-28. doi: 10.1524/anly.2008.0900.  Google Scholar

[39]

IEEE Trans. Automatic Control, AC-11 (1966), 30-35.  Google Scholar

[40]

Opuscules. Ellipses, Paris, 2008. Google Scholar

[41]

SIAM J. Control, 4 (1966), 263-275. doi: 10.1137/0304023.  Google Scholar

[42]

Cambridge Studies in Advanced Mathematics, 52, Cambridge University Press, 1997.  Google Scholar

[43]

John Wiley, New York, 1967.  Google Scholar

[44]

Nature, 261 (1976), 459-467. Google Scholar

[45]

Volumes 330 and 331 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006.  Google Scholar

[46]

Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.  Google Scholar

[47]

Interdisciplinary Applied Mathematics, Vol. 38, Springer, 2012. doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[48]

North-Holland Publishing Co., Amsterdam, Volume 24, 1987.  Google Scholar

[49]

Kluwer Academic Publishers, Boston, MA, second edition, 2000.  Google Scholar

[50]

Mathématiques Concrètes. Vuibert, Paris, 2005.  Google Scholar

[51]

J. Optim. Theory Appl., 154 (2012), 713-758. doi: 10.1007/s10957-012-0050-5.  Google Scholar

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