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

Optimal sampled-data control, and generalizations on time scales

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.
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
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

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

[1]

Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041

[2]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[3]

Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021076

[4]

Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373

[5]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

[6]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014

[7]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012

[8]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[9]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007

[10]

Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215

[11]

Muhammad Ajmal, Xiande Zhang. New optimal error-correcting codes for crosstalk avoidance in on-chip data buses. Advances in Mathematics of Communications, 2021, 15 (3) : 487-506. doi: 10.3934/amc.2020078

[12]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073

[13]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011

[14]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[15]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040

[16]

John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026

[17]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009

[18]

Christian Meyer, Stephan Walther. Optimal control of perfect plasticity part I: Stress tracking. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021022

[19]

Jaouad Danane. Optimal control of viral infection model with saturated infection rate. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 363-375. doi: 10.3934/naco.2020031

[20]

Vladimir Gaitsgory, Ilya Shvartsman. Linear programming estimates for Cesàro and Abel limits of optimal values in optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021102

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]