Optimal boundary control for hyperdiffusion equation

Hanif Heidari; Alaeddin Malek

Kybernetika (2010)

  • Volume: 46, Issue: 5, page 907-925
  • ISSN: 0023-5954

Abstract

top
In this paper, we consider the solution of optimal control problem for hyperdiffusion equation involving boundary function of continuous time variable in its cost function. A specific direct approach based on infinite series of Fourier expansion in space and temporal integration by parts for analytical solution is proposed to solve optimal boundary control for hyperdiffusion equation. The time domain is divided into number of finite subdomains and optimal function is estimated at each subdomain to obtain desired state with minimum energy. Proposed method has high flexibility so that decision makers are able to trace optimal control in a prescribed subinterval. The implementation of the theory is presented and the effectiveness of the boundary control is investigated by some numerical examples.

How to cite

top

Heidari, Hanif, and Malek, Alaeddin. "Optimal boundary control for hyperdiffusion equation." Kybernetika 46.5 (2010): 907-925. <http://eudml.org/doc/196455>.

@article{Heidari2010,
abstract = {In this paper, we consider the solution of optimal control problem for hyperdiffusion equation involving boundary function of continuous time variable in its cost function. A specific direct approach based on infinite series of Fourier expansion in space and temporal integration by parts for analytical solution is proposed to solve optimal boundary control for hyperdiffusion equation. The time domain is divided into number of finite subdomains and optimal function is estimated at each subdomain to obtain desired state with minimum energy. Proposed method has high flexibility so that decision makers are able to trace optimal control in a prescribed subinterval. The implementation of the theory is presented and the effectiveness of the boundary control is investigated by some numerical examples.},
author = {Heidari, Hanif, Malek, Alaeddin},
journal = {Kybernetika},
keywords = {hyperdiffusion equation; optimal boundary control; swimming at microscale; swimming at microscale; Fourier expansion in space; temporal integration by parts; numerical examples},
language = {eng},
number = {5},
pages = {907-925},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Optimal boundary control for hyperdiffusion equation},
url = {http://eudml.org/doc/196455},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Heidari, Hanif
AU - Malek, Alaeddin
TI - Optimal boundary control for hyperdiffusion equation
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 5
SP - 907
EP - 925
AB - In this paper, we consider the solution of optimal control problem for hyperdiffusion equation involving boundary function of continuous time variable in its cost function. A specific direct approach based on infinite series of Fourier expansion in space and temporal integration by parts for analytical solution is proposed to solve optimal boundary control for hyperdiffusion equation. The time domain is divided into number of finite subdomains and optimal function is estimated at each subdomain to obtain desired state with minimum energy. Proposed method has high flexibility so that decision makers are able to trace optimal control in a prescribed subinterval. The implementation of the theory is presented and the effectiveness of the boundary control is investigated by some numerical examples.
LA - eng
KW - hyperdiffusion equation; optimal boundary control; swimming at microscale; swimming at microscale; Fourier expansion in space; temporal integration by parts; numerical examples
UR - http://eudml.org/doc/196455
ER -

References

top
  1. Boyd, S., Vandenberghe, L., Convex Optimization, Cambridge University Press 2004. (2004) Zbl1058.90049MR2061575
  2. Brennen, C., Winet, H., 10.1146/annurev.fl.09.010177.002011, Ann. Rev. Fluid Mech. 9 (1977), 339–398. (1977) Zbl0431.76100DOI10.1146/annurev.fl.09.010177.002011
  3. Burk, F., Lebesgue Measure and Integration: An Itroduction, John Wiley Sons, 1998. (1998) MR1478419
  4. Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A., Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, 2006. (2006) Zbl1093.76002MR2223552
  5. Dimitriu, G., 10.1080/00207169808804746, Internat. J. Comput. Math. 70 (1998), 197–209. (1998) Zbl0915.65069MR1712493DOI10.1080/00207169808804746
  6. Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A., Bibette, J., 10.1038/nature04090, Nature 437 (2005), 862–865. (2005) DOI10.1038/nature04090
  7. Fahroo, F., Optimal placement of controls for a one-dimensional active noise control problem, Kybernetika 34 (1998), 655–665. (1998) MR1695369
  8. Farahi, M. H., Rubio, J. E., Wilson, D. A., 10.1080/00207179608921871, Internat. J. Control 63 (1996), 833–848. (1996) Zbl0841.49001MR1652573DOI10.1080/00207179608921871
  9. Heidari, H., Malek, A., Null boundary controllability for hyperdiffusion equation, Internat. J. Appl. Math. 22 (2009), 615–626. (2009) Zbl1177.93018MR2537040
  10. Ji, G., Martin, C., 10.1016/S0096-3003(01)00011-X, Appl. Math. Comput. 127 (2002), 335–345. (2002) Zbl1040.49037MR1883857DOI10.1016/S0096-3003(01)00011-X
  11. Kim, Y. W., Netz, R. R., 10.1103/PhysRevLett.96.158101, Phys. Rev. Lett. 96 (2006), 158101. (2006) DOI10.1103/PhysRevLett.96.158101
  12. Lauga, E., 10.1103/PhysRevE.75.041916, Phys. Rev. E. 75 (2007), 041916. (2007) MR2358588DOI10.1103/PhysRevE.75.041916
  13. Lauga, E., Powers, T. R., 10.1088/0034-4885/72/9/096601, Rep. Prog. Phys. 72 (2009), 096601. (2009) MR2539632DOI10.1088/0034-4885/72/9/096601
  14. Lions, J. L., Magenes, E., Non-homogeneous Boundary Value Problem and Applications, Springer-Verlag, 1972. (1972) 
  15. Machin, K. E., 10.1098/rspb.1963.0036, Proc. Roy. Soc. B. 158 (1963), 88–104. (1963) DOI10.1098/rspb.1963.0036
  16. Machin, K. E., Wave propagation along flagella, J. Exp. Biol. 35 (1985), 796–806. (1985) 
  17. Mordukhovich, B. S., Raymond, J. P., 10.1016/j.na.2004.12.017, Nonlinear Analysis 63 (2005), 823–830. (2005) Zbl1153.49315MR2188155DOI10.1016/j.na.2004.12.017
  18. Park, H. M., Lee, M. W., Jang, Y. D., 10.1016/S0045-7825(98)00092-9, Comput. Meth. Appl. Mech. Engrg. 166 (1998), 289–308. (1998) Zbl0949.76024MR1659187DOI10.1016/S0045-7825(98)00092-9
  19. Purcell, E. M., 10.1119/1.10903, Amer. J. Phys. 45 (1977), 3–11. (1977) DOI10.1119/1.10903
  20. Reju, S. A., Evans, D. J., 10.1080/00207160008804980, Internat. J. Comput Math. 75 (2000), 247–258. (2000) Zbl0961.65064MR1787282DOI10.1080/00207160008804980
  21. Rektorys, K., Variational Methods in Mathematics, Sciences and Engineering, D. Reidel Publishing Company, 1977. (1977) MR0487653
  22. Sakthivel, K., Balachandran, K., Sowrirajan, R., Kim, J-H., On exact null controllability of black scholes equation, Kybernetika 44 (2008), 685–704. (2008) Zbl1177.93021MR2479312
  23. Wiggins, C. H., Riveline, D., Ott, A., Goldstein, R. E., 10.1016/S0006-3495(98)74029-9, Biophys. J. 74 (1998), 1043–1060. (1998) DOI10.1016/S0006-3495(98)74029-9
  24. Williams, P., A Gauss–Lobatto quadrature method for solving optimal control problems, ANZIAM 47 (2006), C101–C115. (2006) MR2242566
  25. Yu, T. S., Lauga, E., Hosoi, A. E., 10.1063/1.2349585, Phys. Fluids 18 (2006), 091701. (2006) DOI10.1063/1.2349585

NotesEmbed ?

top

You must be logged in to post comments.