Kybernetika 46 no. 5, 907-925, 2010

Optimal boundary control for hyperdiffusion equation

Hanif Heidari and Alaeddin Malek

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.

Keywords:

hyperdiffusion equation, optimal boundary control, swimming at microscale

Classification:

35K35, 35B37, 49J20

paper.pdf

References:

  1. S. Boyd and L. Vandenberghe: Convex Optimization. Cambridge University Press 2004.   CrossRef
  2. C. Brennen and H. Winet: Fluid mechanics of propulsion by cilia and flagella. Ann. Rev. Fluid Mech.   CrossRef
  3. F. Burk: Lebesgue Measure and Integration: An Itroduction. John Wiley $&$ Sons, 1998.   CrossRef
  4. C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang: Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, 2006.   CrossRef
  5. G. Dimitriu: Numerical approximation of the optimal inputs for an identification problem. Internat. J. Comput. Math.   CrossRef
  6. R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone and J. Bibette: Microscopic artificial swimmers. Nature 437 (2005), 862-865.   CrossRef
  7. F. Fahroo: Optimal placement of controls for a one-dimensional active noise control problem. Kybernetika 34 (1998), 655-665.   CrossRef
  8. M. H. Farahi, J. E. Rubio and D. A. Wilson: The optimal control of the linear wave equation. Internat. J. Control 63 (1996), 833-848.   CrossRef
  9. H. Heidari and A. Malek: Null boundary controllability for hyperdiffusion equation. Internat. J. Appl. Math. {\mi 22} (2009), 615-626.   CrossRef
  10. G. Ji and C. Martin: Optimal boundary control of the heat equation with target function at terminal time. Appl. Math. Comput. 127 (2002), 335-345.   CrossRef
  11. Y. W. Kim and R. R. Netz: Pumping fluids with periodically beating grafted elastic filaments. Phys. Rev. Lett. 96 (2006), 158101.   CrossRef
  12. E. Lauga: Floppy swimming: Viscous locomotion of actuated elastica. Phys. Rev. E. 75 (2007), 041916.   CrossRef
  13. E. Lauga and T. R. Powers: The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (2009), 096601.   CrossRef