Geodesic
Optimal boundary control for hyperdiffusion equation
Kybernetika, Tome 46 (2010) no. 5, pp. 907-925
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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.
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.
Classification : 35B37, 35K35, 49J20
Keywords: hyperdiffusion equation; optimal boundary control; swimming at microscale 
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Heidari, Hanif; Malek, Alaeddin. Optimal boundary control for hyperdiffusion equation. Kybernetika, Tome 46 (2010) no. 5, pp. 907-925. http://geodesic.mathdoc.fr/item/KYB_2010_46_5_a6/

[1] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press 2004. | MR | Zbl

[2] Brennen, C., Winet, H.: Fluid mechanics of propulsion by cilia and flagella. Ann. Rev. Fluid Mech. 9 (1977), 339–398. | DOI | Zbl

[3] Burk, F.: Lebesgue Measure and Integration: An Itroduction. John Wiley $\&$ Sons, 1998. | MR

[4] Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A.: Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, 2006. | MR | Zbl

[5] Dimitriu, G.: Numerical approximation of the optimal inputs for an identification problem. Internat. J. Comput. Math. 70 (1998), 197–209. | DOI | MR | Zbl

[6] Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A., Bibette, J.: Microscopic artificial swimmers. Nature 437 (2005), 862–865. | DOI

[7] Fahroo, F.: Optimal placement of controls for a one-dimensional active noise control problem. Kybernetika 34 (1998), 655–665. | MR

[8] Farahi, M. H., Rubio, J. E., Wilson, D. A.: The optimal control of the linear wave equation. Internat. J. Control 63 (1996), 833–848. | DOI | MR | Zbl

[9] Heidari, H., Malek, A.: Null boundary controllability for hyperdiffusion equation. Internat. J. Appl. Math. 22 (2009), 615–626. | MR | Zbl

[10] Ji, G., Martin, C.: Optimal boundary control of the heat equation with target function at terminal time. Appl. Math. Comput. 127 (2002), 335–345. | DOI | MR | Zbl

[11] Kim, Y. W., Netz, R. R.: Pumping fluids with periodically beating grafted elastic filaments. Phys. Rev. Lett. 96 (2006), 158101. | DOI

[12] Lauga, E.: Floppy swimming: Viscous locomotion of actuated elastica. Phys. Rev. E. 75 (2007), 041916. | DOI | MR

[13] Lauga, E., Powers, T. R.: The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (2009), 096601. | DOI | MR

[14] Lions, J. L., Magenes, E.: Non-homogeneous Boundary Value Problem and Applications. Springer-Verlag, 1972.

[15] Machin, K. E.: The control and synchronization of flagellar movement. Proc. Roy. Soc. B. 158 (1963), 88–104. | DOI

[16] Machin, K. E.: Wave propagation along flagella. J. Exp. Biol. 35 (1985), 796–806.

[17] Mordukhovich, B. S., Raymond, J. P.: Optimal boundary control of hyperbolic equations with pointwise state constraints. Nonlinear Analysis 63 (2005), 823–830. | DOI | MR | Zbl

[18] Park, H. M., Lee, M. W., Jang, Y. D.: An efficient computational method of boundary optimal control problems for the burgers equation. Comput. Meth. Appl. Mech. Engrg. 166 (1998), 289–308. | DOI | MR | Zbl

[19] Purcell, E. M.: Life at low Reynolds number. Amer. J. Phys. 45 (1977), 3–11. | DOI

[20] Reju, S. A., Evans, D. J.: Computational results of the optimal control of the diffusion equation with the extended conjugate gradient algorithm. Internat. J. Comput Math. 75 (2000), 247–258. | DOI | MR | Zbl

[21] Rektorys, K.: Variational Methods in Mathematics, Sciences and Engineering. D. Reidel Publishing Company, 1977. | MR

[22] Sakthivel, K., Balachandran, K., Sowrirajan, R., Kim, J-H.: On exact null controllability of black scholes equation. Kybernetika 44 (2008), 685–704. | MR | Zbl

[23] Wiggins, C. H., Riveline, D., Ott, A., Goldstein, R. E.: Trapping and wiggling: Elastohydrodynamics of driven microfilaments. Biophys. J. 74 (1998), 1043–1060. | DOI

[24] Williams, P.: A Gauss–Lobatto quadrature method for solving optimal control problems. ANZIAM 47 (2006), C101–C115. | MR

[25] Yu, T. S., Lauga, E., Hosoi, A. E.: A experimental investigations of elastic tail propulsion at low Reynolds number. Phys. Fluids 18 (2006), 091701. | DOI