Previous |  Up |  Next


hyperdiffusion equation; optimal boundary control; swimming at microscale
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.
[1] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press 2004. MR 2061575 | Zbl 1058.90049
[2] Brennen, C., Winet, H.: Fluid mechanics of propulsion by cilia and flagella. Ann. Rev. Fluid Mech. 9 (1977), 339–398. DOI 10.1146/annurev.fl.09.010177.002011 | Zbl 0431.76100
[3] Burk, F.: Lebesgue Measure and Integration: An Itroduction. John Wiley $\&$ Sons, 1998. MR 1478419
[4] Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A.: Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, 2006. MR 2223552 | Zbl 1093.76002
[5] Dimitriu, G.: Numerical approximation of the optimal inputs for an identification problem. Internat. J. Comput. Math. 70 (1998), 197–209. DOI 10.1080/00207169808804746 | MR 1712493 | Zbl 0915.65069
[6] Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A., Bibette, J.: Microscopic artificial swimmers. Nature 437 (2005), 862–865. DOI 10.1038/nature04090
[7] Fahroo, F.: Optimal placement of controls for a one-dimensional active noise control problem. Kybernetika 34 (1998), 655–665. MR 1695369
[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 10.1080/00207179608921871 | MR 1652573 | Zbl 0841.49001
[9] Heidari, H., Malek, A.: Null boundary controllability for hyperdiffusion equation. Internat. J. Appl. Math. 22 (2009), 615–626. MR 2537040 | Zbl 1177.93018
[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 10.1016/S0096-3003(01)00011-X | MR 1883857 | Zbl 1040.49037
[11] Kim, Y. W., Netz, R. R.: Pumping fluids with periodically beating grafted elastic filaments. Phys. Rev. Lett. 96 (2006), 158101. DOI 10.1103/PhysRevLett.96.158101
[12] Lauga, E.: Floppy swimming: Viscous locomotion of actuated elastica. Phys. Rev. E. 75 (2007), 041916. DOI 10.1103/PhysRevE.75.041916 | MR 2358588
[13] Lauga, E., Powers, T. R.: The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (2009), 096601. DOI 10.1088/0034-4885/72/9/096601 | MR 2539632
[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 10.1098/rspb.1963.0036
[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 10.1016/ | MR 2188155 | Zbl 1153.49315
[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 10.1016/S0045-7825(98)00092-9 | MR 1659187 | Zbl 0949.76024
[19] Purcell, E. M.: Life at low Reynolds number. Amer. J. Phys. 45 (1977), 3–11. DOI 10.1119/1.10903
[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 10.1080/00207160008804980 | MR 1787282 | Zbl 0961.65064
[21] Rektorys, K.: Variational Methods in Mathematics, Sciences and Engineering. D. Reidel Publishing Company, 1977. MR 0487653
[22] Sakthivel, K., Balachandran, K., Sowrirajan, R., Kim, J-H.: On exact null controllability of black scholes equation. Kybernetika 44 (2008), 685–704. MR 2479312 | Zbl 1177.93021
[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 10.1016/S0006-3495(98)74029-9
[24] Williams, P.: A Gauss–Lobatto quadrature method for solving optimal control problems. ANZIAM 47 (2006), C101–C115. MR 2242566
[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 10.1063/1.2349585
Partner of
EuDML logo