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Optimal boundary control for hyperdiffusion equation. (English) Zbl 1206.35138
Summary: We consider the solution of optimal control problem for hyperdiffusion equation involving a 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. The 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.

MSC:
35K35 Initial-boundary value problems for higher-order parabolic equations
49J20 Existence theories for optimal control problems involving partial differential equations
35Q93 PDEs in connection with control and optimization
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References:
[1] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press 2004. · Zbl 1058.90049
[2] Brennen, C., Winet, H.: Fluid mechanics of propulsion by cilia and flagella. Ann. Rev. Fluid Mech. 9 (1977), 339-398. · Zbl 0431.76100
[3] Burk, F.: Lebesgue Measure and Integration: An Itroduction. John Wiley \(\&\) Sons, 1998. · Zbl 0886.28001
[4] Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A.: Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, 2006. · Zbl 1093.76002
[5] Dimitriu, G.: Numerical approximation of the optimal inputs for an identification problem. Internat. J. Comput. Math. 70 (1998), 197-209. · 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. · Zbl 1132.76062
[7] Fahroo, F.: Optimal placement of controls for a one-dimensional active noise control problem. Kybernetika 34 (1998), 655-665. · Zbl 1274.49049
[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. · Zbl 0841.49001
[9] Heidari, H., Malek, A.: Null boundary controllability for hyperdiffusion equation. Internat. J. Appl. Math. 22 (2009), 615-626. · 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. · Zbl 1040.49037
[11] Kim, Y. W., Netz, R. R.: Pumping fluids with periodically beating grafted elastic filaments. Phys. Rev. Lett. 96 (2006), 158101.
[12] Lauga, E.: Floppy swimming: Viscous locomotion of actuated elastica. Phys. Rev. E. 75 (2007), 041916.
[13] Lauga, E., Powers, T. R.: The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (2009), 096601.
[14] Lions, J. L., Magenes, E.: Non-homogeneous Boundary Value Problem and Applications. Springer-Verlag, 1972. · Zbl 0223.35039
[15] Machin, K. E.: The control and synchronization of flagellar movement. Proc. Roy. Soc. B. 158 (1963), 88-104.
[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. · 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. · Zbl 0949.76024
[19] Purcell, E. M.: Life at low Reynolds number. Amer. J. Phys. 45 (1977), 3-11.
[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. · Zbl 0961.65064
[21] Rektorys, K.: Variational Methods in Mathematics, Sciences and Engineering. D. Reidel Publishing Company, 1977.
[22] Sakthivel, K., Balachandran, K., Sowrirajan, R., Kim, J-H.: On exact null controllability of black scholes equation. Kybernetika 44 (2008), 685-704. · 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.
[24] Williams, P.: A Gauss-Lobatto quadrature method for solving optimal control problems. ANZIAM 47 (2006), C101-C115.
[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.
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