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Optimal control systems by time-dependent coefficients using CAS wavelets. (English) Zbl 1185.49033

Summary: This paper considers the problem of controlling the solution of an initial boundary-value problem for a wave equation with time-dependent sound speed. The control problem is to determine the optimal sound speed function which damps the vibration of the system by minimizing a given energy performance measure. The minimization of the energy performance measure over sound speed is subjected to the equation of motion of the system with imposed initial and boundary conditions. Using the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of bilinear time-invariant lumped-parameter systems. A wavelet-based method for evaluating the modal optimal control and trajectory of the bilinear system is proposed. The method employs finite CAS wavelets to approximate modal control and state variables. Numerical examples are presented to demonstrate the effectiveness of the method in reducing the energy of the system.

MSC:

49M30 Other numerical methods in calculus of variations (MSC2010)
35L05 Wave equation
35L20 Initial-boundary value problems for second-order hyperbolic equations
70Q05 Control of mechanical systems
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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References:

[1] V. Berdichersky, V. Jikov, and G. Papanicolcou, Eds., Homogenization, vol. 50 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing, River Edge, NJ, USA, 1999. · Zbl 0924.00012
[2] K. A. Lurie, “Control in the coefficients of linear hyperbolic equations via spacio-temporal components,” in Homogenization, vol. 50 of Series on Advances in Mathematics for Applied Sciences, pp. 285-315, World Scientific, River Edge, NJ, USA, 1999. · Zbl 1035.78021
[3] K. A. Lurie, “Some new advances in the theory of dynamic materials,” Journal of Elasticity, vol. 72, no. 1-3, pp. 229-239, 2003. · Zbl 1091.78501 · doi:10.1023/B:ELAS.0000018780.82718.19
[4] S. Hurlebaus and L. Gaul, “Smart structure dynamics,” Mechanical Systems and Signal Processing, vol. 20, no. 2, pp. 255-281, 2006. · doi:10.1016/j.ymssp.2005.08.025
[5] A. Chambolle and F. Santosa, “Control of the wave equation by time-dependent coefficient,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 8, pp. 375-392, 2002. · Zbl 1073.35032 · doi:10.1051/cocv:2002029
[6] H. R. Joshi, “Optimal control of the convective velocity coefficient in a parabolic problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5-7, pp. 1383-1390, 2005. · Zbl 1224.49006 · doi:10.1016/j.na.2005.02.025
[7] S. Lenhart, M. Liang, and V. Protopopescu, “Identification problem for a wave equation via optimal control,” in Control of Distributed Parameter and Stochastic Systems, vol. 141 of IFIP Conference Proceedings, pp. 79-84, Kluwer Academic Publishers, Boston, Mass, USA, 1999. · Zbl 0984.93016
[8] F. Maestre, A. Münch, and P. Pedregal, “A spatio-temporal design problem for a damped wave equation,” SIAM Journal on Applied Mathematics, vol. 68, no. 1, pp. 109-132, 2007. · Zbl 1147.35052 · doi:10.1137/07067965X
[9] M. Tatari and M. Dehghan, “Identifying a control function in parabolic partial differential equations from over specified boundary data,” Computers & Mathematics with Applications, vol. 53, no. 12, pp. 1933-1942, 2007. · Zbl 1121.93019 · doi:10.1016/j.camwa.2006.01.018
[10] M. Razzaghi and S. Yousefi, “Sine-cosine wavelets operational matrix of integration and its applications in the calculus of variations,” International Journal of Systems Science, vol. 33, no. 10, pp. 805-810, 2002. · Zbl 1012.65063 · doi:10.1080/00207720210161768
[11] S. Yousefi and A. Banifatemi, “Numerical solution of Fredholm integral equations by using CAS wavelets,” Applied Mathematics and Computation, vol. 183, no. 1, pp. 458-463, 2006. · Zbl 1109.65121 · doi:10.1016/j.amc.2006.05.081
[12] H. Danfu and S. Xufeng, “Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration,” Applied Mathematics and Computation, vol. 194, no. 2, pp. 460-466, 2007. · Zbl 1193.65216 · doi:10.1016/j.amc.2007.04.048
[13] M. Liang, “Bilinear optimal control for a wave equation,” Mathematical Models & Methods in Applied Sciences, vol. 9, no. 1, pp. 45-68, 1999. · Zbl 0939.49016 · doi:10.1142/S0218202599000051
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