# Optimal control of nonlinear one-dimensional periodic wave equation with x-dependent coefficients

Open Mathematics (2011)

• Volume: 9, Issue: 2, page 269-280
• ISSN: 2391-5455

top

## Abstract

top
This paper is concerned with an optimal control problem governed by the nonlinear one dimensional periodic wave equation with x-dependent coefficients. The control of the system is realized via the outer function of the state. Such a model arises from the propagation of seismic waves in a nonisotropic medium. By investigating some important properties of the linear operator associated with the state equation, we obtain the existence and regularity of the weak solution to the state equation. Furthermore, the existence of the optimal control is proved by means of the Arzelà-Ascoli lemma and Sobolev compact imbedding theorem.

## How to cite

top

Hengyan Li, and Shuguan Ji. "Optimal control of nonlinear one-dimensional periodic wave equation with x-dependent coefficients." Open Mathematics 9.2 (2011): 269-280. <http://eudml.org/doc/269592>.

@article{HengyanLi2011,
abstract = {This paper is concerned with an optimal control problem governed by the nonlinear one dimensional periodic wave equation with x-dependent coefficients. The control of the system is realized via the outer function of the state. Such a model arises from the propagation of seismic waves in a nonisotropic medium. By investigating some important properties of the linear operator associated with the state equation, we obtain the existence and regularity of the weak solution to the state equation. Furthermore, the existence of the optimal control is proved by means of the Arzelà-Ascoli lemma and Sobolev compact imbedding theorem.},
author = {Hengyan Li, Shuguan Ji},
journal = {Open Mathematics},
keywords = {Optimal control; Periodic wave equation; Weak solution; Arzelà-Ascoli lemma; Sobolev compact imbedding theorem; optimal control; periodic wave equation; weak solution},
language = {eng},
number = {2},
pages = {269-280},
title = {Optimal control of nonlinear one-dimensional periodic wave equation with x-dependent coefficients},
url = {http://eudml.org/doc/269592},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Hengyan Li
AU - Shuguan Ji
TI - Optimal control of nonlinear one-dimensional periodic wave equation with x-dependent coefficients
JO - Open Mathematics
PY - 2011
VL - 9
IS - 2
SP - 269
EP - 280
AB - This paper is concerned with an optimal control problem governed by the nonlinear one dimensional periodic wave equation with x-dependent coefficients. The control of the system is realized via the outer function of the state. Such a model arises from the propagation of seismic waves in a nonisotropic medium. By investigating some important properties of the linear operator associated with the state equation, we obtain the existence and regularity of the weak solution to the state equation. Furthermore, the existence of the optimal control is proved by means of the Arzelà-Ascoli lemma and Sobolev compact imbedding theorem.
LA - eng
KW - Optimal control; Periodic wave equation; Weak solution; Arzelà-Ascoli lemma; Sobolev compact imbedding theorem; optimal control; periodic wave equation; weak solution
UR - http://eudml.org/doc/269592
ER -

## References

top
1. [1] Adams R.A., Sobolev Spaces, Pure Appl. Math., 65, Academic Press, New York, 1975
2. [2] Akkouchi M., Bounabat A., Goebel M., Smooth and nonsmooth Lipschitz controls for a class of nonlinear ordinary differential equations of second order, preprint available at http://www.mathematik.uni-halle.de/reports/sources/1998/98-33report.dvi
3. [3] Barbu V, Optimal control of the one-dimensional periodic wave equation, Appl. Math. Optim., 1997, 35(1), 77–90 Zbl0866.49028
4. [4] Barbu V, Pavel N.H., Periodic solutions to one-dimensional wave equation with piece-wise constant coefficients, J. Differential Equations, 1996, 132(2), 319–337 http://dx.doi.org/10.1006/jdeq.1996.0182 Zbl0896.35075
5. [5] Barbu V, Pavel N.H., Determining the acoustic impedance in the 1-D wave equation via an optimal control problem, SIAM J. Control Optim., 1997, 35(5), 1544–1556 http://dx.doi.org/10.1137/S0363012995283698 Zbl0906.49009
6. [6] Barbu V., Pavel N.H., Periodic solutions to nonlinear one dimensional wave equation with x-dependent coefficients, Trans. Amer. Math. Soc, 1997, 349(5), 2035–2048 http://dx.doi.org/10.1090/S0002-9947-97-01714-5 Zbl0880.35073
7. [7] Brézis H., Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc. (N.S.), 1983, 8(3), 409–426 http://dx.doi.org/10.1090/S0273-0979-1983-15105-4 Zbl0515.35060
8. [8] Brown R.C., Hinton D.B., Schwabik Š., Applications of a one-dimensional Sobolev inequality to eigenvalue problems, Differential Integral Equations, 1996, 9(3), 481–498 Zbl0842.34083
9. [9] Goebel M., On smooth and nonsmooth Lipschitz controls, FB Mathematik und Informatik Report, 39, Martin-Luther-Universität Halle-Wittenberg, 1997, available at http://www.mathematik.uni-halle.de/reports/sources/1997/97-39report.dvi
10. [10] Ji S., Smooth and nonsmooth Lipschitz controls for a class of vector differential equations, J. Optim. Theory Appl., 2006, 131(2), 245–264 http://dx.doi.org/10.1007/s10957-006-9138-0 Zbl1139.49301
11. [11] Ji S., Time periodic solutions to a nonlinear wave equation with x-dependent coefficients, Calc. Var. Partial Differential Equations, 2008, 32(2), 137–153 http://dx.doi.org/10.1007/s00526-007-0132-7 Zbl1160.35053
12. [12] Ji S., Periodic solutions on the Sturm-Liouville boundary value problem for two-dimensional wave equation, J. Math. Phys., 2009, 50(11), #113510 Zbl1304.35427
13. [13] Ji S., Time-periodic solutions to a nonlinear wave equation with periodic or anti-periodic boundary conditions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465(2103), 895–913 http://dx.doi.org/10.1098/rspa.2008.0272 Zbl1186.35112
14. [14] Ji S., Li Y., Periodic solutions to one-dimensional wave equation with x-dependent coefficients, J. Differential Equations, 2006, 229(2), 466–493 http://dx.doi.org/10.1016/j.jde.2006.03.020 Zbl1103.35011
15. [15] Ji S., Li Y., Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 2007, 137(2), 349–371 http://dx.doi.org/10.1017/S0308210505001174 Zbl1221.35240
16. [16] Ji S., Li Y., Time periodic solutions to one-dimensional nonlinear wave equation, Arch. Ration. Mech. Anal., DOI: 10.1007/s00205-010-0328-4
17. [17] Rabinowitz PH., Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math., 1978, 31(1), 31–68 http://dx.doi.org/10.1002/cpa.3160310103 Zbl0341.35051
18. [18] Talenti G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 1976, 110(1), 353–372 http://dx.doi.org/10.1007/BF02418013 Zbl0353.46018
19. [19] Wu Z., Li Y., Ordinary Differential Equations, Higher Education Press, Beijing, 2004
20. [20] Wu Z., Yin J., Wang C, Introduction to Partial Differential Equations of Elliptic and Parabolic Type, Science Beijing, 2004
21. [21] Yosida K., Functional Analysis, 6th ed, Grundlehren Math. Wiss., 123, Springer, Berlin-New York, 1980

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.