Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 231-252.

We show that the set of nonnegative equilibrium-like states, namely, like $\left({y}_{d},0\right)$ of the semilinear vibrating string that can be reached from any non-zero initial state $\left({y}_{0},{y}_{1}\right)\in {H}_{0}^{1}\left(0,1\right)×{L}^{2}\left(0,1\right)$, by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace ${L}^{2}\left(0,1\right)×\left\{0\right\}$ of ${L}^{2}\left(0,1\right)×{H}^{-1}\left(0,1\right)$. Our main results deal with nonlinear terms which admit at most the linear growth at infinity in $\phantom{\rule{0.277778em}{0ex}}y\phantom{\rule{0.277778em}{0ex}}$ and satisfy certain restriction on their total impact on $\left(0,\infty \right)$ with respect to the time-variable.

DOI : https://doi.org/10.1051/cocv:2006001
Classification : 93,  35
Mots clés : semilinear wave equation, approximate controllability, multiplicative controls, axial load, damping
@article{COCV_2006__12_2_231_0,
author = {Khapalov, Alexander Y.},
title = {Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {231--252},
publisher = {EDP-Sciences},
volume = {12},
number = {2},
year = {2006},
doi = {10.1051/cocv:2006001},
zbl = {1105.93011},
mrnumber = {2209352},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2006001/}
}
Khapalov, Alexander Y. Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 231-252. doi : 10.1051/cocv:2006001. http://www.numdam.org/articles/10.1051/cocv:2006001/

[1] A. Baciotti, Local Stabilizability of Nonlinear Control Systems. Ser. Adv. Math. Appl. Sci. 8 (1992). | Zbl 0757.93061

[2] J.M. Ball and M. Slemrod, Feedback stabilization of semilinear control systems. Appl. Math. Opt. 5 (1979) 169-179. | Zbl 0405.93030

[3] J.M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems. Comm. Pure. Appl. Math. 32 (1979) 555-587. | Zbl 0394.93041

[4] J.M. Ball, J.E. Mardsen and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Contr. Optim. (1982) 575-597. | Zbl 0485.93015

[5] M.E. Bradley, S. Lenhart and J. Yong, Bilinear optimal control of the velocity term in a Kirchhoff plate equation. J. Math. Anal. Appl. 238 (1999) 451-467. | Zbl 0936.49003

[6] A. Chambolle and F. Santosa, Control of the wave equation by time-dependent coefficient. ESAIM: COCV 8 (2002) 375-392. | Numdam | Zbl 1073.35032

[7] L.A. Fernández, Controllability of some semilinear parabolic problems with multiplicative control, presented at the Fifth SIAM Conference on Control and its applications, held in San Diego, July 11-14 (2001).

[8] A.Y. Khapalov, Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms, the Special volume “Control of Nonlinear Distributed Parameter Systems”, dedicated to David Russell, G. Chen/I. Lasiecka/J. Zhou Eds., Marcel Dekker (2001) 139-155. | Zbl 0983.93023

[9] A.Y. Khapalov, Global non-negative controllability of the semilinear parabolic equation governed by bilinear control. ESAIM: COCV 7 (2002) 269-283. | Numdam | Zbl 1024.93026

[10] A.Y. Khapalov, On bilinear controllability of the parabolic equation with the reaction-diffusion term satisfying Newton's Law. Special issue dedicated to the memory of J.-L. Lions. Computat. Appl. Math. 21 (2002) 1-23. | Zbl 1119.93017

[11] A.Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach. SIAM J. Control. Optim. 41 (2003) 1886-1900. | Zbl 1041.93026

[12] A.Y. Khapalov, Bilinear controllability properties of a vibrating string with variable axial load and damping gain. Dynamics Cont. Discrete. Impulsive Systems 10 (2003) 721-743. | Zbl 1035.35015

[13] A.Y. Khapalov, Controllability properties of a vibrating string with variable axial load. Discrete Control Dynamical Systems 11 (2004) 311-324.

[14] K. Kime, Simultaneous control of a rod equation and a simple Schrödinger equation. Syst. Control Lett. 24 (1995) 301-306. | Zbl 0877.93003

[15] S. Lenhart, Optimal control of convective-diffusive fluid problem. Math. Models Methods Appl. Sci. 5 (1995) 225-237. | Zbl 0828.76066

[16] S. Lenhart and M. Liang, Bilinear optimal control for a wave equation with viscous damping. Houston J. Math. 26 (2000) 575-595. | Zbl 0976.49005

[17] M. Liang, Bilinear optimal control for a wave equation. Math. Models Methods Appl. Sci. 9 (1999) 45-68. | Zbl 0939.49016

[18] S. Müller, Strong convergence and arbitrarily slow decay of energy for a class of bilinear control problems. J. Differ. Equ. 81 (1989) 50-67. | Zbl 0711.35017