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Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping. (English) Zbl 1105.93011
Summary: We show that the set of nonnegative equilibrium-like states, namely, like \((y_d, 0)\) of the semilinear vibrating string that can be reached from any non-zero initial state \( (y_0, y_1) \in H^1_0 (0,1) \times L^2 (0,1)\), by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace \( L^2 (0,1) \times \{0\} \) of \( L^2 (0,1) \times H^{-1} (0,1)\). Our main results deal with nonlinear terms which admit at most the linear growth at infinity in \(y\) and satisfy certain restriction on their total impact on \( (0, \infty)\) with respect to the time-variable.

MSC:
93B05 Controllability
35L70 Second-order nonlinear hyperbolic equations
74H45 Vibrations in dynamical problems in solid mechanics
74K05 Strings
74M05 Control, switches and devices (“smart materials”) in solid mechanics
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References:
[1] A. Baciotti , Local Stabilizability of Nonlinear Control Systems . Ser. Adv. Math. Appl. Sci. 8 ( 1992 ). Zbl 0757.93061 · 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 · 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 · 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 · 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 · 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 · 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 · 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 · 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 · 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 · 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 · 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 . Zbl pre02110763 · Zbl 1175.93032
[14] K. Kime , Simultaneous control of a rod equation and a simple Schrödinger equation . Syst. Control Lett. 24 ( 1995 ) 301 - 306 . Zbl 0877.93003 · 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 · 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 · 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 · 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 · Zbl 0711.35017
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