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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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