Exponential stability of two strings under joint damping with variable coefficients. (English) Zbl 1447.93297
Authors’ abstract: We study the stabilization of two vibrating strings with variable physical coefficients joined by a feedback control at a common endpoint and subject to non-symmetrical boundary conditions. We prove that this system is exponentially stable under sufficient conditions on the physical coefficients. For this result, we show that the system has a sequence of the generalized eigenvectors which forms a Riesz basis with parentheses for the state Hilbert space, and as a consequence the spectrum-determined growth condition holds.
Reviewer: Kaïs Ammari (Monastir)
References:
| [1] | Ammari, K.; Henrot, A.; Tucsnak, M., Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptot. Anal., 28, 3, 215-240 (2001) · Zbl 0994.35030 |
| [2] | Chen, G.; Coleman, M.; West, H. H., Pointwise stabilization in the middle of the span for second order systems, nonuniform and uniform exponential decay of solutions, SIAM J. Appl. Math., 47, 4, 751-780 (1987) · Zbl 0641.93047 |
| [3] | Guo, B. Z.; Jin, F. F., Arbitrary decay rate for two connected strings with joint anti-damping by boundary output feedback, Automatica, 46, 7, 1203-1209 (2010) · Zbl 1194.93180 |
| [4] | Guo, B. Z.; Zhu, W. D., On the energy decay of two coupled strings through a joint damper, J. Sound Vib., 203, 3, 447-455 (1997) |
| [5] | Khapalov, A. Y., Exponential decay for the one-dimensional wave equation with internal pointwise damping, Math. Methods Appl. Sci., 20, 14, 1171-1183 (1997) · Zbl 0886.35088 |
| [6] | Liu, K. S., Energy decay problems in the design of a point stabilizer for coupled string vibrating systems, SIAM J. Control Optim., 26, 1348-1356 (1988) · Zbl 0662.93054 |
| [7] | Luo, Z. H.; Guo, B. Z.; Morgül, Ö., Stability and Stabilization of Infinite Dimensional Systems with Applications (1999), Springer-Verlag: Springer-Verlag London · Zbl 0922.93001 |
| [8] | Rzepnicki, Ł.; Schnaubelt, R., Polynomial stability for a system of coupled strings, Bull. Lond. Math. Soc., 50, 6, 1117-1136 (2018) · Zbl 1406.35185 |
| [9] | Liu, K. S.; Huang, F. L.; Chen, G., Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM J. Appl. Math., 49, 6, 1694-1707 (1989) · Zbl 0685.93054 |
| [10] | Guo, B. Z.; Wang, J. M., Control of Wave and Beam PDEs: The Riesz Basis Approach, Vol. 596 (2019), Springer-Verlag: Springer-Verlag Cham · Zbl 1426.35002 |
| [11] | Avdonin, S.; Edward, J., Exact controllability for string with attached masses, SIAM J. Control Optim., 56, 945-980 (2018) · Zbl 1390.93395 |
| [12] | Ben Amara, J.; Beldi, E., Boundary controllability of two vibrating strings connected by a point mass with variable coefficients, SIAM J. Control Optim., 57, 5, 3360-3387 (2019) · Zbl 1423.35363 |
| [13] | Ben Amara, J.; Beldi, E., Simultaneous controllability of two vibrating strings with variable coefficients, Evol. Equ. Control Theory, 8, 4, 687-694 (2019) · Zbl 1425.93035 |
| [14] | Ben Amara, J.; Bouzidi, H., Null boundary controllability of a one-dimensional heat equation with an internal point mass and variable coefficients, J. Math. Phys., 59, Article 011512 pp. (2018) · Zbl 1381.93023 |
| [15] | Gomilko, A.; Pivovarchik, V., On basis properties of a part of eigenfunctions of the problem of vibrations of a smooth inhomogeneous string damped at the midpoint, Math. Nachr., 245, 1, 72-93 (2002) · Zbl 1023.34023 |
| [16] | Pivovarchik, V., Direct and inverse three-point Sturm-Liouville problem with parameter-dependent boundary conditions, Asymptot. Anal., 26, 219-238 (2001) · Zbl 0999.34021 |
| [17] | Xu, G. Q.; Guo, B. Z., Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM J. Control Optim., 42, 966-984 (2003) · Zbl 1066.93028 |
| [18] | Guo, B. Z.; Xie, Y., A sufficient condition on Riesz basis with parentheses of non-self-adjoint operator and application to a serially connected string system under joint feedbacks, SIAM J. Control Optim., 43, 4, 1234-1252 (2004) · Zbl 1101.93040 |
| [19] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0516.47023 |
| [20] | Levitan, B. M.; Sargsyan, I. C., Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators (1975), AMS · Zbl 0302.47036 |
| [21] | Fedoryuk, M. V., Asymptotic Analysis: Linear Ordinary Differential Equations (1993), Springer-Verlag · Zbl 0782.34001 |
| [22] | Avdonin, S.; Ivanov, S., Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0866.93001 |
| [23] | Young, R. M., An Introduction to Nonharmonic Fourier Series (1980), Academic Press · Zbl 0493.42001 |
| [24] | Shkalikov, A. A., Boundary problems for ordinary differential equations with parameter in the boundary conditions, J. Math., 33, 6, 1311-1342 (1986) · Zbl 0609.34019 |
| [25] | Gohberg, I. C.; Krein, M. G., (Introduction to the Theory of Linear Nonselfadjoint Operators. Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Math. Monogr., vol. 18 (1969), AMS: AMS Providence, RI) · Zbl 0181.13503 |
| [26] | Conway, J., (Functions of One Complex Variable. Functions of One Complex Variable, Graduate Texts in Mathematics, vol. 11 (1978), Springer-Verlag: Springer-Verlag New York, Berlin) |
| [27] | Xu, G. Q.; Han, Z. J.; Yung, S. P., Riesz basis property of serially connected Timoshenko beams, Internat. J. Control, 80, 470-485 (2007) · Zbl 1120.93026 |
| [28] | Xu, G. Q.; Yung, S., The expansion of semigroup and a Riesz basis criterion, J. Differential Equations, 210, 1-24 (2005) · Zbl 1131.47042 |
| [29] | Guo, B. Z.; Xu, G. Q., Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, J. Funct. Anal., 231, 245-268 (2006) · Zbl 1153.35368 |
| [30] | Lyubich, Y. I.; Phóng, V. Q., Asymptotic stability of linear differential equations in banach spaces, Studia Math., 88, 37-42 (1988) · Zbl 0639.34050 |
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