×

Exact controllability of the transmission string-beam equations with a single boundary control. (English) Zbl 1536.93092

By means of the multiplier method, the paper investigates the exact controllability of a transmission string-beam system with control active at the end of string. A boundary observability inequality is presented. Moreover, a controllability result is given by the HUM method.

MSC:

93B05 Controllability
93B07 Observability
93C20 Control/observation systems governed by partial differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text: DOI

References:

[1] I.Lasiecka, Mathematical control theory of coupled PDEs, CBMS‐NSF Regional Conference Series in Applied Mathematics, Vol. 75, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. · Zbl 1032.93002
[2] S. G.Chai and B. Z.Guo, Well‐posedness and regularity of weakly coupled wave‐plate equation with boundary control and observation, J. Dyn. Control Syst.15 (2009), no. 3, 331-358. · Zbl 1203.93055
[3] F.Alabau, P.Cannarsa, and V.Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ.2 (2002), no. 2, 127-150. · Zbl 1011.35018
[4] Z.Liu and B.Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl.335 (2007), no. 2, 860-881. · Zbl 1152.35069
[5] X. Y.Fu and Q.Lü, Stabilization of the weakly coupled wave‐plate system with one internal damping, Vietnam J. Math.49 (2021), no. 3, 767-786. · Zbl 1470.93127
[6] J. H.Hao and P. P.Wang, Uniform stability of transmission of wave‐plate equations with source on Riemannian manifold, J. Differ. Equ.268 (2020), no. 10, 6385-6415. · Zbl 1435.35238
[7] S.Mansouri, Boundary stabilization of coupled plate equations, Palestine J. Math.2 (2013), 233-242. · Zbl 1343.35227
[8] X.Zhang and E.Zuazua, Control, observation and polynomial decay for a coupled heat‐wave system, C.R. Math.336 (2003), no. 10, 823-828. · Zbl 1029.93037
[9] X.Zhang and E.Zuazua, Polynomial decay and control of a
[( 1-d \]\) hyperbolic‐parabolic coupled system, J. Differ. Equ.204 (2004), no. 2, 380-438. · Zbl 1064.93008
[10] E.Zuazua, Null control of a
[( 1-d \]\) model of mixed hyperbolic‐parabolic type, Optim. Control Partial Differ. Equ.198 (2001), 198-207. · Zbl 1054.35009
[11] F.Alabau‐Boussouira and M.Léautaud, Indirect controllability of locally coupled wave‐type systems and applications, J. Math. Pures Appl.99 (2013), no. 5, 544-576. · Zbl 1293.35167
[12] D.D’Alessandro and R.Romano, Indirect controllability of quantum systems; a study of two interacting quantum bits, IEEE Trans. Autom. Control57 (2012), no. 8, 2009-2020. · Zbl 1369.81043
[13] S.Gerbi, C.Kassem, A.Mortada, and A.Wehbe, Exact controllability and stabilization of locally coupled wave equations: theoretical results, Z. Anal. Anwend.40 (2021), no. 1, 67-96. · Zbl 1467.35216
[14] T.Liard and P.Lissy, A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups, Math. Control Sig. Syst.29 (2017), no. 2, 9. · Zbl 1366.93056
[15] F.Alabau‐Boussouira, A two‐level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim.42 (2003), no. 3, 871-906. · Zbl 1125.93311
[16] F.Alabau‐Boussouira and M.Léautaud, Indirect controllability of locally coupled systems under geometric conditions, C.R. Math.349 (2011), no. 7-8, 395-400. · Zbl 1217.35113
[17] B.Dehman, J. L.Rousseau, and M.Léautaud, Controllability of two coupled wave equations on a compact manifold, Arch. Rational Mech. Anal.211 (2014), no. 1, 113-187. · Zbl 1290.35278
[18] Y. F.Li, Z. J.Han, and G. Q.Xu, Explicit decay rate for coupled string‐beam system with localized frictional damping, Appl. Math. Lett.78 (2017), 51-58. · Zbl 1383.35028
[19] F.Hassine, Asymptotic behavior of the transmission Euler‐Bernoulli plate and wave equation with a localized Kelvin‐Voigt damping. arXiv preprint arXiv:1812.10420. · Zbl 1350.35031
[20] Y. P.Guo, J. M.Wang, and D. X.Zhao, Energy decay estimates for a two‐dimensional coupled wave‐plate system with localized frictional damping, Z. Angew. Math. Mech.100 (2019), no. 2, e201900030. · Zbl 07794849
[21] C.Castro and E.Zuazua, Exact boundary controllability of two Euler‐Bernoulli beams connected by a point mass, Math. Comput. Model.32 (2000), no. 9, 955-969. · Zbl 1005.74041
[22] J. L.Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Vol. 8, Masson, Paris, 1988. · Zbl 0653.93002
[23] E.Zuazua, Controllability and observability of partial differential equations: some results and open problems, Handbook of differential equations: evolutionary equations, Vol. 3, North‐Holland, Amsterdam, pp. 527-621, 2007. · Zbl 1193.35234
[24] F. D.Araruna and E.Zuazua, Controllability of the Kirchhoff system for beams as a limit of the Mindlin‐Timoshenko system, SIAM J. Control Optim.47 (2008), no. 4, 1909-1938. · Zbl 1170.74351
[25] S.Hansen and E.Zuazua, Exact controllability and stabilization of a vibrating string with an interior point mass, SIAM J. Control Optim.33 (1995), no. 5, 1357-1391. · Zbl 0853.93018
[26] Z. J.Yang and Y. Q.Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differ. Equ.249 (2010), no. 12, 3258-3278. · Zbl 1213.35126
[27] Y. L.Zhang and J. M.Wang, Exact controllability of a micro beam with boundary bending moment, Int. J. Control92 (2019), no. 6, 1335-1343. · Zbl 1416.93036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.