Abstract
By a procedure of successive projections, the authors decompose a coupled system of wave equations into a sequence of sub-systems. Then, they can clarify the indirect controls and the total number of controls. Moreover, the authors give a uniqueness theorem of solution to the system of wave equations under Kalman’s rank condition.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alabau-Boussouira, F., A hierarchic multi-level energy method for the control of bidiagonal and mixed n-coupled cascade systems of PDE’s by a reduced number of controls, Adv. Diff. Equ., 18, 2013, 1005–1073.
Alabau-Boussouira, F., Cannarsa, P. and Komornik, V., Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2, 2002, 127–150.
Dehman, B., Le Rousseau, J. and Léautaud, M., Controllability of two coupled wave equations on a compact manifold, Arch. Rat. Mech. Anal., 211, 2014, 113–187.
Hao, J. and Rao, B., Influence of the hidden regularity on the stability of partially damped systems of wave equations, J. Math. Pures Appl., 143, 2020, 257–286.
Kalman, R. E., Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana, 5, 1960, 102–119.
Li, T.-T. and Rao, B., A note on the exact synchronization by groups for a coupled system of wave equations, Math. Meth. Appl. Sci., 38, 2015, 2803–2808.
Li, T.-T. and Rao, B., Criteria of Kalman’s type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls, SIAM J. Control Optim., 54, 2016, 49–72.
Li, T.-T. and Rao, B., On the approximate boundary synchronization for a coupled system of wave equations: Direct and indirect controls, ESIAM: COCV, 24, 2018, 1675–1704.
Li, T.-T. and Rao, B., Boundary Synchronization for Hyperbolic Systems, Progress in Non Linear Differential Equations, Subseries in Control, 94, Birkhaüser, Switzerland, 2019.
Lions, J.-L., Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Vol. 1, Masson, Paris, 1988.
Lions, J.-L. and Magenes, E., Problèmes aux Limites non Homogènes et Applications, Vol. 1, Dunod, Paris 1968.
Pazy, A., Semi-Groups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
Rao, B., On the sensitivity of the transmission of boundary dissipation for strongly coupled and indirectly damped systems of wave equations, Z. Angew. Math. Phys., 70, 2019, Paper No. 75, 25pp.
Rauch, J., Zhang, X. and Zuazua, E., Polynomial decay for a hyperbolic-parabolic coupled system, J. Math. Pures Appl., 84, 2005, 407–470.
Rosier, L. and de Teresa, L., Exact controllability of a cascade system of conservative equations, C. R. Math. Acad. Sci. Paris, 349, 2011, 291–295.
Russell, D. L., A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173, 1993, 339–358.
Zu, Ch., Li, T.-T. and Rao, B., Sufficiency of Kalman’s rank condition for the approximate boundary controllability on spherical domain, Math. Methods Appl. Sci., 47, 2021, 13509–13525.
Acknowledgement
The authors would like to thank the reviewer’s valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (No. 11831011).
Rights and permissions
About this article
Cite this article
Li, T., Rao, B. A Note on the Indirect Controls for a Coupled System of Wave Equations. Chin. Ann. Math. Ser. B 43, 359–372 (2022). https://doi.org/10.1007/s11401-022-0328-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-022-0328-3