A Note on the Indirect Controls for a Coupled System of Wave Equations

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.

Keywords

Indirect controls / Approximate controllability / Coupled system of wave equations

References

[1] 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., 2013, 18: 1005-1073

[2] Alabau-Boussouira F, Cannarsa P, Komornik V. Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equ., 2002, 2: 127-150

[3] Dehman B, Le Rousseau J, Léautaud M. Controllability of two coupled wave equations on a compact manifold. Arch. Rat. Mech. Anal., 2014, 211: 113-187

[4] Hao J, Rao B. Influence of the hidden regularity on the stability of partially damped systems of wave equations. J. Math. Pures Appl., 2020, 143: 257-286

[5] Kalman R E. Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana, 1960, 5: 102-119

[6] Li T-T, Rao B. A note on the exact synchronization by groups for a coupled system of wave equations. Math. Meth. Appl. Sci., 2015, 38: 2803-2808

[7] Li T-T, 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., 2016, 54: 49-72

[8] Li T-T, Rao B. On the approximate boundary synchronization for a coupled system of wave equations: Direct and indirect controls. ESIAM: COCV, 2018, 24: 1675-1704

[9] Li T-T, Rao B. Boundary Synchronization for Hyperbolic Systems, 2019, Switzerland: Birkhaüser

[10] Lions J-L. Contrôlabilité Exacte, 1988, Paris: Masson

[11] Lions J-L, Magenes E. Problèmes aux Limites non Homogènes et Applications, 1968, Paris: Dunod Vol. 1

[12] Pazy A. Semi-Groups of Linear Operators and Applications to Partial Differential Equations, 1983, New York: Applied Mathematical Sciences, Springer-Verlag

[13] 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.

[14] Rauch J, Zhang X, Zuazua E. Polynomial decay for a hyperbolic-parabolic coupled system. J. Math. Pures Appl., 2005, 84: 407-470

[15] Rosier L, de Teresa L. Exact controllability of a cascade system of conservative equations. C. R. Math. Acad. Sci. Paris, 2011, 349: 291-295

[16] Russell D L. A general framework for the study of indirect damping mechanisms in elastic systems. J. Math. Anal. Appl., 1993, 173: 339-358

[17] Zu Ch, Li T-T, Rao B. Sufficiency of Kalman’s rank condition for the approximate boundary controllability on spherical domain. Math. Methods Appl. Sci., 2021, 47: 13509-13525