Uniqueness of solution to systems of elliptic operators and application to asymptotic synchronization of linear dissipative systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 117.

We show that under Kalman’s rank condition on the coupling matrices, the uniqueness of solution to a complex system of elliptic operators can be reduced to the observability of a scalar problem. Based on this result, we establish the asymptotic stability and the asymptotic synchronization for a large class of linear dissipative systems.

DOI : 10.1051/cocv/2020062
Classification : 93B05, 93C20, 35L53
Mots-clés : uniqueness, elliptic systems, asymptotic synchronization, condition of compatibility, Kalman’s rank condition 
@article{COCV_2020__26_1_A117_0,
     author = {Li, Tatsien and Rao, Bopeng},
     editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinsk, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
     title = {Uniqueness of solution to systems of elliptic operators and application to asymptotic synchronization of linear dissipative systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2020062},
     mrnumber = {4188830},
     zbl = {1461.93416},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2020062/}
}
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Li, Tatsien; Rao, Bopeng. Uniqueness of solution to systems of elliptic operators and application to asymptotic synchronization of linear dissipative systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 117. doi : 10.1051/cocv/2020062. http://www.numdam.org/articles/10.1051/cocv/2020062/

[1] F. Ammar-Khodja, A. Benabdallah, J.E. Munoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type. J. Diff. Eqs. 194 (2003) 82–115. | DOI | MR | Zbl

[2] W. Arendt and C.J. Batty, Tauberian theorems and stability of one-parameter semi-groups. Trans. Amer. Math. Soc. 306 (1988) 837–852. | DOI | MR | Zbl

[3] C.D. Benchimol, A note on weak stabilization of contraction semi-groups. SIAM J. Control Optim. 16 (1978) 373–379. | DOI | MR | Zbl

[4] L. De Teresa and E. Zuazua, Controllability of the linear system of thermoelastic plates. Adv. Diff. Eqs. 1 (1996) 369–402. | MR | Zbl

[5] N. Garofalo and F. Lin, Uniqueness of solution for elliptic operators. A geometric-variational approach. Commun. Pure Appl. Math. 40 (1987) 347–366. | DOI | MR | Zbl

[6] S.W. Hansen and B. Zhang, Boundary control of a linear thermo-elastic beam. J. Math. Anal. Appl. 210 (1997) 182–205. | DOI | MR | Zbl

[7] J. Hao, B. Rao, Influence of the hidden regularity on the stability of partially damped systems of wave equations. J. Math. Pure Appl. 143 (2020) 257–286. | DOI | MR | Zbl

[8] B.V. Kapitonov, Stabilization and exact boundary controllability for Maxwell’s equations. SIAM J. Control Optim. 32 (1994) 408–420. | DOI | MR | Zbl

[9] J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25 (1987) 1417–1429. | DOI | MR | Zbl

[10] H. Koch and D. Tataru, Carleman estimates and uniqueness of solution for second-order elliptic equations with nonsmooth coefficients. Commun. Pure Appl. Math. 54 (2001) 339–360. | DOI | MR | Zbl

[11] J.E. Lagnese, Boundary Stabilization of Thin Plates. SIAM, Study in applied mathematics. Philadelphia (1989). | MR | Zbl

[12] J.E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées, Masson, Paris (1988). | MR | Zbl

[13] I. Lasiecka and R. Triggiani, Uniform stabilization of a shallow shell model with nonlinear boundary dampings. J. Math. Anal. Appl. 269 (2002) 642–688. | DOI | MR | Zbl

[14] F. Li and G. Du, General energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback. J. Appl. Anal. Comp. 8 (2018) 390–401. | MR | Zbl

[15] F. Li and Z. Jia, Global existence and stability of a class of nonlinear evolution equations with hereditary memory and variable density Bound. Value Probl. 37 (2019) 23. | MR | Zbl

[16] T.-T. Li and B. Rao, 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. | DOI | MR | Zbl

[17] T.-T. Li and B. Rao, On the approximate boundary synchronization for a coupled system of wave equations: Direct and indirect controls. ESAIM: COCV 24 (2018) 1675–1704. | Numdam | MR | Zbl

[18] T.T. Li and B. Rao, Kalman’s criterion on the uniqueness of continuation for the nilpotent system of wave equations. C. R. Acad. Sci. Paris, Ser. I 356 (2018) 1188–1194. | DOI | MR | Zbl

[19] T.-T. Li and B. Rao, On the approximate boundary synchronization for a coupled system of wave equations: Direct and indirect boundary controls. ESAIM: COCV 24 (2018) 1675–1704. | Numdam | MR | Zbl

[20] T.-T. Li and B. Rao, Boundary Synchronization for Hyperbolic Systems. Progress in Non Linear Differential Equations and Their Applications, Subseries in Control, Vol. 94. Birkhaüser (2019). | MR | Zbl

[21] T.-T. Li and B. Rao, Uniqueness theorem for partially observed elliptic systems and application to asymptotic synchronization. C. R. Math. 358 (2020) 285–295. | DOI | MR | Zbl

[22] T.-T. Li and B. Rao, Uniform synchronization of second order evolution equations. Inpreparation 2020.

[23] T.-T. Li, B. Rao and Y.M. Wei, Generalized exact boundary synchronization for second order evolution systems. Disc. Contin. Dyn. Syst. 34 (2014) 2893–2905. | DOI | MR | Zbl

[24] F. Li, Sh. Xi, K. Xu and X. Xue, Dynamic properties for nonlinear viscoelastic Kirchhoff-type equation with acoustic control boundary conditions II. J. Appl. Anal. Comput. 9 (2019) 2318–2332. | MR | Zbl

[25] Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system. Z. Angew. Math. Phys. 60 (2009) 54–69. | DOI | MR | Zbl

[26] A. Pazy, Semi-groups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences. Springer-Verlag (1983). | MR | Zbl

[27] B. Rao, 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) 25. | MR | Zbl

[28] L. Ren and J. Xin, Almost global existence for the Neumann problem of quasilinear wave equations outside star-shaped domains in 3D. Electr. J. Diff. Eqs. 312 (2018) 1–22. | MR | Zbl

[29] F. Trèves, Basic Linear Partial Differential Equations. Pure and Applied Mathematics, Vol. 62. Academic Press, New York/London (1975). | MR | Zbl

[30] X. Zheng, J. Xin and X. Peng, Orbital stability of periodic traveling wave solutions to the generalized long-short wave equations. J. Appl. Anal. Comput. 9 (2019) 2389–2408. | MR | Zbl

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