Carleman estimates for the structurally damped plate equations with Robin boundary conditions and applications

References

• Fung YC. Foundations of solid mechanics. Englewood Cliffs: Prentice-Hall; 1965.
   Google Scholar
• Scanlon RH, Rosenbaum R. Introduction to the study of aircraft vibration and flutter. New York: Macmillan; 1951.
   Google Scholar
• Chen G, Russell DL. A mathematical model for linear elastic systems with structural damping. Quart Appl Math. 1982;39:433–454.
   Web of Science ®Google Scholar
• Favini A, Obrecht E. Conditions for parabolicity of second order abstract differential equations. Differ Integral Equ. 1991;4:1005–1022.
   Google Scholar
• Hörmander L. The analysis of linear partial differential operators III. Berlin: Springer; 1985. (Pseudodifferential Operators).
   Google Scholar
• Kenig CE. Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems. International congress of mathematicians; Vol. II, Berkeley, California, USA: 1986. p. 948–960.
   Google Scholar
• Bellassoued M, Yamamoto M. Carleman estimates and applications to inverse problems for Hyperbolic systems. Japan KK: Springer; 2017. (Springer Monographs in Mathematics).
   Google Scholar
• Isakov V. Inverse problem for partial differential equations. 2nd ed. Springer: New York; 2006.
   Google Scholar
• Klibanov MV. Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J Inverse Ill-Posed Probl. 2013;21:477–560.
   Web of Science ®Google Scholar
• Yamamoto M. Carleman estimates for parabolic equations and applications. Inverse Problem. 2009;25:123013.
   Web of Science ®Google Scholar
• Fu XY, Lü Q, Zhang X. Carleman estimates for second order partial differential operators an applications: a unified approach. Springer: Switzerland; 2019.
   Google Scholar
• Fursikov AV, Yu. Imanuvilov O. Controllability of evolution equations. Seoul, Korea: Research Institute of Mathematics, Seoul National University; 1996. (Lecture Notes Series; 34).
   Google Scholar
• Zuazua E. Controllability and observability of partial differential equations: some results and open problems. Handbook of Differential Equations: Evolutionary Differential Equations. Vol. 3, Elsevier Science; 2006. p. 527–621.
   Google Scholar
• Guerrero S, Kassab K. Carleman estimate and null controllability of a fourth order parabolic equation in dimension n≥2. J Math Pures Appl. 2019;121:135–161.
   Web of Science ®Google Scholar
• Zhou ZC. Observability estimate and null controllability for one-dimensional fourth order parabolic equation. Taiwan J Math. 2012;16:1991–2017.
   Web of Science ®Google Scholar
• Fu XY, Zhang X, Zuazua E. On the optimality of the observability inequalities for plate systems with potentials. In: Bove A, Colombini F, Del Santo D, editors, Phase space analysis of PDEs. Birkhäuser; 2006. p. 117–132.
   Google Scholar
• Gao P. Global Carleman estimate for the plate equation and applications to inverse problems. Electron J Differ Equ. 2016;2016:1–13.
   Google Scholar
• Mitra S. Carleman estimate for an adjoint of a damped beam equation and an application to null controllability. J Math Anal Appl. 2020;484:123718.
   Google Scholar
• Fu XY. Null controllability for the parabolic equation with a complex principal part. J Funct Anal. 2009;257:1333–1354.
   Web of Science ®Google Scholar
• Imanuvilov OYu, Yamamoto M. Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Prob. 1998;14:1229–1245.
   Web of Science ®Google Scholar
• Vancostenoble J. Sharp Carleman estimates for singular parabolic equations and application to Lipschitz stability in inverse source problems. C R Acad Sci Paris Ser I. 2010;348:801–805.
   Google Scholar
• Vessella S. Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates. Inverse Prob. 2008;24:023001.
   Web of Science ®Google Scholar
• Leis R. Initial-boundary value problems in mathematical physics. Stuttgart/Chichester: B.G Teubner/John Wiley & Sons, Ltd.; 1986.
   Google Scholar
• Russell D. Mathematical models for the elastic beam and their control-theoretic implications. Semigroups, theory and applications, vol. II, Trieste, 1984. Harlow: Longman Sci. Tech; 1986. p. 177–216. (Pitman Res. Notes Math. Ser.; 152).
   Google Scholar
• Wang YH. Global uniqueness and stability for an inverse plate problem. J Optim Theor App. 2007;132:161–173.
   Web of Science ®Google Scholar
• Yuan GH, Yamamoto M. Lipschitz stability in inverse problems for a Kirchhoff plate equation. Asymptotic Anal. 2007;53:29–60.
   Web of Science ®Google Scholar
• Lasiecka I, Triggiani R. Exact controllability of the Euler–Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: a nonconservative case. SIAM J Control Optim. 1989;27:330–373.
   Web of Science ®Google Scholar
• Lasiecka I, Triggiani R. Exact controllability and uniform stabilization of Kirchoff plates with only one active control in Δw|Σ. J Differ Equ. 1991;93:62–101.
   Web of Science ®Google Scholar
• Zhang X. Exact controllability of the semilinear plate equations. Asympt Anal. 2001;27:95–125.
   Web of Science ®Google Scholar
• Edward J, Tebou L L. Internal null-controllability for a structurally damped beam equation. Asympt Anal. 47;2006:55–83.
   Google Scholar
• Miller L. Non-structural controllability of linear elastic systems with structural damping. J Funct Anal. 2006;236:592–608.
   Web of Science ®Google Scholar