# American Institute of Mathematical Sciences

March  2018, 7(1): 61-77. doi: 10.3934/eect.2018004

## Inverse observability inequalities for integrodifferential equations in square domains

 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa 16,00161 Roma, Italy

* Corresponding author: Daniela Sforza.

Received  January 2017 Revised  September 2017 Published  January 2018

In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality.

Citation: Paola Loreti, Daniela Sforza. Inverse observability inequalities for integrodifferential equations in square domains. Evolution Equations and Control Theory, 2018, 7 (1) : 61-77. doi: 10.3934/eect.2018004
##### References:
 [1] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055. [2] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [3] C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4. [4] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426. [5] J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110.  doi: 10.1137/0331008. [6] V. Komornik and P. Loreti, Fourier Series in Control Theory Springer Monogr. Math., Springer-Verlag, New York, 2005. doi: 10.1007/b139040. [7] V. Komornik and P. Loreti, Observability of rectangular membranes and plates on small sets, Evol. Equ. Control Theory, 3 (2014), 287-304.  doi: 10.3934/eect.2014.3.287. [8] V. Komornik and P. Loreti, Observability of square membranes by Fourier series methods, Bulletin SUSU MMCS, 8 (2015), 127-140.  doi: 10.14529/mmp150308. [9] G. Lebon, C. Perez-Garcia and J. Casas-Vazquez, On the thermodynamic foundations of viscoelasticity, J. Chem. Phys., 88 (1988), 5068-5075.  doi: 10.1063/1.454660. [10] J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, Siam Rev., 30 (1988), 1-68.  doi: 10.1137/1030001. [11] P. Loreti and D. Sforza, Exact reachability for second order integro-differential equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 1153-1158.  doi: 10.1016/j.crma.2009.08.007. [12] P. Loreti and D. Sforza, Reachability problems for a class of integro-differential equations, J. Differential Equations, 248 (2010), 1711-1755.  doi: 10.1016/j.jde.2009.09.016. [13] P. Loreti and D. Sforza, Multidimensional controllability problems with memory, in Modern Aspects of the Theory of Partial Differential Equations (eds. M. Ruzhansky and J. Wirth), Operator Theory: Advances and Applications 216, Birkhäuser/Springer, Basel, (2011), 261-274. doi: 10.1007/978-3-0348-0069-3_15. [14] M. Mehrenberger, An Ingham type proof for the boundary observability of a $N$-d wave equation, C. R. Math. Acad. Sci. Paris, 347 (2009), 63-68.  doi: 10.1016/j.crma.2008.11.002. [15] J. Prüss, Evolutionary Integral Equations and Applications Monographs in Mathematics, 87 Birkhäuser Verlag, Basel, 1993. doi: 10. 1007/978-3-0348-8570-6. [16] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity Pitman Monogr. Pure Appl. Math., 35 Longman Sci. Tech., Harlow, Essex, 1987. [17] M. Renardy, Are viscoelastic flows under control or out of control?, Systems Control Lett., 54 (2005), 1183-1193.  doi: 10.1016/j.sysconle.2005.04.006.

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##### References:
 [1] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055. [2] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [3] C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations, 7 (1970), 554-569.  doi: 10.1016/0022-0396(70)90101-4. [4] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426. [5] J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110.  doi: 10.1137/0331008. [6] V. Komornik and P. Loreti, Fourier Series in Control Theory Springer Monogr. Math., Springer-Verlag, New York, 2005. doi: 10.1007/b139040. [7] V. Komornik and P. Loreti, Observability of rectangular membranes and plates on small sets, Evol. Equ. Control Theory, 3 (2014), 287-304.  doi: 10.3934/eect.2014.3.287. [8] V. Komornik and P. Loreti, Observability of square membranes by Fourier series methods, Bulletin SUSU MMCS, 8 (2015), 127-140.  doi: 10.14529/mmp150308. [9] G. Lebon, C. Perez-Garcia and J. Casas-Vazquez, On the thermodynamic foundations of viscoelasticity, J. Chem. Phys., 88 (1988), 5068-5075.  doi: 10.1063/1.454660. [10] J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, Siam Rev., 30 (1988), 1-68.  doi: 10.1137/1030001. [11] P. Loreti and D. Sforza, Exact reachability for second order integro-differential equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 1153-1158.  doi: 10.1016/j.crma.2009.08.007. [12] P. Loreti and D. Sforza, Reachability problems for a class of integro-differential equations, J. Differential Equations, 248 (2010), 1711-1755.  doi: 10.1016/j.jde.2009.09.016. [13] P. Loreti and D. Sforza, Multidimensional controllability problems with memory, in Modern Aspects of the Theory of Partial Differential Equations (eds. M. Ruzhansky and J. Wirth), Operator Theory: Advances and Applications 216, Birkhäuser/Springer, Basel, (2011), 261-274. doi: 10.1007/978-3-0348-0069-3_15. [14] M. Mehrenberger, An Ingham type proof for the boundary observability of a $N$-d wave equation, C. R. Math. Acad. Sci. Paris, 347 (2009), 63-68.  doi: 10.1016/j.crma.2008.11.002. [15] J. Prüss, Evolutionary Integral Equations and Applications Monographs in Mathematics, 87 Birkhäuser Verlag, Basel, 1993. doi: 10. 1007/978-3-0348-8570-6. [16] M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity Pitman Monogr. Pure Appl. Math., 35 Longman Sci. Tech., Harlow, Essex, 1987. [17] M. Renardy, Are viscoelastic flows under control or out of control?, Systems Control Lett., 54 (2005), 1183-1193.  doi: 10.1016/j.sysconle.2005.04.006.
Plot of the function $\beta\to\Lambda^{+}_{11}(\beta)+\Lambda^{-}_{11}(\beta)$
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