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

Inverse observability inequalities for integrodifferential equations in square domains

Paola Loreti and  Daniela Sforza ,

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.

Keywords: Observability, Fourier series, Ingham type estimates.
Mathematics Subject Classification: Primary: 45K05; Secondary: 93C20.
Citation: Paola Loreti, Daniela Sforza. Inverse observability inequalities for integrodifferential equations in square domains. Evolution Equations & 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.  Google Scholar

[2]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[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.  Google Scholar

[4]

A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

[5]

J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110.  doi: 10.1137/0331008.  Google Scholar

[6]

V. Komornik and P. Loreti, Fourier Series in Control Theory Springer Monogr. Math., Springer-Verlag, New York, 2005. doi: 10.1007/b139040.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, Siam Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Prüss, Evolutionary Integral Equations and Applications Monographs in Mathematics, 87 Birkhäuser Verlag, Basel, 1993. doi: 10. 1007/978-3-0348-8570-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[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.  Google Scholar

[4]

A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379.  doi: 10.1007/BF01180426.  Google Scholar

[5]

J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim., 31 (1993), 101-110.  doi: 10.1137/0331008.  Google Scholar

[6]

V. Komornik and P. Loreti, Fourier Series in Control Theory Springer Monogr. Math., Springer-Verlag, New York, 2005. doi: 10.1007/b139040.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, Siam Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Prüss, Evolutionary Integral Equations and Applications Monographs in Mathematics, 87 Birkhäuser Verlag, Basel, 1993. doi: 10. 1007/978-3-0348-8570-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Plot of the function $\beta\to\Lambda^{+}_{11}(\beta)+\Lambda^{-}_{11}(\beta)$
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