Figure 1.
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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 |
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.
Mathematics Subject Classification:
Primary: 45K05; Secondary: 93C20.
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
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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. |
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. |
| [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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