# American Institute of Mathematical Sciences

June  2016, 9(3): 745-757. doi: 10.3934/dcdss.2016026

## Observability of $N$-dimensional integro-differential systems

 1 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sezione di Matematica, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma 2 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa 16 I-00161 Roma, Italy

Received  March 2015 Revised  September 2015 Published  April 2016

The aim of the paper is to show a reachability result for the solution of a multidimensional coupled Petrovsky and wave system when a non local term, expressed as a convolution integral, is active. Motivations to the study are in linear acoustic theory in three dimensions. To achieve that, we prove observability estimates by means of Ingham type inequalities applied to the Fourier series expansion of the solution.
Citation: Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026
##### References:
 [1] G. Gripenberg, S. O. Londen and O. J. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl. 34, Cambridge Univ. Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805. [2] A. Hanyga, Dispersion and attenuation for an acoustic wave equation consistent with viscoelasticity, J. Comput. Acoust., 22 (2014), 1450006, 22 pp. doi: 10.1142/S0218396X14500064. [3] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426. [4] V. Komornik and P. Loreti, Ingham type theorems for vector-valued functions and observability of coupled linear system, SIAM J. Control Optim., 37 (1999), 461-485. doi: 10.1137/S0363012997317505. [5] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monogr. Math., Springer-Verlag, New York, 2005. doi: 10.1007/b139040. [6] J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates, Rech. Math. Appl., Masson, Paris, 1988. [7] I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290. doi: 10.1007/BF01448201. [8] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, with appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 8, Masson, Paris, 1988. [9] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 9, Masson, Paris, 1988. [10] 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. [11] 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. [12] P. Loreti and D. Sforza, Control problems for weakly coupled systems with memory, J. Differential Equations, 257 (2014), 1879-1938. doi: 10.1016/j.jde.2014.05.016. [13] J. E. McDonald, Maxwellian Intepretation of the Laplacian, Am. J. Phys., 33 (1965), 706-711. [14] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [15] 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. [16] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095. [17] R. Triggiani, Exact boundary controllability on $L_2(\Omega)\times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial\Omega$, and related problems, Appl. Math. Optim., 18 (1988), 241-277. doi: 10.1007/BF01443625.

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##### References:
 [1] G. Gripenberg, S. O. Londen and O. J. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl. 34, Cambridge Univ. Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805. [2] A. Hanyga, Dispersion and attenuation for an acoustic wave equation consistent with viscoelasticity, J. Comput. Acoust., 22 (2014), 1450006, 22 pp. doi: 10.1142/S0218396X14500064. [3] A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426. [4] V. Komornik and P. Loreti, Ingham type theorems for vector-valued functions and observability of coupled linear system, SIAM J. Control Optim., 37 (1999), 461-485. doi: 10.1137/S0363012997317505. [5] V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monogr. Math., Springer-Verlag, New York, 2005. doi: 10.1007/b139040. [6] J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates, Rech. Math. Appl., Masson, Paris, 1988. [7] I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290. doi: 10.1007/BF01448201. [8] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, with appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 8, Masson, Paris, 1988. [9] J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 9, Masson, Paris, 1988. [10] 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. [11] 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. [12] P. Loreti and D. Sforza, Control problems for weakly coupled systems with memory, J. Differential Equations, 257 (2014), 1879-1938. doi: 10.1016/j.jde.2014.05.016. [13] J. E. McDonald, Maxwellian Intepretation of the Laplacian, Am. J. Phys., 33 (1965), 706-711. [14] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [15] 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. [16] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095. [17] R. Triggiani, Exact boundary controllability on $L_2(\Omega)\times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial\Omega$, and related problems, Appl. Math. Optim., 18 (1988), 241-277. doi: 10.1007/BF01443625.
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