Reachability problems for a wave-wave system with a memory term

Paola Loreti; Daniela Sforza

doi:10.3934/eect.2020018

Article Contents

2020, Volume 9, Issue 1: 87-130. Doi: 10.3934/eect.2020018

This issue Previous Article On quasilinear parabolic equations and continuous maximal regularity Next Article Optimization of the blood pressure with the control in coefficients

Reachability problems for a wave-wave system with a memory term

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: Paola Loreti
^* Corresponding author: Paola Loreti

Received: August 31, 2018

Revised: May 31, 2019

Early access: October 2019

Published: March 2020

Abstract Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

We solve the reachability problem for a coupled wave-wave system with an integro-differential term. The control functions act on one side of the boundary. The estimates on the time is given in terms of the parameters of the problem and they are explicitly computed thanks to Ingham type results. Nevertheless some restrictions appear in our main results. The Hilbert Uniqueness Method is briefly recalled. Our findings can be applied to concrete examples in viscoelasticity theory.

Keywords:

Boundary observability,
reachability,
Fourier series,
hyperbolic integro-differential systems,
abstract linear evolution equations.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

Figure 1. $ P'(x) $ when $ P'(0)>0 $ and $ P'(x_0)<0 $

Download: Full-size image PowerPoint slide

Figure 2. $ P(x) $ when $ P(0)>0 $, $ P(x_1)>0 $ and $ P(x_2)>0 $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	F. Alabau-Boussouira, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., 42 (2003), 871-906. doi: 10.1137/S0363012902402608.
[2]	S. Avdonin, A. Choque Rivero and L. de Teresa, Exact boundary controllability of coupled hyperbolic equations, Int. J. Appl. Math. Comput. Sci., 23 (2013), 701-709. doi: 10.2478/amcs-2013-0052.
[3]	L. Boltzmann, Zur Theorie der elastichen Nachwirkung, Wiener Berichte, 70 (1874), 275-306.
[4]	E. Cesàro, Sur la convergence des séries, Nouvelles Annales de Mathématiques, 7 (1888), 49-59.
[5]	B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys., 33 (1961), 239-249. doi: 10.1103/RevModPhys.33.239.
[6]	C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. doi: 10.1007/BF00251609.
[7]	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.
[8]	F. Dell'Oro, I. Lasiecka and V. Pata, On the MGT equation with memory of type Ⅱ, preprint, arXiv: 1904.08203.
[9]	G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.
[10]	A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465.
[11]	A. E. Ingham, Some trigonometrical inequalities with applications to the theory of series, Math. Z., 41 (1936), 367-379. doi: 10.1007/BF01180426.
[12]	V. Komornik, A new method of exact controllability in short time and applications, Ann. Fac. Sci. Toulouse Math., 10 (1989), 415-464. doi: 10.5802/afst.685.
[13]	V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.
[14]	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.
[15]	V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.
[16]	V. Komornik and P. Loreti, Observability of compactly perturbed systems, J. Math. Anal. Appl., 243 (2000), 409-428. doi: 10.1006/jmaa.1999.6678.
[17]	I. Lasiecka, Mathematical Control Theory of Coupled PDEs, CBMS-NSF Regional Conference Series in Applied Mathematics, 75. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.
[18]	I. Lasiecka, Controllability of a viscoelastic Kirchhoff plate Control and Estimation of Distributed Parameter Systems (Vorau, 1988), Internat. Ser. Numer. Math., Birkhäuser, Basel, 91 (1989), 237-247.
[19]	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.
[20]	G. Lebon, C. Perez-Garcia and J. Casas-Vázquez, On the thermodynamic foundations of viscoelasticity, J. Chem. Phys., 88 (1988), 5068-5075. doi: 10.1063/1.454660.
[21]	G. Leugering, Exact boundary controllability of an integro-differential equation, Appl. Math. Optim., 15 (1987), 223-250. doi: 10.1007/BF01442653.
[22]	G. Leugering, Boundary controllability of a viscoelastic string, in Volterra Integrodifferential Equations in Banach Spaces and Applications, Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 190 (1989), 258-270.
[23]	T. Li and B. P. Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chin. Ann. Math. Ser. B, 34 (2013), 139-160. doi: 10.1007/s11401-012-0754-8.
[24]	J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.
[25]	J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8. Masson, Paris, 1988.
[26]	J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées, 9. Masson, Paris, 1988.
[27]	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.
[28]	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.
[29]	P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim., 35 (1997), 641-653. doi: 10.1137/S036301299526962X.
[30]	J. E. Muñoz Rivera and M. G. Naso, Exact controllability for hyperbolic thermoelastic systems with large memory, Adv. Differential Equations, 9 (2004), 1369-1394.
[31]	M. Najafi, G. R. Sarhangi and H. Wang, The study of the stabilizability of the coupled wave equations under various end conditions, in Proceedings of the 31st IEEE Conference on Decision and Control, Vol. I (Tucson Arizona 1992), New York, (1992), 374-379. doi: 10.1109/CDC.1992.371711.
[32]	J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87. Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.
[33]	Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs, New Series, 26. The Clarendon Press, Oxford University Press, Oxford, 2002.
[34]	M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs Pure Appl.Math, 35. Longman Sci. Tech., Harlow, John Wiley & Sons, Inc., New York, 1987.
[35]	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.
[36]	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.
[37]	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.