March  2020, 9(1): 87-130. doi: 10.3934/eect.2020018

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

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

Received  September 2018 Revised  June 2019 Published  October 2019

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.

Citation: Paola Loreti, Daniela Sforza. Reachability problems for a wave-wave system with a memory term. Evolution Equations & Control Theory, 2020, 9 (1) : 87-130. doi: 10.3934/eect.2020018
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.  Google Scholar

[2]

S. AvdoninA. 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.  Google Scholar

[3]

L. Boltzmann, Zur Theorie der elastichen Nachwirkung, Wiener Berichte, 70 (1874), 275-306.   Google Scholar

[4]

E. Cesàro, Sur la convergence des séries, Nouvelles Annales de Mathématiques, 7 (1888), 49-59.   Google Scholar

[5]

B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys., 33 (1961), 239-249.  doi: 10.1103/RevModPhys.33.239.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

F. Dell'Oro, I. Lasiecka and V. Pata, On the MGT equation with memory of type Ⅱ, preprint, arXiv: 1904.08203. Google Scholar

[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.  Google Scholar

[10]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465.   Google Scholar

[11]

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

[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.  Google Scholar

[13]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[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.  Google Scholar

[15]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

G. LebonC. Perez-Garcia and J. Casas-Vázquez, On the thermodynamic foundations of viscoelasticity, J. Chem. Phys., 88 (1988), 5068-5075.  doi: 10.1063/1.454660.  Google Scholar

[21]

G. Leugering, Exact boundary controllability of an integro-differential equation, Appl. Math. Optim., 15 (1987), 223-250.  doi: 10.1007/BF01442653.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8. Masson, Paris, 1988.  Google Scholar

[26]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées, 9. Masson, Paris, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim., 35 (1997), 641-653.  doi: 10.1137/S036301299526962X.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

S. AvdoninA. 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.  Google Scholar

[3]

L. Boltzmann, Zur Theorie der elastichen Nachwirkung, Wiener Berichte, 70 (1874), 275-306.   Google Scholar

[4]

E. Cesàro, Sur la convergence des séries, Nouvelles Annales de Mathématiques, 7 (1888), 49-59.   Google Scholar

[5]

B. D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys., 33 (1961), 239-249.  doi: 10.1103/RevModPhys.33.239.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

F. Dell'Oro, I. Lasiecka and V. Pata, On the MGT equation with memory of type Ⅱ, preprint, arXiv: 1904.08203. Google Scholar

[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.  Google Scholar

[10]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire, J. Math. Pures Appl., 68 (1989), 457-465.   Google Scholar

[11]

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

[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.  Google Scholar

[13]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar

[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.  Google Scholar

[15]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

G. LebonC. Perez-Garcia and J. Casas-Vázquez, On the thermodynamic foundations of viscoelasticity, J. Chem. Phys., 88 (1988), 5068-5075.  doi: 10.1063/1.454660.  Google Scholar

[21]

G. Leugering, Exact boundary controllability of an integro-differential equation, Appl. Math. Optim., 15 (1987), 223-250.  doi: 10.1007/BF01442653.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8. Masson, Paris, 1988.  Google Scholar

[26]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées, 9. Masson, Paris, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

P. Loreti and V. Valente, Partial exact controllability for spherical membranes, SIAM J. Control Optim., 35 (1997), 641-653.  doi: 10.1137/S036301299526962X.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  $ P'(x) $ when $ P'(0)>0 $ and $ P'(x_0)<0 $
Figure 2.  $ P(x) $ when $ P(0)>0 $, $ P(x_1)>0 $ and $ P(x_2)>0 $
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