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  March 2020 Early access  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 and 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.

[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.

[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. 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.

[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.

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.

[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.

[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. 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.

[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.

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 $
[1]

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

[2]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

[3]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977

[4]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057

[5]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021051

[6]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems on degenerate integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022025

[7]

Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517

[8]

Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 433-444. doi: 10.3934/naco.2019042

[9]

Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations and Control Theory, 2022, 11 (1) : 1-24. doi: 10.3934/eect.2020100

[10]

Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053

[11]

Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160

[12]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053

[13]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191

[14]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541

[15]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065

[16]

Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems and Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693

[17]

Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015

[18]

Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial and Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119

[19]

Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control and Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044

[20]

Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 639-657. doi: 10.3934/dcdsb.2021059

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (274)
  • HTML views (237)
  • Cited by (0)

Other articles
by authors

[Back to Top]