August  2022, 11(4): 1071-1086. doi: 10.3934/eect.2021036

Exact boundary null controllability for a coupled system of plate equations with variable coefficients

College of Science, Beijing Forestry University, Beijing, 100083, China

Received  January 2021 Revised  May 2021 Published  August 2022 Early access  July 2021

This paper studies a coupled system of plate equations with variable coefficients, subject to the clamped boundary conditions. By the Riemannian geometry approach, the duality method, the multiplier technique and a compact perturbation method, we establish exact boundary null controllability of the system under verifiable assumptions.

Citation: Fengyan Yang. Exact boundary null controllability for a coupled system of plate equations with variable coefficients. Evolution Equations and Control Theory, 2022, 11 (4) : 1071-1086. doi: 10.3934/eect.2021036
References:
[1]

B. Allibert and S. Micu, Controllability of analytic functions for a wave equation coupled with a beam, Rev. Mat. Iberoamericana, 15 (1999), 547-592.  doi: 10.4171/RMI/265.

[2]

G. Avalos, The exponential stability of a coupled hyerbolic/parabolic system arising in structural acoustics, Appl. Abstr. Anal., 1 (1996), 203-217.  doi: 10.1155/S1085337596000103.

[3]

G. AvalosI. Lasiecka and R. Rebarber, Boundary controllability of a coupled wave/Kirchoff system, Systems Control Lett., 50 (2003), 331-341.  doi: 10.1016/S0167-6911(03)00179-8.

[4]

K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.

[5]

G. AvalosI. Lasiecka and R. Triggiani, Heat-wave interaction in 2-3 dimensions: Optimal rational decay rate, J. Math. Anal. Appl., 437 (2016), 782-815.  doi: 10.1016/j.jmaa.2015.12.051.

[6]

S. G. Chai and K. S. Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser. A, 26 (2005), 605-612. 

[7]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220.  doi: 10.1137/0315015.

[8]

L. Deng and Z. F. Zhang, Controllability for transmission wave/plate equations on Riemannian manifolds, Systems Control Lett., 91 (2016), 48-54.  doi: 10.1016/j.sysconle.2016.02.016.

[9]

B. Z. Guo and Z. X. Zhang, Well-posedness and regularity for an Euler-Bernoulli plate with variable coefficients and boundary control and observation, Math. Control Signals Systems, 19 (2007), 337-360.  doi: 10.1007/s00498-007-0017-5.

[10]

Y. X. Guo and P. F. Yao, Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback, J. Math. Anal. Appl., 317 (2006), 50-70.  doi: 10.1016/j.jmaa.2005.12.006.

[11]

B. Z. Guo and Z. C. Shao, On well-posedness, regularity and exact controllability for problems of transmission of plate equation with variable coefficients, Quart. Appl. Math., 65 (2007), 705-736.  doi: 10.1090/S0033-569X-07-01069-9.

[12]

F. Hassine, Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping, Discrete Contin. Dyn. Syst. Ser. B., 21 (2016), 1757-1774.  doi: 10.3934/dcdsb.2016021.

[13]

M. A. Horn, Exact controllability of the Euler-Bernoulli plate via bending moments only on the space of optimal regularity, J. Math. Anal. Appl., 167 (1992), 557-581.  doi: 10.1016/0022-247X(92)90224-2.

[14]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, 1994.

[15]

J. E. Lagnese and J. L. Lions, Modeling, Analysis and Control of Thin Plates, Masson, Paris, 1988.

[16]

I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers, Commun. Pure Appl. Anal., 15 (2016), 1515-1543.  doi: 10.3934/cpaa.2016001.

[17]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: A non-conservative case, SIAM J. Control Optim., 27 (1989), 330-373.  doi: 10.1137/0327018.

[18]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment, J. Math. Anal. Appl., 146 (1990), 1-33.  doi: 10.1016/0022-247X(90)90330-I.

[19]

I. Lasiecka and J. T. Webster, Stabilization of a nonlinear flow-plate interaction via component-wise decomposition, Bull. Braz. Math. Soc. (N.S.)., 47 (2016), 489-506.  doi: 10.1007/s00574-016-0164-8.

[20]

J. Li and S. G. Chai, Stabilization of the variable-coefficient structural acoustic model with curved middle surface and delay effects in the structural component, J. Math. Anal. Appl., 454 (2017), 510-532.  doi: 10.1016/j.jmaa.2017.05.001.

[21]

T.-T. Li and B. P. Rao, Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls, Asym. Anal., 86 (2014), 199-226.  doi: 10.3233/ASY-131193.

[22]

T.-T. Li and B. P. Rao, Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls, Chin. Ann. Math. Ser. B, 38 (2007), 473-488.  doi: 10.1007/s11401-017-1078-5.

[23]

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

[24]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer Verlag, New York, 1972.

[25]

W. Liu and G. H. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Aust. Math. Soc., 57 (1998), 305-327.  doi: 10.1017/S0004972700031683.

[26]

W. Liu and G. H. Williams, Exact controllability for problems of transmission of the plate equation with lower-order terms, Quart. Appl. Math., 58 (2000), 37-68.  doi: 10.1090/qam/1738557.

[27]

S. Micu and E. Zuazua, Boundary controllability of a linear hybrid system arising in the control of noise, SIAM J. Control Optim., 35 (1997), 1614-1637.  doi: 10.1137/S0363012996297972.

[28]

N. Ourada and R. Triggiani, Uniform stabilization of the EulerBernoulli equation with feedback operator only in the Neumann boundary condition, Differential Integral Equations, 4 (1991), 277-292. 

[29]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[30]

R. N. Pederson, On the unique continuation theorem for certain second and fourth order elliptic equations, Comm. Pure Appl. Math., 11 (1958), 67-80.  doi: 10.1002/cpa.3160110104.

[31]

R. Szilard, Theories and Applications of Plate Analysis: Classical Numerical and Engineering Methods, John Wiley & Sons, Inc., Hoboken, New Jersey, 2004. doi: 10.1115/1.1849175.

[32]

T. Shirota, A remark on the unique continuation theorem for certain fourth order elliptic equations, Proc. Japan Acad., 36 (1960), 571-573. 

[33]

D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295.  doi: 10.1007/BF01215993.

[34]

E. Ventsel and T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, Inc., New York, 2001. doi: 10.1201/9780203908723.

[35]

R. L. WenS. G. Chai and B. Z. Guo, Well-posedness and regularity of Euler-Bernoulli equation with variable coefficient and Dirichlet boundary control and collocated observation, Math. Methods Appl. Sci., 37 (2014), 2889-2905.  doi: 10.1002/mma.3028.

[36]

P. F. Yao, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, in Differential Geometric Methods in the Control of Partial Differential Equations, Contemp. Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000,383–406. doi: 10.1090/conm/268/04320.

[37]

P. F. Yao, Modeling and Control in Vibrational and Structual Dynamics: A Differential Geometric Approach, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011. doi: 10.1201/b11042.

[38]

F. Y. Yang, Exact controllability of the Euler-Bernoulli plate with variable coefficients and simply supported boundary condition, Electron. J. Diff. Eq., 257 (2016), 1-19. 

[39]

F. Y. YangB. Bin-MohsinG. Chen and P. F. Yao, Exact-approximate boundary controllability of the thermoelastic plate with a curved middle surface, J. Math. Anal. Appl., 451 (2017), 405-433.  doi: 10.1016/j.jmaa.2017.02.005.

[40]

F. Y. YangP. F. Yao and G. Chen, Boundary controllability of structural acoustic systems with variable coefficients and curved walls, Math. Control Signals Syst., 30 (2018), 1-28.  doi: 10.1007/s00498-018-0211-7.

[41]

W. Zhang and Z. F. Zhang, Stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by nonlinear feedbacks, J. Math. Anal. Appl., 422 (2015), 1504-1526.  doi: 10.1016/j.jmaa.2014.09.044.

[42]

X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184 (2007), 49-120.  doi: 10.1007/s00205-006-0020-x.

[43]

E. Zuazua, Controlabilite exacte d'un modele de plaques vibrantes en un temps arbitrairement petit, C. R. Acad. Sci. Paris Ser. I Math., 304 (1987), 173-176. 

show all references

References:
[1]

B. Allibert and S. Micu, Controllability of analytic functions for a wave equation coupled with a beam, Rev. Mat. Iberoamericana, 15 (1999), 547-592.  doi: 10.4171/RMI/265.

[2]

G. Avalos, The exponential stability of a coupled hyerbolic/parabolic system arising in structural acoustics, Appl. Abstr. Anal., 1 (1996), 203-217.  doi: 10.1155/S1085337596000103.

[3]

G. AvalosI. Lasiecka and R. Rebarber, Boundary controllability of a coupled wave/Kirchoff system, Systems Control Lett., 50 (2003), 331-341.  doi: 10.1016/S0167-6911(03)00179-8.

[4]

K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.

[5]

G. AvalosI. Lasiecka and R. Triggiani, Heat-wave interaction in 2-3 dimensions: Optimal rational decay rate, J. Math. Anal. Appl., 437 (2016), 782-815.  doi: 10.1016/j.jmaa.2015.12.051.

[6]

S. G. Chai and K. S. Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser. A, 26 (2005), 605-612. 

[7]

S. Dolecki and D. L. Russell, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220.  doi: 10.1137/0315015.

[8]

L. Deng and Z. F. Zhang, Controllability for transmission wave/plate equations on Riemannian manifolds, Systems Control Lett., 91 (2016), 48-54.  doi: 10.1016/j.sysconle.2016.02.016.

[9]

B. Z. Guo and Z. X. Zhang, Well-posedness and regularity for an Euler-Bernoulli plate with variable coefficients and boundary control and observation, Math. Control Signals Systems, 19 (2007), 337-360.  doi: 10.1007/s00498-007-0017-5.

[10]

Y. X. Guo and P. F. Yao, Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback, J. Math. Anal. Appl., 317 (2006), 50-70.  doi: 10.1016/j.jmaa.2005.12.006.

[11]

B. Z. Guo and Z. C. Shao, On well-posedness, regularity and exact controllability for problems of transmission of plate equation with variable coefficients, Quart. Appl. Math., 65 (2007), 705-736.  doi: 10.1090/S0033-569X-07-01069-9.

[12]

F. Hassine, Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping, Discrete Contin. Dyn. Syst. Ser. B., 21 (2016), 1757-1774.  doi: 10.3934/dcdsb.2016021.

[13]

M. A. Horn, Exact controllability of the Euler-Bernoulli plate via bending moments only on the space of optimal regularity, J. Math. Anal. Appl., 167 (1992), 557-581.  doi: 10.1016/0022-247X(92)90224-2.

[14]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris, 1994.

[15]

J. E. Lagnese and J. L. Lions, Modeling, Analysis and Control of Thin Plates, Masson, Paris, 1988.

[16]

I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers, Commun. Pure Appl. Anal., 15 (2016), 1515-1543.  doi: 10.3934/cpaa.2016001.

[17]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: A non-conservative case, SIAM J. Control Optim., 27 (1989), 330-373.  doi: 10.1137/0327018.

[18]

I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment, J. Math. Anal. Appl., 146 (1990), 1-33.  doi: 10.1016/0022-247X(90)90330-I.

[19]

I. Lasiecka and J. T. Webster, Stabilization of a nonlinear flow-plate interaction via component-wise decomposition, Bull. Braz. Math. Soc. (N.S.)., 47 (2016), 489-506.  doi: 10.1007/s00574-016-0164-8.

[20]

J. Li and S. G. Chai, Stabilization of the variable-coefficient structural acoustic model with curved middle surface and delay effects in the structural component, J. Math. Anal. Appl., 454 (2017), 510-532.  doi: 10.1016/j.jmaa.2017.05.001.

[21]

T.-T. Li and B. P. Rao, Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls, Asym. Anal., 86 (2014), 199-226.  doi: 10.3233/ASY-131193.

[22]

T.-T. Li and B. P. Rao, Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls, Chin. Ann. Math. Ser. B, 38 (2007), 473-488.  doi: 10.1007/s11401-017-1078-5.

[23]

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

[24]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer Verlag, New York, 1972.

[25]

W. Liu and G. H. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Aust. Math. Soc., 57 (1998), 305-327.  doi: 10.1017/S0004972700031683.

[26]

W. Liu and G. H. Williams, Exact controllability for problems of transmission of the plate equation with lower-order terms, Quart. Appl. Math., 58 (2000), 37-68.  doi: 10.1090/qam/1738557.

[27]

S. Micu and E. Zuazua, Boundary controllability of a linear hybrid system arising in the control of noise, SIAM J. Control Optim., 35 (1997), 1614-1637.  doi: 10.1137/S0363012996297972.

[28]

N. Ourada and R. Triggiani, Uniform stabilization of the EulerBernoulli equation with feedback operator only in the Neumann boundary condition, Differential Integral Equations, 4 (1991), 277-292. 

[29]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[30]

R. N. Pederson, On the unique continuation theorem for certain second and fourth order elliptic equations, Comm. Pure Appl. Math., 11 (1958), 67-80.  doi: 10.1002/cpa.3160110104.

[31]

R. Szilard, Theories and Applications of Plate Analysis: Classical Numerical and Engineering Methods, John Wiley & Sons, Inc., Hoboken, New Jersey, 2004. doi: 10.1115/1.1849175.

[32]

T. Shirota, A remark on the unique continuation theorem for certain fourth order elliptic equations, Proc. Japan Acad., 36 (1960), 571-573. 

[33]

D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295.  doi: 10.1007/BF01215993.

[34]

E. Ventsel and T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, Marcel Dekker, Inc., New York, 2001. doi: 10.1201/9780203908723.

[35]

R. L. WenS. G. Chai and B. Z. Guo, Well-posedness and regularity of Euler-Bernoulli equation with variable coefficient and Dirichlet boundary control and collocated observation, Math. Methods Appl. Sci., 37 (2014), 2889-2905.  doi: 10.1002/mma.3028.

[36]

P. F. Yao, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, in Differential Geometric Methods in the Control of Partial Differential Equations, Contemp. Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000,383–406. doi: 10.1090/conm/268/04320.

[37]

P. F. Yao, Modeling and Control in Vibrational and Structual Dynamics: A Differential Geometric Approach, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011. doi: 10.1201/b11042.

[38]

F. Y. Yang, Exact controllability of the Euler-Bernoulli plate with variable coefficients and simply supported boundary condition, Electron. J. Diff. Eq., 257 (2016), 1-19. 

[39]

F. Y. YangB. Bin-MohsinG. Chen and P. F. Yao, Exact-approximate boundary controllability of the thermoelastic plate with a curved middle surface, J. Math. Anal. Appl., 451 (2017), 405-433.  doi: 10.1016/j.jmaa.2017.02.005.

[40]

F. Y. YangP. F. Yao and G. Chen, Boundary controllability of structural acoustic systems with variable coefficients and curved walls, Math. Control Signals Syst., 30 (2018), 1-28.  doi: 10.1007/s00498-018-0211-7.

[41]

W. Zhang and Z. F. Zhang, Stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by nonlinear feedbacks, J. Math. Anal. Appl., 422 (2015), 1504-1526.  doi: 10.1016/j.jmaa.2014.09.044.

[42]

X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Ration. Mech. Anal., 184 (2007), 49-120.  doi: 10.1007/s00205-006-0020-x.

[43]

E. Zuazua, Controlabilite exacte d'un modele de plaques vibrantes en un temps arbitrairement petit, C. R. Acad. Sci. Paris Ser. I Math., 304 (1987), 173-176. 

[1]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations and Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020

[2]

Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665

[3]

Ali Wehbe, Marwa Koumaiha, Layla Toufaily. Boundary observability and exact controllability of strongly coupled wave equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1269-1305. doi: 10.3934/dcdss.2021091

[4]

Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure and Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639

[5]

M. Eller, Roberto Triggiani. Exact/approximate controllability of thermoelastic plates with variable thermal coefficients. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 283-302. doi: 10.3934/dcds.2001.7.283

[6]

Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2893-2905. doi: 10.3934/dcds.2014.34.2893

[7]

Scott W. Hansen, Oleg Yu Imanuvilov. Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions. Mathematical Control and Related Fields, 2011, 1 (2) : 189-230. doi: 10.3934/mcrf.2011.1.189

[8]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020

[9]

Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations and Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052

[10]

Long Hu, Tatsien Li, Bopeng Rao. Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type. Communications on Pure and Applied Analysis, 2014, 13 (2) : 881-901. doi: 10.3934/cpaa.2014.13.881

[11]

Brahim Allal, Abdelkarim Hajjaj, Jawad Salhi, Amine Sbai. Boundary controllability for a coupled system of degenerate/singular parabolic equations. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021055

[12]

Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control and Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31

[13]

Shikuan Mao, Yongqin Liu. Decay property for solutions to plate type equations with variable coefficients. Kinetic and Related Models, 2017, 10 (3) : 785-797. doi: 10.3934/krm.2017031

[14]

Stephanie Flores, Elijah Hight, Everardo Olivares-Vargas, Tamer Oraby, Jose Palacio, Erwin Suazo, Jasang Yoon. Exact and numerical solution of stochastic Burgers equations with variable coefficients. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2735-2750. doi: 10.3934/dcdss.2020224

[15]

Orazio Arena. A problem of boundary controllability for a plate. Evolution Equations and Control Theory, 2013, 2 (4) : 557-562. doi: 10.3934/eect.2013.2.557

[16]

Abdelaziz Bennour, Farid Ammar Khodja, Djamel Teniou. Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evolution Equations and Control Theory, 2017, 6 (4) : 487-516. doi: 10.3934/eect.2017025

[17]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699

[18]

Benzion Shklyar. Exact null-controllability of interconnected abstract evolution equations with unbounded input operators. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 463-479. doi: 10.3934/dcds.2021124

[19]

Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control and Related Fields, 2020, 10 (2) : 217-256. doi: 10.3934/mcrf.2019037

[20]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control and Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (410)
  • HTML views (403)
  • Cited by (0)

Other articles
by authors

[Back to Top]