American Institute of Mathematical Sciences

Advanced
December  2013, 2(4): 557-562. doi: 10.3934/eect.2013.2.557

A problem of boundary controllability for a plate

Orazio Arena 1,

1. 

Dipartimento di Architettura DIDA, Università degli Studi di Firenze, piazza Brunelleschi, 6 - 50121 Firenze, Italy

Received  March 2013 Revised  September 2013 Published  November 2013

The boundary controllability problem, here discussed, might be described by a two-dimensional space equation modeling, at the same time $t$, different physical phenomena in a composite solid made of different materials. These phenomena may be governed, at the same time $t$, for example, by the heat equation and by the Schrödinger equation in two separate regions. Interface transmission conditions are imposed.
Keywords: interface condition, Null boundary control, Schrödinger equation, Gevrey class., heat equation.
Mathematics Subject Classification: 35, 9.
Citation: Orazio Arena. A problem of boundary controllability for a plate. Evolution Equations & Control Theory, 2013, 2 (4) : 557-562. doi: 10.3934/eect.2013.2.557
References:
[1]

O. Arena and W. Littman, Boundary Control of Two PDE's Separated by Interface Conditions,, J. Syst. Sci. Complex, 23 (2010), 431.  doi: 10.1007/s11424-010-0138-7.  Google Scholar

[2]

O. Arena and W. Littman, Null Boundary Controllability of the Schrödinger Equation with a Potential,, Proceedings $7^{th}$ Int. ISAAC Congress (July 2009), (2009).  doi: 10.1142/9789814313179_0046.  Google Scholar

[3]

G. Avalos and I. Lasiecka, The null controllability of thermo-elastic plates and singularity of the associated minimal energy function,, J. Math. Anal. Appl., 294 (2004), 34.  doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[4]

L. Hörmander, Linear Partial Differential Operators,, Academic Press, (1963).   Google Scholar

[5]

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 Opt., 27 (1989), 330.  doi: 10.1137/0327018.  Google Scholar

[6]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control,, Differential and Integral Equations, 5 (1992), 521.   Google Scholar

[7]

W. Littman, Boundary control Theory for Beams and Plates,, Proceedings, (1985), 2007.  doi: 10.1109/CDC.1985.268511.  Google Scholar

[8]

W. Littman and S. Taylor, Smoothing Evolution Equations and Boundary control theory. Festschrift on the occasion of the 70th birthday of Shmuel Agmon,, Journal d'Analyse. Mathématique, 59 (1992), 117.  doi: 10.1007/BF02790221.  Google Scholar

[9]

W. Littman and S. Taylor, The heat and schrödinger equation boundary control with one shot,, Control Methods in PDE-Dynamical Systems, (2007), 293.  doi: 10.1090/conm/426/08194.  Google Scholar

[10]

W. Littman and S. Taylor, The balayage method: Boundary control of a thermo-elastic plate,, Applicationes Math., 35 (2008), 467.  doi: 10.4064/am35-4-5.  Google Scholar

[11]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Appl. Math. Sci. 44, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[12]

S. Taylor, Gevrey smoothing properties of the schrödinger evolution group in weighted sobodev spaces,, Journal of Math. Anal. and Appl., 194 (1995), 14.  doi: 10.1006/jmaa.1995.1284.  Google Scholar

[13]

F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators,, Notas de Matemática, (1968).   Google Scholar

[14]

X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system,, Journal of Diff. Eq., 204 (2004), 380.  doi: 10.1016/j.jde.2004.02.004.  Google Scholar

[15]

E. Zuazua, Null Control of a 1-d Model of Mixed Hyperbolic-Parabolic Type,, in: J. L. Menaldi et al., (2001).   Google Scholar

[16]

J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1984).  doi: 10.1007/978-1-4612-5208-5.  Google Scholar

show all references

References:
[1]

O. Arena and W. Littman, Boundary Control of Two PDE's Separated by Interface Conditions,, J. Syst. Sci. Complex, 23 (2010), 431.  doi: 10.1007/s11424-010-0138-7.  Google Scholar

[2]

O. Arena and W. Littman, Null Boundary Controllability of the Schrödinger Equation with a Potential,, Proceedings $7^{th}$ Int. ISAAC Congress (July 2009), (2009).  doi: 10.1142/9789814313179_0046.  Google Scholar

[3]

G. Avalos and I. Lasiecka, The null controllability of thermo-elastic plates and singularity of the associated minimal energy function,, J. Math. Anal. Appl., 294 (2004), 34.  doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[4]

L. Hörmander, Linear Partial Differential Operators,, Academic Press, (1963).   Google Scholar

[5]

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 Opt., 27 (1989), 330.  doi: 10.1137/0327018.  Google Scholar

[6]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control,, Differential and Integral Equations, 5 (1992), 521.   Google Scholar

[7]

W. Littman, Boundary control Theory for Beams and Plates,, Proceedings, (1985), 2007.  doi: 10.1109/CDC.1985.268511.  Google Scholar

[8]

W. Littman and S. Taylor, Smoothing Evolution Equations and Boundary control theory. Festschrift on the occasion of the 70th birthday of Shmuel Agmon,, Journal d'Analyse. Mathématique, 59 (1992), 117.  doi: 10.1007/BF02790221.  Google Scholar

[9]

W. Littman and S. Taylor, The heat and schrödinger equation boundary control with one shot,, Control Methods in PDE-Dynamical Systems, (2007), 293.  doi: 10.1090/conm/426/08194.  Google Scholar

[10]

W. Littman and S. Taylor, The balayage method: Boundary control of a thermo-elastic plate,, Applicationes Math., 35 (2008), 467.  doi: 10.4064/am35-4-5.  Google Scholar

[11]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Appl. Math. Sci. 44, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[12]

S. Taylor, Gevrey smoothing properties of the schrödinger evolution group in weighted sobodev spaces,, Journal of Math. Anal. and Appl., 194 (1995), 14.  doi: 10.1006/jmaa.1995.1284.  Google Scholar

[13]

F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators,, Notas de Matemática, (1968).   Google Scholar

[14]

X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system,, Journal of Diff. Eq., 204 (2004), 380.  doi: 10.1016/j.jde.2004.02.004.  Google Scholar

[15]

E. Zuazua, Null Control of a 1-d Model of Mixed Hyperbolic-Parabolic Type,, in: J. L. Menaldi et al., (2001).   Google Scholar

[16]

J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1984).  doi: 10.1007/978-1-4612-5208-5.  Google Scholar

[1]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161
[2]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791
[3]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878
[4]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147
[5]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627
[6]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143
[7]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[8]

Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016
[9]

Vyacheslav A. Trofimov, Evgeny M. Trykin. A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation. Conference Publications, 2015, 2015 (special) : 1070-1078. doi: 10.3934/proc.2015.1070
[10]

Luz de Teresa, Enrique Zuazua. Identification of the class of initial data for the insensitizing control of the heat equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 457-471. doi: 10.3934/cpaa.2009.8.457
[11]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179
[12]

Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033
[13]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[14]

Haoyue Cui, Dongyi Liu, Genqi Xu. Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback. Mathematical Control & Related Fields, 2018, 8 (2) : 383-395. doi: 10.3934/mcrf.2018015
[15]

Seda İǧret Araz, Murat Subașı. On the optimal coefficient control in a heat equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020124
[16]

Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481
[17]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689
[18]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[19]

Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations & Control Theory, 2014, 3 (1) : 35-58. doi: 10.3934/eect.2014.3.35
[20]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences