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), 431-437. 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), Progress in Analysis and its Applications, (M. Ruzhansky and J. Wirth Eds.) 2010. 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), 34-61. doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[4]

L. Hörmander, Linear Partial Differential Operators, Academic Press, New York, 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), 330-373. 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), 521-535.  Google Scholar

[7]

W. Littman, Boundary control Theory for Beams and Plates, Proceedings, 24th Conference on Decision and Control, Ft. Lauderdale, FL, 2007-2009, December 1985. 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), 117-131. 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, Contemporary Math., 426, AMS, Providence, RI, (2007), 293-305. 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), 467-479. 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, Springer, New York, 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), 14-38. doi: 10.1006/jmaa.1995.1284.  Google Scholar

[13]

F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators, Notas de Matemática, No. 46 Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro 1968 iii+238 pp.  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), 380-438. 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., (Eds), Optimal Control and PDE, IOS Press, Amsterdam, 2001. Google Scholar

[16]

J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 262. Springer-Verlag, New York Inc., 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), 431-437. 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), Progress in Analysis and its Applications, (M. Ruzhansky and J. Wirth Eds.) 2010. 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), 34-61. doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[4]

L. Hörmander, Linear Partial Differential Operators, Academic Press, New York, 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), 330-373. 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), 521-535.  Google Scholar

[7]

W. Littman, Boundary control Theory for Beams and Plates, Proceedings, 24th Conference on Decision and Control, Ft. Lauderdale, FL, 2007-2009, December 1985. 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), 117-131. 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, Contemporary Math., 426, AMS, Providence, RI, (2007), 293-305. 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), 467-479. 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, Springer, New York, 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), 14-38. doi: 10.1006/jmaa.1995.1284.  Google Scholar

[13]

F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators, Notas de Matemática, No. 46 Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro 1968 iii+238 pp.  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), 380-438. 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., (Eds), Optimal Control and PDE, IOS Press, Amsterdam, 2001. Google Scholar

[16]

J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 262. Springer-Verlag, New York Inc., 1984. doi: 10.1007/978-1-4612-5208-5.  Google Scholar

[1]

Abdelaziz Khoutaibi, Lahcen Maniar. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evolution Equations & Control Theory, 2020, 9 (2) : 535-559. doi: 10.3934/eect.2020023
[2]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034
[3]

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

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791
[5]

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
[6]

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
[7]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627
[8]

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
[9]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[10]

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
[11]

Idriss Boutaayamou, Lahcen Maniar, Omar Oukdach. Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021044
[12]

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
[13]

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
[14]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179
[15]

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
[16]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, Johannes Lankeit. The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6529-6546. doi: 10.3934/dcds.2020289
[17]

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
[18]

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
[19]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021028
[20]

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

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy