June  2016, 9(3): 613-618. doi: 10.3934/dcdss.2016015

On some boundary control problems

1. 

Università degli Studi di Firenze, piazza Brunelleschi 6, 50121 Firenze, Italy

Received  March 2015 Revised  July 2015 Published  April 2016

The boundary controllability problems firstly discussed, in this paper, might be described by a one-dimensional $x$-space equation and $t>0$, modeling - at same time $t$ - different physical phenomena in a composite solid made of different materials. These phenomena may be governed, at same time $t$, for example, by the heat equation and by the Schrödinger equation in separate regions. Interface conditions are assumed. Extensions of such boundary controllability problems to two-dimensional $(x,y)$-space are also investigated.
Citation: Orazio Arena. On some boundary control problems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 613-618. doi: 10.3934/dcdss.2016015
References:
[1]

O. Arena, Some problems on boundary controllability for PDE's, Boll. Acad. Gioenia (CT), 46 (2013), 12-17.

[2]

O. Arena, A problem of boundary controllability for a plate, Evol. Equ. and Control Theory, 2 (2013), 557-562. doi: 10.3934/eect.2013.2.557.

[3]

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.

[4]

O. Arena and W. Littman, Null boundary controllability of the Schrödinger equation with a potential, in Progress in Analys and its applications, World Sci. Publ., Hackensack, NJ, 2010, 357-362. doi: 10.1142/9789814313179_0046.

[5]

G. Avalos and I. Lasiecka, The null controllability of thermoelastic 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.

[6]

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.

[7]

L. Hörmander, Linear Partial Differential Operators, Academy Press, New York, 1963.

[8]

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.

[9]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control, Differential Integral Equations, 5 (1992), 521-535.

[10]

W. Littman, Boundary control theory for beams and plates, in Proceedings, 24th Conference on Decision and Control, Ft. Lauderdale, FL, (1985), 2007-2009. doi: 10.1109/CDC.1985.268511.

[11]

W. Littman and S. Taylor, Smoothing evolution equations and boundary control theory, Festschrift on the occasion of the $70^{th}$ birthday of Samuel Agmon, Journal d'Analyse. Mathématique, 59 (1992), 117-131. doi: 10.1007/BF02790221.

[12]

W. Littman and S. Taylor, The heat and Schrödinger equation: Boundary control with one shot, in Control Methods in PDE-Dynamical Systems, Contemp. Math., 426, Amer. Math. Soc., Providence, RI (2007), 293-305. doi: 10.1090/conm/426/08194.

[13]

W. Littman and S. Taylor, The Balayage method: Boundary control of a thermo-elastic plate, Appl. Math. (Warsaw), 35 (2008), 467-479. doi: 10.4064/am35-4-5.

[14]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[15]

I. N. Sneddon, Fourier Transforms, Dover Publ. inc., New York, 1995.

[16]

S. Taylor, Gevrey smoothing properties of the Schrödinger evolution group in weighted Sobodev spaces, J. Math. Anal. Appl., 194 (1985), 14-38. doi: 10.1006/jmaa.1995.1284.

[17]

F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators, Notas de Matemática, 46, Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968.

[18]

X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Differential Equations, 204 (2004), 380-438. doi: 10.1016/j.jde.2004.02.004.

[19]

E. Zuazua, Null control of a 1-d model of mixed hyperbolic-Parabolic Type, in: Optimal Control and PDE (eds. J.L. Menaldi et al.), IOS Press, Amsterdam, 2001.

show all references

References:
[1]

O. Arena, Some problems on boundary controllability for PDE's, Boll. Acad. Gioenia (CT), 46 (2013), 12-17.

[2]

O. Arena, A problem of boundary controllability for a plate, Evol. Equ. and Control Theory, 2 (2013), 557-562. doi: 10.3934/eect.2013.2.557.

[3]

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.

[4]

O. Arena and W. Littman, Null boundary controllability of the Schrödinger equation with a potential, in Progress in Analys and its applications, World Sci. Publ., Hackensack, NJ, 2010, 357-362. doi: 10.1142/9789814313179_0046.

[5]

G. Avalos and I. Lasiecka, The null controllability of thermoelastic 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.

[6]

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.

[7]

L. Hörmander, Linear Partial Differential Operators, Academy Press, New York, 1963.

[8]

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.

[9]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control, Differential Integral Equations, 5 (1992), 521-535.

[10]

W. Littman, Boundary control theory for beams and plates, in Proceedings, 24th Conference on Decision and Control, Ft. Lauderdale, FL, (1985), 2007-2009. doi: 10.1109/CDC.1985.268511.

[11]

W. Littman and S. Taylor, Smoothing evolution equations and boundary control theory, Festschrift on the occasion of the $70^{th}$ birthday of Samuel Agmon, Journal d'Analyse. Mathématique, 59 (1992), 117-131. doi: 10.1007/BF02790221.

[12]

W. Littman and S. Taylor, The heat and Schrödinger equation: Boundary control with one shot, in Control Methods in PDE-Dynamical Systems, Contemp. Math., 426, Amer. Math. Soc., Providence, RI (2007), 293-305. doi: 10.1090/conm/426/08194.

[13]

W. Littman and S. Taylor, The Balayage method: Boundary control of a thermo-elastic plate, Appl. Math. (Warsaw), 35 (2008), 467-479. doi: 10.4064/am35-4-5.

[14]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[15]

I. N. Sneddon, Fourier Transforms, Dover Publ. inc., New York, 1995.

[16]

S. Taylor, Gevrey smoothing properties of the Schrödinger evolution group in weighted Sobodev spaces, J. Math. Anal. Appl., 194 (1985), 14-38. doi: 10.1006/jmaa.1995.1284.

[17]

F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators, Notas de Matemática, 46, Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968.

[18]

X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system, J. Differential Equations, 204 (2004), 380-438. doi: 10.1016/j.jde.2004.02.004.

[19]

E. Zuazua, Null control of a 1-d model of mixed hyperbolic-Parabolic Type, in: Optimal Control and PDE (eds. J.L. Menaldi et al.), IOS Press, Amsterdam, 2001.

[1]

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

[2]

Abdelaziz Khoutaibi, Lahcen Maniar, Omar Oukdach. Null controllability for semilinear heat equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1525-1546. doi: 10.3934/dcdss.2022087

[3]

Salah-Eddine Chorfi, Ghita El Guermai, Lahcen Maniar, Walid Zouhair. Impulse null approximate controllability for heat equation with dynamic boundary conditions. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022026

[4]

Idriss Boutaayamou, Lahcen Maniar, Omar Oukdach. Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions. Evolution Equations and Control Theory, 2022, 11 (4) : 1285-1307. doi: 10.3934/eect.2021044

[5]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations and Control Theory, 2022, 11 (3) : 837-867. doi: 10.3934/eect.2021028

[6]

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

[7]

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

[8]

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

[9]

Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic and Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151

[10]

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

[11]

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

[12]

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

[13]

Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. The homogenization of the heat equation with mixed conditions on randomly subsets of the boundary. Conference Publications, 2013, 2013 (special) : 85-94. doi: 10.3934/proc.2013.2013.85

[14]

Huicong Li. Effective boundary conditions of the heat equation on a body coated by functionally graded material. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1415-1430. doi: 10.3934/dcds.2016.36.1415

[15]

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

[16]

Jean-Paul Chehab, Alejandro A. Franco, Youcef Mammeri. Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (1) : 87-100. doi: 10.3934/dcdss.2017005

[17]

Die Hu, Xianhua Tang, Qi Zhang. Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type. Communications on Pure and Applied Analysis, 2022, 21 (3) : 1071-1091. doi: 10.3934/cpaa.2022010

[18]

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

[19]

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

[20]

Maicon Sônego. Stable solution induced by domain geometry in the heat equation with nonlinear boundary conditions on surfaces of revolution. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5981-5988. doi: 10.3934/dcdsb.2019116

2021 Impact Factor: 1.865

Metrics

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

Other articles
by authors

[Back to Top]