On some boundary control problems

Previous Article
Preface to the special issue in memory of Alfredo Lorenzi
DCDS-S Home
This Issue
Next Article
A fractional eigenvalue problem in $\mathbb{R}^N$

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

Abstract
Related Papers

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.

Keywords: Null boundary control, heat equation, Gevrey class., Schrödinger equation, interfaces conditions.

Mathematics Subject Classification: 35, 9.

Citation: Orazio Arena. On some boundary control problems. Discrete & 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. Google Scholar
[2]	O. Arena, A problem of boundary controllability for a plate,, Evol. Equ. and Control Theory, 2 (2013), 557. doi: 10.3934/eect.2013.2.557. Google Scholar
[3]	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
[4]	O. Arena and W. Littman, Null boundary controllability of the Schrödinger equation with a potential,, in Progress in Analys and its applications, (2010), 357. doi: 10.1142/9789814313179_0046. Google Scholar
[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. doi: 10.1016/j.jmaa.2004.01.035. Google Scholar*
[6]	J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 262, 262 (1984). doi: 10.1007/978-1-4612-5208-5. Google Scholar
[7]	L. Hörmander, Linear Partial Differential Operators,, Academy Press, (1963). Google Scholar
[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. doi: 10.1137/0327018. Google Scholar
[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. Google Scholar
[10]	W. Littman, Boundary control theory for beams and plates,, in Proceedings, (1985), 2007. doi: 10.1109/CDC.1985.268511. Google Scholar
[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. doi: 10.1007/BF02790221. Google Scholar
[12]	W. Littman and S. Taylor, The heat and Schrödinger equation: Boundary control with one shot,, in Control Methods in PDE-Dynamical Systems, (2007), 293. doi: 10.1090/conm/426/08194. Google Scholar
[13]	W. Littman and S. Taylor, The Balayage method: Boundary control of a thermo-elastic plate,, Appl. Math. (Warsaw), 35 (2008), 467. doi: 10.4064/am35-4-5. Google Scholar
[14]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[15]	I. N. Sneddon, Fourier Transforms,, Dover Publ. inc., (1995). Google Scholar
[16]	S. Taylor, Gevrey smoothing properties of the Schrödinger evolution group in weighted Sobodev spaces,, J. Math. Anal. Appl., 194 (1985), 14. doi: 10.1006/jmaa.1995.1284. Google Scholar
[17]	F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators,, Notas de Matemática, 46 (1968). Google Scholar
[18]	X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system,, J. Differential Equations, 204 (2004), 380. doi: 10.1016/j.jde.2004.02.004. Google Scholar
[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.), (2001). Google Scholar

show all references

References:

[1]	O. Arena, Some problems on boundary controllability for PDE's,, Boll. Acad. Gioenia (CT), 46 (2013), 12. Google Scholar
[2]	O. Arena, A problem of boundary controllability for a plate,, Evol. Equ. and Control Theory, 2 (2013), 557. doi: 10.3934/eect.2013.2.557. Google Scholar
[3]	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
[4]	O. Arena and W. Littman, Null boundary controllability of the Schrödinger equation with a potential,, in Progress in Analys and its applications, (2010), 357. doi: 10.1142/9789814313179_0046. Google Scholar
[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. doi: 10.1016/j.jmaa.2004.01.035. Google Scholar*
[6]	J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 262, 262 (1984). doi: 10.1007/978-1-4612-5208-5. Google Scholar
[7]	L. Hörmander, Linear Partial Differential Operators,, Academy Press, (1963). Google Scholar
[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. doi: 10.1137/0327018. Google Scholar
[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. Google Scholar
[10]	W. Littman, Boundary control theory for beams and plates,, in Proceedings, (1985), 2007. doi: 10.1109/CDC.1985.268511. Google Scholar
[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. doi: 10.1007/BF02790221. Google Scholar
[12]	W. Littman and S. Taylor, The heat and Schrödinger equation: Boundary control with one shot,, in Control Methods in PDE-Dynamical Systems, (2007), 293. doi: 10.1090/conm/426/08194. Google Scholar
[13]	W. Littman and S. Taylor, The Balayage method: Boundary control of a thermo-elastic plate,, Appl. Math. (Warsaw), 35 (2008), 467. doi: 10.4064/am35-4-5. Google Scholar
[14]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[15]	I. N. Sneddon, Fourier Transforms,, Dover Publ. inc., (1995). Google Scholar
[16]	S. Taylor, Gevrey smoothing properties of the Schrödinger evolution group in weighted Sobodev spaces,, J. Math. Anal. Appl., 194 (1985), 14. doi: 10.1006/jmaa.1995.1284. Google Scholar
[17]	F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators,, Notas de Matemática, 46 (1968). Google Scholar
[18]	X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system,, J. Differential Equations, 204 (2004), 380. doi: 10.1016/j.jde.2004.02.004. Google Scholar
[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.), (2001). 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]	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
[4]	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
[5]	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 & Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151
[6]	Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[7]	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
[8]	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
[9]	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
[10]	Huicong Li. Effective boundary conditions of the heat equation on a body coated by functionally graded material. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1415-1430. doi: 10.3934/dcds.2016.36.1415
[11]	Jean-Paul Chehab, Alejandro A. Franco, Youcef Mammeri. Boundary control of the number of interfaces for the one-dimensional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 87-100. doi: 10.3934/dcdss.2017005
[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]	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
[14]	Maicon Sônego. Stable solution induced by domain geometry in the heat equation with nonlinear boundary conditions on surfaces of revolution. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5981-5988. doi: 10.3934/dcdsb.2019116
[15]	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
[16]	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
[17]	Igor Shevchenko, Barbara Kaltenbacher. Absorbing boundary conditions for the Westervelt equation. Conference Publications, 2015, 2015 (special) : 1000-1008. doi: 10.3934/proc.2015.1000
[18]	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
[19]	Zhongwei Tang. Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5299-5323. doi: 10.3934/dcds.2014.34.5299
[20]	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

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences