On some boundary control problems

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Preface to the special issue in memory of Alfredo Lorenzi

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

On some boundary control problems

Orazio Arena ^1,

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

Received March 2015 Revised July 2015 Published April 2016

Abstract
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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 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.

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