American Institute of Mathematical Sciences

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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

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), 12-17. Google Scholar

[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.  Google Scholar

[3]

O. Arena and W. Littman, Boundary control of two PDE's separated by interface conditions, J. Syst. Sci. Complex, 23 (2010), 431-437. 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, World Sci. Publ., Hackensack, NJ, 2010, 357-362. 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), 34-61. 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, Springer-Verlag, New York Inc., 1984. doi: 10.1007/978-1-4612-5208-5.  Google Scholar

[7]

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

[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.  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), 117-131. 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, Contemp. Math., 426, Amer. Math. Soc., Providence, RI (2007), 293-305. 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), 467-479. 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, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[15]

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

[16]

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

[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.  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), 380-438. 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.), IOS Press, Amsterdam, 2001. Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

O. Arena and W. Littman, Boundary control of two PDE's separated by interface conditions, J. Syst. Sci. Complex, 23 (2010), 431-437. 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, World Sci. Publ., Hackensack, NJ, 2010, 357-362. 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), 34-61. 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, Springer-Verlag, New York Inc., 1984. doi: 10.1007/978-1-4612-5208-5.  Google Scholar

[7]

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

[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.  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), 117-131. 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, Contemp. Math., 426, Amer. Math. Soc., Providence, RI (2007), 293-305. 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), 467-479. 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, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[15]

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

[16]

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

[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.  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), 380-438. 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.), IOS Press, Amsterdam, 2001. Google Scholar

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