A problem of boundary controllability for a plate

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December 2013, 2(4): 557-562. doi: 10.3934/eect.2013.2.557

A problem of boundary controllability for a plate

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

Abstract
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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. 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), (2009). 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. doi: 10.1016/j.jmaa.2004.01.035. Google Scholar*
[4]	L. Hörmander, Linear Partial Differential Operators,, Academic Press, (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. 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. Google Scholar
[7]	W. Littman, Boundary control Theory for Beams and Plates,, Proceedings, (1985), 2007. 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. 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, (2007), 293. 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. 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, (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. doi: 10.1006/jmaa.1995.1284. Google Scholar*
[13]	F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators,, Notas de Matemática, (1968). 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. 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., (2001). Google Scholar
[16]	J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1984). doi: 10.1007/978-1-4612-5208-5. Google Scholar

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References:

[1]	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
[2]	O. Arena and W. Littman, Null Boundary Controllability of the Schrödinger Equation with a Potential,, Proceedings $7^{th}$ Int. ISAAC Congress (July 2009), (2009). 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. doi: 10.1016/j.jmaa.2004.01.035. Google Scholar*
[4]	L. Hörmander, Linear Partial Differential Operators,, Academic Press, (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. 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. Google Scholar
[7]	W. Littman, Boundary control Theory for Beams and Plates,, Proceedings, (1985), 2007. 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. 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, (2007), 293. 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. 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, (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. doi: 10.1006/jmaa.1995.1284. Google Scholar*
[13]	F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators,, Notas de Matemática, (1968). 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. 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., (2001). Google Scholar
[16]	J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1984). doi: 10.1007/978-1-4612-5208-5. Google Scholar

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