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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

Orazio Arena 1,

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

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), 431-437. 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), Progress in Analysis and its Applications, (M. Ruzhansky and J. Wirth Eds.) 2010. 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), 34-61. doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[4]

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

[7]

W. Littman, Boundary control Theory for Beams and Plates, Proceedings, 24th Conference on Decision and Control, Ft. Lauderdale, FL, 2007-2009, December 1985. 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), 117-131. 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, Contemporary Math., 426, AMS, Providence, RI, (2007), 293-305. 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), 467-479. 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, Springer, New York, 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), 14-38. doi: 10.1006/jmaa.1995.1284.  Google Scholar

[13]

F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators, Notas de Matemática, No. 46 Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro 1968 iii+238 pp.  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), 380-438. 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., (Eds), Optimal Control and PDE, IOS Press, Amsterdam, 2001. Google Scholar

[16]

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

show all references

References:
[1]

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

[2]

O. Arena and W. Littman, Null Boundary Controllability of the Schrödinger Equation with a Potential, Proceedings $7^{th}$ Int. ISAAC Congress (July 2009), Progress in Analysis and its Applications, (M. Ruzhansky and J. Wirth Eds.) 2010. 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), 34-61. doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[4]

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

[7]

W. Littman, Boundary control Theory for Beams and Plates, Proceedings, 24th Conference on Decision and Control, Ft. Lauderdale, FL, 2007-2009, December 1985. 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), 117-131. 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, Contemporary Math., 426, AMS, Providence, RI, (2007), 293-305. 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), 467-479. 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, Springer, New York, 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), 14-38. doi: 10.1006/jmaa.1995.1284.  Google Scholar

[13]

F. Trèves, Ovcyannikov Theorem and Hyperdifferential Operators, Notas de Matemática, No. 46 Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro 1968 iii+238 pp.  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), 380-438. 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., (Eds), Optimal Control and PDE, IOS Press, Amsterdam, 2001. Google Scholar

[16]

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

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