December  2013, 2(4): 733-740. doi: 10.3934/eect.2013.2.733

Locally smooth unitary groups and applications to boundary control of PDEs

1. 

Mathematics Department, The University of Auckland, Private Bag 92019, Auckland, New Zealand

Received  July 2013 Revised  September 2013 Published  November 2013

Let $\mathcal{P}$ be the projection operator for a closed subspace $\mathcal{S}$ of a Hilbert space $\mathcal{H}$ and let $U$ be a unitary operator on $\mathcal{H}$. We consider the questions
    1. Under what conditions is $\mathcal{P}U\mathcal{P}$ a strict contraction?
    2. If $g$, $h\in \mathcal{S}$, can we find $f\in \mathcal{H}$ such that $\mathcal{P}f=g$ and $\mathcal{P}Uf=h$?
The results are abstract versions and generalisations of results developed for boundary control of partial differential equations. We discuss how these results can be used as tools in the direct construction of boundary controls.
Citation: Stephen W. Taylor. Locally smooth unitary groups and applications to boundary control of PDEs. Evolution Equations and Control Theory, 2013, 2 (4) : 733-740. doi: 10.3934/eect.2013.2.733
References:
[1]

M. Ben-Artzi and S. Klainerman, Decay and Regularity for the Schrödinger equation, J. Anal. Math., 58 (1992), 25-37. doi: 10.1007/BF02790356.

[2]

N. Burq, Smoothing Effect for Schrödinger boundary value problems, Duke Mathematical Journal, 123 (2004), 403-427. doi: 10.1215/S0012-7094-04-12326-7.

[3]

P. Constantin and J.-C. Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J., 38 (1989), 791-810. doi: 10.1512/iumj.1989.38.38037.

[4]

S.-I. Doi, Remarks on the Cauchy problem for Schrödinger-type equations, Comm. Partial Differential Equations, 21 (1996), 163-178. doi: 10.1080/03605309608821178.

[5]

S.-I. Doi, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds, Duke Math. J., 82 (1996), 679-706. doi: 10.1215/S0012-7094-96-08228-9.

[6]

S.-I. Doi, Smoothing effects for Schrödinger equation and global behaviour of geodesic flow, Math. Ann., 318 (2000), 355-389. doi: 10.1007/s002080000128.

[7]

M. A. Horn and W. Littman, Boundary control of a Schrödinger Equation with nonconstant principal part, Control of Partial Differential Equations and Applications, Lecture Notes in Pure and Applied Mathematics, 174 (1996), Dekker, New York, 101-106.

[8]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of the Schrödinger equation with Dirichlet control, Differential and Integral Equations, 5 (1992), 521-535.

[9]

I. Lasiecka, R. Triggiani and X. Zhang, Global Uniqueness, Observability and Stabilization of Nonconservative Schrödinger Equations via Pointwise Carleman Estimates. Part I: $H^1(\Omega)$-estimates, J. Inverse and Ill-Posed Problems, 12 1-81. Part II: J. Inverse and Ill-Posed Problems, 12 (2004), 182-231. doi: 10.1163/156939404773972761.

[10]

W. Littman, Boundary Control Theory for Beams and Plates, Proceedings, 24th Conference on Decision and Control (December, 1985), Ft Lauderdale, FL. doi: 10.1109/CDC.1985.268511.

[11]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Archive for Rational Mechanics and Analysis, 103 (1988), 193-236. doi: 10.1007/BF00251758.

[12]

W. Littman and S. W. Taylor, Smoothing evolution equations and boundary control theory, J. d'Analyse Math., 59 (1992), 117-131. doi: 10.1007/BF02790221.

[13]

W. Littman and S. W. Taylor, Local Smoothing and Energy Decay for a Semi-Infinite Beam Pinned at Several Points and Applications to Boundary Control, Differential Equations, Dynamical Systems and Control Science, Lecture Notes in Pure and Applied Mathematics, 152 (1994), Dekker, NY, 683-704.

[14]

W. Littman and S. W. Taylor, The Heat and Schrödinger Equations: Boundary Control with One Shot, Control methods in PDE-dynamical systems, Contemp. Math., 426 (2007), Amer. Math. Soc., Providence, RI, 293-305. doi: 10.1090/conm/426/08194.

[15]

W. Littman and S. W. Taylor, The balayage method: Boundary control of a thermo-elastic plate, Applicationes Mathematicae, 35 (2008), 467-479. doi: 10.4064/am35-4-5.

[16]

E. Machtyngier, Controlabilité exact et stabilisation frontiere de l'equation de Schrödinger, (French) [Exact boundary controllability and stabilizability for the Schr?dinger equation], C. R. Acad. Sc. Paris, 310 (1990), 801-806.

[17]

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.

[18]

D. Tataru, A priori estimates of Carleman's type in domains with boundary, J. Math. Pures Appl., 73 (1994), 355-387.

[19]

D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295. doi: 10.1007/BF01215993.

[20]

D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl., (9) 75 (1996), 367-408.

[21]

D. Tataru, Carleman estimates, unique continuation and controllability for anizotropic PDEs, Optimization methods in partial differential equations (South Hadley, MA, 1996, S. Cox and I. Lasiecka, Editors), 267-279, Contemp. Math., 209 (1997), Amer. Math. Soc., Providence, RI. doi: 10.1090/conm/209/02771.

[22]

S. W. Taylor, Gevrey smoothing properties of the Schrödinger evolution group in weighted Sobolev spaces, Journal of Mathematical Analysis and Applications, 194 (1995), 14-38. doi: 10.1006/jmaa.1995.1284.

[23]

S. W. Taylor, Exact Boundary Controllability of a Beam and Mass System, Computation and Control IV, Progress in Systems and Control Theory, Bowers and Lund, editors, Birkhauser, Boston, 1995.

[24]

S. W. Taylor, A smoothing property of a hyperbolic system and boundary controllability, Journal of Computational and Applied Mathematics, 114 (2000), 23-40. doi: 10.1016/S0377-0427(99)00286-1.

[25]

S. W. Taylor and S. Yau, Boundary Control of a Rotating Timoshenko Beam, ANZIAM Journal, 44 (2003), E143-E184.

[26]

R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled non-conservative Schrödinger equations, Dedicated to the memory of Pierre Grisvard. Rend. Istit. Mat. Univ. Trieste, 28 (1996), suppl. (1997), 453-504.

[27]

R. Triggiani and X. Xu, Pointwise Carleman Estimates, Global Uniqueness, Observability, and Stabilization for Schrödinger Equations on Riemannian Manifolds at the $H^1(\Omega)$-Level, Control methods in PDE-dynamical systems, 339-404, Contemp. Math., 426 (2007), Amer. Math. Soc., Providence, RI. doi: 10.1090/conm/426/08197.

[28]

R. Triggiani and P. -F. Yao, Inverse/observability estimates for Schrödinger equations with variable coefficients, Recent advances in control of PDEs. Control Cybernet, 28 (1999), 627-664.

show all references

References:
[1]

M. Ben-Artzi and S. Klainerman, Decay and Regularity for the Schrödinger equation, J. Anal. Math., 58 (1992), 25-37. doi: 10.1007/BF02790356.

[2]

N. Burq, Smoothing Effect for Schrödinger boundary value problems, Duke Mathematical Journal, 123 (2004), 403-427. doi: 10.1215/S0012-7094-04-12326-7.

[3]

P. Constantin and J.-C. Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J., 38 (1989), 791-810. doi: 10.1512/iumj.1989.38.38037.

[4]

S.-I. Doi, Remarks on the Cauchy problem for Schrödinger-type equations, Comm. Partial Differential Equations, 21 (1996), 163-178. doi: 10.1080/03605309608821178.

[5]

S.-I. Doi, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds, Duke Math. J., 82 (1996), 679-706. doi: 10.1215/S0012-7094-96-08228-9.

[6]

S.-I. Doi, Smoothing effects for Schrödinger equation and global behaviour of geodesic flow, Math. Ann., 318 (2000), 355-389. doi: 10.1007/s002080000128.

[7]

M. A. Horn and W. Littman, Boundary control of a Schrödinger Equation with nonconstant principal part, Control of Partial Differential Equations and Applications, Lecture Notes in Pure and Applied Mathematics, 174 (1996), Dekker, New York, 101-106.

[8]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of the Schrödinger equation with Dirichlet control, Differential and Integral Equations, 5 (1992), 521-535.

[9]

I. Lasiecka, R. Triggiani and X. Zhang, Global Uniqueness, Observability and Stabilization of Nonconservative Schrödinger Equations via Pointwise Carleman Estimates. Part I: $H^1(\Omega)$-estimates, J. Inverse and Ill-Posed Problems, 12 1-81. Part II: J. Inverse and Ill-Posed Problems, 12 (2004), 182-231. doi: 10.1163/156939404773972761.

[10]

W. Littman, Boundary Control Theory for Beams and Plates, Proceedings, 24th Conference on Decision and Control (December, 1985), Ft Lauderdale, FL. doi: 10.1109/CDC.1985.268511.

[11]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Archive for Rational Mechanics and Analysis, 103 (1988), 193-236. doi: 10.1007/BF00251758.

[12]

W. Littman and S. W. Taylor, Smoothing evolution equations and boundary control theory, J. d'Analyse Math., 59 (1992), 117-131. doi: 10.1007/BF02790221.

[13]

W. Littman and S. W. Taylor, Local Smoothing and Energy Decay for a Semi-Infinite Beam Pinned at Several Points and Applications to Boundary Control, Differential Equations, Dynamical Systems and Control Science, Lecture Notes in Pure and Applied Mathematics, 152 (1994), Dekker, NY, 683-704.

[14]

W. Littman and S. W. Taylor, The Heat and Schrödinger Equations: Boundary Control with One Shot, Control methods in PDE-dynamical systems, Contemp. Math., 426 (2007), Amer. Math. Soc., Providence, RI, 293-305. doi: 10.1090/conm/426/08194.

[15]

W. Littman and S. W. Taylor, The balayage method: Boundary control of a thermo-elastic plate, Applicationes Mathematicae, 35 (2008), 467-479. doi: 10.4064/am35-4-5.

[16]

E. Machtyngier, Controlabilité exact et stabilisation frontiere de l'equation de Schrödinger, (French) [Exact boundary controllability and stabilizability for the Schr?dinger equation], C. R. Acad. Sc. Paris, 310 (1990), 801-806.

[17]

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.

[18]

D. Tataru, A priori estimates of Carleman's type in domains with boundary, J. Math. Pures Appl., 73 (1994), 355-387.

[19]

D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295. doi: 10.1007/BF01215993.

[20]

D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl., (9) 75 (1996), 367-408.

[21]

D. Tataru, Carleman estimates, unique continuation and controllability for anizotropic PDEs, Optimization methods in partial differential equations (South Hadley, MA, 1996, S. Cox and I. Lasiecka, Editors), 267-279, Contemp. Math., 209 (1997), Amer. Math. Soc., Providence, RI. doi: 10.1090/conm/209/02771.

[22]

S. W. Taylor, Gevrey smoothing properties of the Schrödinger evolution group in weighted Sobolev spaces, Journal of Mathematical Analysis and Applications, 194 (1995), 14-38. doi: 10.1006/jmaa.1995.1284.

[23]

S. W. Taylor, Exact Boundary Controllability of a Beam and Mass System, Computation and Control IV, Progress in Systems and Control Theory, Bowers and Lund, editors, Birkhauser, Boston, 1995.

[24]

S. W. Taylor, A smoothing property of a hyperbolic system and boundary controllability, Journal of Computational and Applied Mathematics, 114 (2000), 23-40. doi: 10.1016/S0377-0427(99)00286-1.

[25]

S. W. Taylor and S. Yau, Boundary Control of a Rotating Timoshenko Beam, ANZIAM Journal, 44 (2003), E143-E184.

[26]

R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled non-conservative Schrödinger equations, Dedicated to the memory of Pierre Grisvard. Rend. Istit. Mat. Univ. Trieste, 28 (1996), suppl. (1997), 453-504.

[27]

R. Triggiani and X. Xu, Pointwise Carleman Estimates, Global Uniqueness, Observability, and Stabilization for Schrödinger Equations on Riemannian Manifolds at the $H^1(\Omega)$-Level, Control methods in PDE-dynamical systems, 339-404, Contemp. Math., 426 (2007), Amer. Math. Soc., Providence, RI. doi: 10.1090/conm/426/08197.

[28]

R. Triggiani and P. -F. Yao, Inverse/observability estimates for Schrödinger equations with variable coefficients, Recent advances in control of PDEs. Control Cybernet, 28 (1999), 627-664.

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