# American Institute of Mathematical Sciences

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 & 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.  doi: 10.1007/BF02790356.  Google Scholar [2] N. Burq, Smoothing Effect for Schrödinger boundary value problems,, Duke Mathematical Journal, 123 (2004), 403.  doi: 10.1215/S0012-7094-04-12326-7.  Google Scholar [3] P. Constantin and J.-C. Saut, Local smoothing properties of Schrödinger equations,, Indiana Univ. Math. J., 38 (1989), 791.  doi: 10.1512/iumj.1989.38.38037.  Google Scholar [4] S.-I. Doi, Remarks on the Cauchy problem for Schrödinger-type equations,, Comm. Partial Differential Equations, 21 (1996), 163.  doi: 10.1080/03605309608821178.  Google Scholar [5] S.-I. Doi, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds,, Duke Math. J., 82 (1996), 679.  doi: 10.1215/S0012-7094-96-08228-9.  Google Scholar [6] S.-I. Doi, Smoothing effects for Schrödinger equation and global behaviour of geodesic flow,, Math. Ann., 318 (2000), 355.  doi: 10.1007/s002080000128.  Google Scholar [7] M. A. Horn and W. Littman, Boundary control of a Schrödinger Equation with nonconstant principal part,, Control of Partial Differential Equations and Applications, 174 (1996), 101.   Google Scholar [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.   Google Scholar [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 (2004), 1.  doi: 10.1163/156939404773972761.  Google Scholar [10] W. Littman, Boundary Control Theory for Beams and Plates,, Proceedings, (1985).  doi: 10.1109/CDC.1985.268511.  Google Scholar [11] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Archive for Rational Mechanics and Analysis, 103 (1988), 193.  doi: 10.1007/BF00251758.  Google Scholar [12] W. Littman and S. W. Taylor, Smoothing evolution equations and boundary control theory,, J. d'Analyse Math., 59 (1992), 117.  doi: 10.1007/BF02790221.  Google Scholar [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, 152 (1994), 683.   Google Scholar [14] W. Littman and S. W. Taylor, The Heat and Schrödinger Equations: Boundary Control with One Shot,, Control methods in PDE-dynamical systems, 426 (2007), 293.  doi: 10.1090/conm/426/08194.  Google Scholar [15] W. Littman and S. W. Taylor, The balayage method: Boundary control of a thermo-elastic plate,, Applicationes Mathematicae, 35 (2008), 467.  doi: 10.4064/am35-4-5.  Google Scholar [16] E. Machtyngier, Controlabilité exact et stabilisation frontiere de l'equation de Schrödinger,, (French) [Exact boundary controllability and stabilizability for the Schr?dinger equation], 310 (1990), 801.   Google Scholar [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [18] D. Tataru, A priori estimates of Carleman's type in domains with boundary,, J. Math. Pures Appl., 73 (1994), 355.   Google Scholar [19] D. Tataru, Boundary controllability for conservative PDEs,, Appl. Math. Optim., 31 (1995), 257.  doi: 10.1007/BF01215993.  Google Scholar [20] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems,, J. Math. Pures Appl., 75 (1996), 367.   Google Scholar [21] D. Tataru, Carleman estimates, unique continuation and controllability for anizotropic PDEs,, Optimization methods in partial differential equations (South Hadley, 209 (1997), 267.  doi: 10.1090/conm/209/02771.  Google Scholar [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.  doi: 10.1006/jmaa.1995.1284.  Google Scholar [23] S. W. Taylor, Exact Boundary Controllability of a Beam and Mass System,, Computation and Control IV, (1995).   Google Scholar [24] S. W. Taylor, A smoothing property of a hyperbolic system and boundary controllability,, Journal of Computational and Applied Mathematics, 114 (2000), 23.  doi: 10.1016/S0377-0427(99)00286-1.  Google Scholar [25] S. W. Taylor and S. Yau, Boundary Control of a Rotating Timoshenko Beam,, ANZIAM Journal, 44 (2003).   Google Scholar [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), 453.   Google Scholar [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, 426 (2007), 339.  doi: 10.1090/conm/426/08197.  Google Scholar [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.   Google Scholar

show all references

##### References:
 [1] M. Ben-Artzi and S. Klainerman, Decay and Regularity for the Schrödinger equation,, J. Anal. Math., 58 (1992), 25.  doi: 10.1007/BF02790356.  Google Scholar [2] N. Burq, Smoothing Effect for Schrödinger boundary value problems,, Duke Mathematical Journal, 123 (2004), 403.  doi: 10.1215/S0012-7094-04-12326-7.  Google Scholar [3] P. Constantin and J.-C. Saut, Local smoothing properties of Schrödinger equations,, Indiana Univ. Math. J., 38 (1989), 791.  doi: 10.1512/iumj.1989.38.38037.  Google Scholar [4] S.-I. Doi, Remarks on the Cauchy problem for Schrödinger-type equations,, Comm. Partial Differential Equations, 21 (1996), 163.  doi: 10.1080/03605309608821178.  Google Scholar [5] S.-I. Doi, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds,, Duke Math. J., 82 (1996), 679.  doi: 10.1215/S0012-7094-96-08228-9.  Google Scholar [6] S.-I. Doi, Smoothing effects for Schrödinger equation and global behaviour of geodesic flow,, Math. Ann., 318 (2000), 355.  doi: 10.1007/s002080000128.  Google Scholar [7] M. A. Horn and W. Littman, Boundary control of a Schrödinger Equation with nonconstant principal part,, Control of Partial Differential Equations and Applications, 174 (1996), 101.   Google Scholar [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.   Google Scholar [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 (2004), 1.  doi: 10.1163/156939404773972761.  Google Scholar [10] W. Littman, Boundary Control Theory for Beams and Plates,, Proceedings, (1985).  doi: 10.1109/CDC.1985.268511.  Google Scholar [11] W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity,, Archive for Rational Mechanics and Analysis, 103 (1988), 193.  doi: 10.1007/BF00251758.  Google Scholar [12] W. Littman and S. W. Taylor, Smoothing evolution equations and boundary control theory,, J. d'Analyse Math., 59 (1992), 117.  doi: 10.1007/BF02790221.  Google Scholar [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, 152 (1994), 683.   Google Scholar [14] W. Littman and S. W. Taylor, The Heat and Schrödinger Equations: Boundary Control with One Shot,, Control methods in PDE-dynamical systems, 426 (2007), 293.  doi: 10.1090/conm/426/08194.  Google Scholar [15] W. Littman and S. W. Taylor, The balayage method: Boundary control of a thermo-elastic plate,, Applicationes Mathematicae, 35 (2008), 467.  doi: 10.4064/am35-4-5.  Google Scholar [16] E. Machtyngier, Controlabilité exact et stabilisation frontiere de l'equation de Schrödinger,, (French) [Exact boundary controllability and stabilizability for the Schr?dinger equation], 310 (1990), 801.   Google Scholar [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [18] D. Tataru, A priori estimates of Carleman's type in domains with boundary,, J. Math. Pures Appl., 73 (1994), 355.   Google Scholar [19] D. Tataru, Boundary controllability for conservative PDEs,, Appl. Math. Optim., 31 (1995), 257.  doi: 10.1007/BF01215993.  Google Scholar [20] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems,, J. Math. Pures Appl., 75 (1996), 367.   Google Scholar [21] D. Tataru, Carleman estimates, unique continuation and controllability for anizotropic PDEs,, Optimization methods in partial differential equations (South Hadley, 209 (1997), 267.  doi: 10.1090/conm/209/02771.  Google Scholar [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.  doi: 10.1006/jmaa.1995.1284.  Google Scholar [23] S. W. Taylor, Exact Boundary Controllability of a Beam and Mass System,, Computation and Control IV, (1995).   Google Scholar [24] S. W. Taylor, A smoothing property of a hyperbolic system and boundary controllability,, Journal of Computational and Applied Mathematics, 114 (2000), 23.  doi: 10.1016/S0377-0427(99)00286-1.  Google Scholar [25] S. W. Taylor and S. Yau, Boundary Control of a Rotating Timoshenko Beam,, ANZIAM Journal, 44 (2003).   Google Scholar [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), 453.   Google Scholar [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, 426 (2007), 339.  doi: 10.1090/conm/426/08197.  Google Scholar [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.   Google Scholar
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