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December  2013, 2(4): 621-630. doi: 10.3934/eect.2013.2.621

A remark on Littman's method of boundary controllability

Matthias Eller 1,

1. 

Department of Mathematics, Georgetown University, Washington, DC 20057

Received  December 2012 Revised  June 2013 Published  November 2013

We extend the method of exact boundary controllability of strictly hyperbolic equations developed by W. Littman [22,23] to a large class of hyperbolic systems with constant coefficients. Our approach is based on the knowledge of the singularities of the fundamental solution of hyperbolic operators.
Keywords: multiple characteristics, Hyperbolic systems, singularities of fundamental solutions., boundary control.
Mathematics Subject Classification: Primary: 35Q93; Secondary: 53L9.
Citation: Matthias Eller. A remark on Littman's method of boundary controllability. Evolution Equations & Control Theory, 2013, 2 (4) : 621-630. doi: 10.3934/eect.2013.2.621
References:
[1]

M. Atiyah, R. Bott and L. G$\dota$rding, Lacunas for hyperbolic differential operators with constant coefficients. I,, Acta Mathematica, 124 (1970), 109.  doi: 10.1007/BF02394570.  Google Scholar

[2]

M. Atiyah, R. Bott and L. G$\dota$rding, Lacunas for hyperbolic differential operators with constant coefficients. II,, Acta Mathematica, 131 (1973), 145.  doi: 10.1007/BF02392039.  Google Scholar

[3]

L. Ahlfors, Complex Analysis,, An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., (1978).   Google Scholar

[4]

F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems,, SIAM Journal on Control and Optimization, 37 (1999), 521.  doi: 10.1137/S0363012996313835.  Google Scholar

[5]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM Journal on Control and Optimization, 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar

[6]

M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems,, in Studies in Mathematics and its Applications, 31 (2002), 329.  doi: 10.1016/S0168-2024(02)80016-9.  Google Scholar

[7]

M. Eller, Continuous observability for the anisotropic Maxwell system,, Applied Mathematics and Optimization, 55 (2007), 185.  doi: 10.1007/s00245-006-0886-x.  Google Scholar

[8]

M. Eller and D. Toundykov, A global Holmgren theorem for multidimensional hyperbolic partial differential equations,, Applicable Analysis, 91 (2012), 69.  doi: 10.1080/00036811.2010.538685.  Google Scholar

[9]

R. Gulliver and W. Littman, Chord uniqueness and controllability: The view from the boundary. I,, in Contemporary Mathematics, 268 (2000), 145.  doi: 10.1090/conm/268/04312.  Google Scholar

[10]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order,, second edition, (1983).   Google Scholar

[11]

L. Ho, Observabilité frontière de l'équation des ondes,, Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 302 (1986), 443.   Google Scholar

[12]

L. Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations,, L'Enseignement Mathématique. Revue Internationale. IIe Série, 17 (1971), 99.   Google Scholar

[13]

L. Hörmander, The Analysis Of Linear Partial Differential Operators. I,, Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1983).   Google Scholar

[14]

L. Hörmander, The Analysis Of Linear Partial Differential Operators. II,, Differential operators with constant coefficients. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1983).  doi: 10.1007/978-3-642-96750-4.  Google Scholar

[15]

M. A. Horn, Exact controllability of the Euler-Bernoulli plate via bending moments only on the space of optimal regularity,, Journal of Mathematical Analysis and Applications, 167 (1992), 557.  doi: 10.1016/0022-247X(92)90224-2.  Google Scholar

[16]

F. John, Partial Differential Equations,, fourth edition, (1991).   Google Scholar

[17]

J. E. Lagnese, Exact boundary controllability of Maxwell's equations in a general region,, SIAM Journal on Control and Optimization, 27 (1989), 374.  doi: 10.1137/0327019.  Google Scholar

[18]

I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control,, Applied Mathematics and Optimization, 19 (1989), 243.  doi: 10.1007/BF01448201.  Google Scholar

[19]

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions,, Applied Mathematics and Optimization, 25 (1992), 189.  doi: 10.1007/BF01182480.  Google Scholar

[20]

I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot,, in Contemporary Mathematics, 268 (2000), 227.  doi: 10.1090/conm/268/04315.  Google Scholar

[21]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Review, 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar

[22]

W. Littman, Near optimal time boundary controllability for a class of hyperbolic equations,, in Lecture Notes in Control and Information Science, 97 (1987), 307.  doi: 10.1007/BFb0038763.  Google Scholar

[23]

W. Littman, A remark on boundary control on manifolds,, in Lecture Notes in Pure and Applied Mathematics, 242 (2005), 175.  doi: 10.1201/9781420028317.ch11.  Google Scholar

[24]

W. Littman and S. Taylor, Smoothing evolution equations and boundary control theory,, Journal d'Analyse Mathématique, 59 (1992), 117.  doi: 10.1007/BF02790221.  Google Scholar

[25]

R. Melrose and G. Uhlmann, Microlocal structure of involutive conical refraction,, Duke Mathematical Journal, 46 (1979), 571.  doi: 10.1215/S0012-7094-79-04630-1.  Google Scholar

[26]

N. Ortner and P. Wagner, On conical refraction in hexagonal and cubic media,, SIAM Journal on Applied Mathematics, 70 (2009), 1239.  doi: 10.1137/080736636.  Google Scholar

[27]

D. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions,, SIAM Review, 20 (1978), 639.  doi: 10.1137/1020095.  Google Scholar

[28]

R. Sakamoto, Hyperbolic Boundary Value Problems,, Translated from the Japanese by Katsumi Miyahara. Cambridge University Press, (1982).   Google Scholar

[29]

D. Tataru, Boundary controllability for conservative PDEs,, Applied Mathematics and Optimization, 31 (1995), 257.   Google Scholar

[30]

D. Tataru, On the regularity of boundary traces for the wave equation,, Annali della Scuola Normale Superiore di Pisa, 26 (1998), 185.   Google Scholar

[31]

M. Taylor, Pseudodifferential Operators,, Princeton Mathematical Series, (1981).   Google Scholar

[32]

R. Triggiani, Regularity theory, exact controllability, and optimal quadratic cost problem for spherical shells with physical boundary controls,, Control and Cybernetics, 25 (1996), 553.   Google Scholar

show all references

References:
[1]

M. Atiyah, R. Bott and L. G$\dota$rding, Lacunas for hyperbolic differential operators with constant coefficients. I,, Acta Mathematica, 124 (1970), 109.  doi: 10.1007/BF02394570.  Google Scholar

[2]

M. Atiyah, R. Bott and L. G$\dota$rding, Lacunas for hyperbolic differential operators with constant coefficients. II,, Acta Mathematica, 131 (1973), 145.  doi: 10.1007/BF02392039.  Google Scholar

[3]

L. Ahlfors, Complex Analysis,, An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., (1978).   Google Scholar

[4]

F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems,, SIAM Journal on Control and Optimization, 37 (1999), 521.  doi: 10.1137/S0363012996313835.  Google Scholar

[5]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,, SIAM Journal on Control and Optimization, 30 (1992), 1024.  doi: 10.1137/0330055.  Google Scholar

[6]

M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems,, in Studies in Mathematics and its Applications, 31 (2002), 329.  doi: 10.1016/S0168-2024(02)80016-9.  Google Scholar

[7]

M. Eller, Continuous observability for the anisotropic Maxwell system,, Applied Mathematics and Optimization, 55 (2007), 185.  doi: 10.1007/s00245-006-0886-x.  Google Scholar

[8]

M. Eller and D. Toundykov, A global Holmgren theorem for multidimensional hyperbolic partial differential equations,, Applicable Analysis, 91 (2012), 69.  doi: 10.1080/00036811.2010.538685.  Google Scholar

[9]

R. Gulliver and W. Littman, Chord uniqueness and controllability: The view from the boundary. I,, in Contemporary Mathematics, 268 (2000), 145.  doi: 10.1090/conm/268/04312.  Google Scholar

[10]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order,, second edition, (1983).   Google Scholar

[11]

L. Ho, Observabilité frontière de l'équation des ondes,, Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 302 (1986), 443.   Google Scholar

[12]

L. Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations,, L'Enseignement Mathématique. Revue Internationale. IIe Série, 17 (1971), 99.   Google Scholar

[13]

L. Hörmander, The Analysis Of Linear Partial Differential Operators. I,, Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1983).   Google Scholar

[14]

L. Hörmander, The Analysis Of Linear Partial Differential Operators. II,, Differential operators with constant coefficients. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1983).  doi: 10.1007/978-3-642-96750-4.  Google Scholar

[15]

M. A. Horn, Exact controllability of the Euler-Bernoulli plate via bending moments only on the space of optimal regularity,, Journal of Mathematical Analysis and Applications, 167 (1992), 557.  doi: 10.1016/0022-247X(92)90224-2.  Google Scholar

[16]

F. John, Partial Differential Equations,, fourth edition, (1991).   Google Scholar

[17]

J. E. Lagnese, Exact boundary controllability of Maxwell's equations in a general region,, SIAM Journal on Control and Optimization, 27 (1989), 374.  doi: 10.1137/0327019.  Google Scholar

[18]

I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control,, Applied Mathematics and Optimization, 19 (1989), 243.  doi: 10.1007/BF01448201.  Google Scholar

[19]

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions,, Applied Mathematics and Optimization, 25 (1992), 189.  doi: 10.1007/BF01182480.  Google Scholar

[20]

I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot,, in Contemporary Mathematics, 268 (2000), 227.  doi: 10.1090/conm/268/04315.  Google Scholar

[21]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Review, 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar

[22]

W. Littman, Near optimal time boundary controllability for a class of hyperbolic equations,, in Lecture Notes in Control and Information Science, 97 (1987), 307.  doi: 10.1007/BFb0038763.  Google Scholar

[23]

W. Littman, A remark on boundary control on manifolds,, in Lecture Notes in Pure and Applied Mathematics, 242 (2005), 175.  doi: 10.1201/9781420028317.ch11.  Google Scholar

[24]

W. Littman and S. Taylor, Smoothing evolution equations and boundary control theory,, Journal d'Analyse Mathématique, 59 (1992), 117.  doi: 10.1007/BF02790221.  Google Scholar

[25]

R. Melrose and G. Uhlmann, Microlocal structure of involutive conical refraction,, Duke Mathematical Journal, 46 (1979), 571.  doi: 10.1215/S0012-7094-79-04630-1.  Google Scholar

[26]

N. Ortner and P. Wagner, On conical refraction in hexagonal and cubic media,, SIAM Journal on Applied Mathematics, 70 (2009), 1239.  doi: 10.1137/080736636.  Google Scholar

[27]

D. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions,, SIAM Review, 20 (1978), 639.  doi: 10.1137/1020095.  Google Scholar

[28]

R. Sakamoto, Hyperbolic Boundary Value Problems,, Translated from the Japanese by Katsumi Miyahara. Cambridge University Press, (1982).   Google Scholar

[29]

D. Tataru, Boundary controllability for conservative PDEs,, Applied Mathematics and Optimization, 31 (1995), 257.   Google Scholar

[30]

D. Tataru, On the regularity of boundary traces for the wave equation,, Annali della Scuola Normale Superiore di Pisa, 26 (1998), 185.   Google Scholar

[31]

M. Taylor, Pseudodifferential Operators,, Princeton Mathematical Series, (1981).   Google Scholar

[32]

R. Triggiani, Regularity theory, exact controllability, and optimal quadratic cost problem for spherical shells with physical boundary controls,, Control and Cybernetics, 25 (1996), 553.   Google Scholar

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