A remark on Littman's method of boundary controllability

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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

1.	Department of Mathematics, Georgetown University, Washington, DC 20057

Received December 2012 Revised June 2013 Published November 2013

Abstract
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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

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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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