American Institute of Mathematical Sciences

Advanced
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

[1]

K. Q. Lan. Multiple positive eigenvalues of conjugate boundary value problems with singularities. Conference Publications, 2003, 2003 (Special) : 501-506. doi: 10.3934/proc.2003.2003.501
[2]

G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377
[3]

Mahesh G. Nerurkar, Héctor J. Sussmann. Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form. Journal of Modern Dynamics, 2007, 1 (2) : 205-253. doi: 10.3934/jmd.2007.1.205
[4]

Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155
[5]

Weishi Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 871-884. doi: 10.3934/dcds.2004.10.871
[6]

Chang-Shou Lin, Lei Zhang. Classification of radial solutions to Liouville systems with singularities. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2617-2637. doi: 10.3934/dcds.2014.34.2617
[7]

Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure & Applied Analysis, 2003, 2 (1) : 77-89. doi: 10.3934/cpaa.2003.2.77
[8]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447
[9]

Felix Sadyrbaev, Inara Yermachenko. Multiple solutions of nonlinear boundary value problems for two-dimensional differential systems. Conference Publications, 2009, 2009 (Special) : 659-668. doi: 10.3934/proc.2009.2009.659
[10]

Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717
[11]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[12]

Dongsheng Kang, Fen Yang. Semilinear elliptic systems involving multiple critical exponents and singularities in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4247-4263. doi: 10.3934/dcds.2012.32.4247
[13]

Volodymyr O. Kapustyan, Ivan O. Pyshnograiev, Olena A. Kapustian. Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014
[14]

Pedro J. Torres. Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 693-698. doi: 10.3934/dcds.2004.11.693
[15]

Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004
[16]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83
[17]

Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501
[18]

Atte Aalto, Jarmo Malinen. Compositions of passive boundary control systems. Mathematical Control & Related Fields, 2013, 3 (1) : 1-19. doi: 10.3934/mcrf.2013.3.1
[19]

Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249
[20]

Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences