# American Institute of Mathematical Sciences

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

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.
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] Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315 [2] Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029 [3] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [4] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434 [5] Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001 [6] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [7] Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021014 [8] Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571 [9] Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 [10] Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294 [11] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [12] Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399 [13] Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179 [14] Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003 [15] Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021004 [16] Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108 [17] Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002 [18] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [19] Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 [20] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

2019 Impact Factor: 0.953

## Metrics

• HTML views (0)
• Cited by (0)

• on AIMS