A remark on Littman's method of boundary controllability

Previous Article
Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions
EECT Home
This Issue
Next Article
On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics

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

Abstract
Related Papers

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 and 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), 109-189. doi: 10.1007/BF02394570.
[2]	M. Atiyah, R. Bott and L. G$\dota$rding, Lacunas for hyperbolic differential operators with constant coefficients. II, Acta Mathematica, 131 (1973), 145-206. doi: 10.1007/BF02392039.
[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., New York, 1978.
[4]	F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems, SIAM Journal on Control and Optimization, 37 (1999), 521-542. doi: 10.1137/S0363012996313835.
[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), 1024-1065. doi: 10.1137/0330055.
[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, North-Holland, (2002), 329-349. doi: 10.1016/S0168-2024(02)80016-9.
[7]	M. Eller, Continuous observability for the anisotropic Maxwell system, Applied Mathematics and Optimization, 55 (2007), 185-201. doi: 10.1007/s00245-006-0886-x.
[8]	M. Eller and D. Toundykov, A global Holmgren theorem for multidimensional hyperbolic partial differential equations, Applicable Analysis, 91 (2012), 69-90. doi: 10.1080/00036811.2010.538685.
[9]	R. Gulliver and W. Littman, Chord uniqueness and controllability: The view from the boundary. I, in Contemporary Mathematics, 268, American Mathematical Society, (2000), 145-175. doi: 10.1090/conm/268/04312.
[10]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order, second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983.
[11]	L. Ho, Observabilité frontière de l'équation des ondes, Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 302 (1986), 443-446.
[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), 99-163.
[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], 256. Springer-Verlag, Berlin, 1983.
[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], 257. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.
[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), 557-581. doi: 10.1016/0022-247X(92)90224-2.
[16]	F. John, Partial Differential Equations, fourth edition, Springer-Verlag, New York, 1991.
[17]	J. E. Lagnese, Exact boundary controllability of Maxwell's equations in a general region, SIAM Journal on Control and Optimization, 27 (1989), 374-388. doi: 10.1137/0327019.
[18]	I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Applied Mathematics and Optimization, 19 (1989), 243-290. doi: 10.1007/BF01448201.
[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), 189-224. doi: 10.1007/BF01182480.
[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, American Mathematical Society, (2000), 227-325. doi: 10.1090/conm/268/04315.
[21]	J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001.
[22]	W. Littman, Near optimal time boundary controllability for a class of hyperbolic equations, in Lecture Notes in Control and Information Science, 97, Springer-Verlag, (1987), 307-312. doi: 10.1007/BFb0038763.
[23]	W. Littman, A remark on boundary control on manifolds, in Lecture Notes in Pure and Applied Mathematics, 242, Chapman & Hall/CRC, (2005), 175-181. doi: 10.1201/9781420028317.ch11.
[24]	W. Littman and S. Taylor, Smoothing evolution equations and boundary control theory, Journal d'Analyse Mathématique, 59 (1992), 117-131. doi: 10.1007/BF02790221.
[25]	R. Melrose and G. Uhlmann, Microlocal structure of involutive conical refraction, Duke Mathematical Journal, 46 (1979), 571-582. doi: 10.1215/S0012-7094-79-04630-1.
[26]	N. Ortner and P. Wagner, On conical refraction in hexagonal and cubic media, SIAM Journal on Applied Mathematics, 70 (2009), 1239-1259. doi: 10.1137/080736636.
[27]	D. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Review, 20 (1978), 639-739. doi: 10.1137/1020095.
[28]	R. Sakamoto, Hyperbolic Boundary Value Problems, Translated from the Japanese by Katsumi Miyahara. Cambridge University Press, Cambridge-New York, 1982.
[29]	D. Tataru, Boundary controllability for conservative PDEs, Applied Mathematics and Optimization, 31 (1995), 257-295.
[30]	D. Tataru, On the regularity of boundary traces for the wave equation, Annali della Scuola Normale Superiore di Pisa, 26 (1998), 185-206.
[31]	M. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34. Princeton University Press, Princeton, N.J., 1981.
[32]	R. Triggiani, Regularity theory, exact controllability, and optimal quadratic cost problem for spherical shells with physical boundary controls, Control and Cybernetics, 25 (1996), 553-568.

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), 109-189. doi: 10.1007/BF02394570.
[2]	M. Atiyah, R. Bott and L. G$\dota$rding, Lacunas for hyperbolic differential operators with constant coefficients. II, Acta Mathematica, 131 (1973), 145-206. doi: 10.1007/BF02392039.
[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., New York, 1978.
[4]	F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems, SIAM Journal on Control and Optimization, 37 (1999), 521-542. doi: 10.1137/S0363012996313835.
[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), 1024-1065. doi: 10.1137/0330055.
[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, North-Holland, (2002), 329-349. doi: 10.1016/S0168-2024(02)80016-9.
[7]	M. Eller, Continuous observability for the anisotropic Maxwell system, Applied Mathematics and Optimization, 55 (2007), 185-201. doi: 10.1007/s00245-006-0886-x.
[8]	M. Eller and D. Toundykov, A global Holmgren theorem for multidimensional hyperbolic partial differential equations, Applicable Analysis, 91 (2012), 69-90. doi: 10.1080/00036811.2010.538685.
[9]	R. Gulliver and W. Littman, Chord uniqueness and controllability: The view from the boundary. I, in Contemporary Mathematics, 268, American Mathematical Society, (2000), 145-175. doi: 10.1090/conm/268/04312.
[10]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order, second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983.
[11]	L. Ho, Observabilité frontière de l'équation des ondes, Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 302 (1986), 443-446.
[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), 99-163.
[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], 256. Springer-Verlag, Berlin, 1983.
[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], 257. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.
[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), 557-581. doi: 10.1016/0022-247X(92)90224-2.
[16]	F. John, Partial Differential Equations, fourth edition, Springer-Verlag, New York, 1991.
[17]	J. E. Lagnese, Exact boundary controllability of Maxwell's equations in a general region, SIAM Journal on Control and Optimization, 27 (1989), 374-388. doi: 10.1137/0327019.
[18]	I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Applied Mathematics and Optimization, 19 (1989), 243-290. doi: 10.1007/BF01448201.
[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), 189-224. doi: 10.1007/BF01182480.
[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, American Mathematical Society, (2000), 227-325. doi: 10.1090/conm/268/04315.
[21]	J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001.
[22]	W. Littman, Near optimal time boundary controllability for a class of hyperbolic equations, in Lecture Notes in Control and Information Science, 97, Springer-Verlag, (1987), 307-312. doi: 10.1007/BFb0038763.
[23]	W. Littman, A remark on boundary control on manifolds, in Lecture Notes in Pure and Applied Mathematics, 242, Chapman & Hall/CRC, (2005), 175-181. doi: 10.1201/9781420028317.ch11.
[24]	W. Littman and S. Taylor, Smoothing evolution equations and boundary control theory, Journal d'Analyse Mathématique, 59 (1992), 117-131. doi: 10.1007/BF02790221.
[25]	R. Melrose and G. Uhlmann, Microlocal structure of involutive conical refraction, Duke Mathematical Journal, 46 (1979), 571-582. doi: 10.1215/S0012-7094-79-04630-1.
[26]	N. Ortner and P. Wagner, On conical refraction in hexagonal and cubic media, SIAM Journal on Applied Mathematics, 70 (2009), 1239-1259. doi: 10.1137/080736636.
[27]	D. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Review, 20 (1978), 639-739. doi: 10.1137/1020095.
[28]	R. Sakamoto, Hyperbolic Boundary Value Problems, Translated from the Japanese by Katsumi Miyahara. Cambridge University Press, Cambridge-New York, 1982.
[29]	D. Tataru, Boundary controllability for conservative PDEs, Applied Mathematics and Optimization, 31 (1995), 257-295.
[30]	D. Tataru, On the regularity of boundary traces for the wave equation, Annali della Scuola Normale Superiore di Pisa, 26 (1998), 185-206.
[31]	M. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34. Princeton University Press, Princeton, N.J., 1981.
[32]	R. Triggiani, Regularity theory, exact controllability, and optimal quadratic cost problem for spherical shells with physical boundary controls, Control and Cybernetics, 25 (1996), 553-568.

[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]	Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155
[4]	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
[5]	Chang-Shou Lin, Lei Zhang. Classification of radial solutions to Liouville systems with singularities. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2617-2637. doi: 10.3934/dcds.2014.34.2617
[6]	Weishi Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 871-884. doi: 10.3934/dcds.2004.10.871
[7]	Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure and 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 and Continuous Dynamical Systems, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447
[9]	Marc Puche, Timo Reis, Felix L. Schwenninger. Funnel control for boundary control systems. Evolution Equations and Control Theory, 2021, 10 (3) : 519-544. doi: 10.3934/eect.2020079
[10]	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
[11]	Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks and Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717
[12]	Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[13]	Dongsheng Kang, Fen Yang. Semilinear elliptic systems involving multiple critical exponents and singularities in $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4247-4263. doi: 10.3934/dcds.2012.32.4247
[14]	Pedro J. Torres. Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 693-698. doi: 10.3934/dcds.2004.11.693
[15]	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 and Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014
[16]	Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501
[17]	Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control and Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004
[18]	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
[19]	Atte Aalto, Jarmo Malinen. Compositions of passive boundary control systems. Mathematical Control and Related Fields, 2013, 3 (1) : 1-19. doi: 10.3934/mcrf.2013.3.1
[20]	Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249

2021 Impact Factor: 1.169

Tools

Metrics

Other articles
by authors

on AIMS
Matthias Eller
on Google Scholar
Matthias Eller

[Back to Top]