Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control

Larissa V. Fardigola

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 748-773
  • ISSN: 1292-8119

Abstract

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In this paper necessary and sufficient conditions of L∞-controllability and approximate L∞-controllability are obtained for the control system wtt = wxx − q2w, w(0,t) = u(t), x > 0, t ∈ (0,T), where q ≥ 0, T > 0, u ∈ L∞(0,T) is a control. This system is considered in the Sobolev spaces.

How to cite

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Fardigola, Larissa V.. "Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 748-773. <http://eudml.org/doc/277826>.

@article{Fardigola2012,
abstract = {In this paper necessary and sufficient conditions of L∞-controllability and approximate L∞-controllability are obtained for the control system wtt = wxx − q2w, w(0,t) = u(t), x > 0, t ∈ (0,T), where q ≥ 0, T > 0, u ∈ L∞(0,T) is a control. This system is considered in the Sobolev spaces. },
author = {Fardigola, Larissa V.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Wave equation; half-axis; controllability problem; influence operator; Fourier transform; Sobolev space; Moore-Penrose inverse; wave equation},
language = {eng},
month = {11},
number = {3},
pages = {748-773},
publisher = {EDP Sciences},
title = {Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control},
url = {http://eudml.org/doc/277826},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Fardigola, Larissa V.
TI - Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 748
EP - 773
AB - In this paper necessary and sufficient conditions of L∞-controllability and approximate L∞-controllability are obtained for the control system wtt = wxx − q2w, w(0,t) = u(t), x > 0, t ∈ (0,T), where q ≥ 0, T > 0, u ∈ L∞(0,T) is a control. This system is considered in the Sobolev spaces.
LA - eng
KW - Wave equation; half-axis; controllability problem; influence operator; Fourier transform; Sobolev space; Moore-Penrose inverse; wave equation
UR - http://eudml.org/doc/277826
ER -

References

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  1. M.I. Belishev and A.F. Vakulenko, On a control problem for the wave equation in R3. Zapiski Nauchnykh Seminarov POMI332 (2006) 19–37(in Russian); English translation : J. Math. Sci.142 (2007) 2528–2539.  
  2. I. Erdelyi, A generalized inverse for arbitrary operators between Hilbert spaces. Proc. Camb. Philos. Soc.71 (1972) 43–50.  
  3. L.V. Fardigola, On controllability problems for the wave equation on a half-plane. J. Math. Phys. Anal., Geom.1 (2005) 93–115.  
  4. L.V. Fardigola, Controllability problems for the string equation on a half-axis with a boundary control bounded by a hard constant. SIAM J. Control Optim.47 (2008) 2179–2199.  
  5. L.V. Fardigola, Neumann boundary control problem for the string equation on a half-axis. Dopovidi Natsionalnoi Akademii Nauk Ukrainy (2009) 36–41 (in Ukrainian).  
  6. L.V. Fardigola and K.S. Khalina, Controllability problems for the wave equation. Ukr. Mat. Zh.59 (2007) 939–952(in Ukrainian), English translation : Ukr. Math. J.59 (2007) 1040–1058.  
  7. S.G. Gindikin and L.R. Volevich, Distributions and convolution equations. Gordon and Breach Sci. Publ., Philadelphia (1992).  
  8. M. Gugat, Optimal switching boundary control of a string to rest in finite time. ZAMM Angew. Math. Mech.88 (2008) 283–305.  
  9. M. Gugat and G. Leugering, L∞-norm minimal control of the wave equation : on the weakness of the bang-bang principle. ESAIM : COCV14 (2008) 254–283.  
  10. M. Gugat, G. Leugering and G.M. Sklyar, Lp-optimal boundary control for the wave equation. SIAM J. Control Optim.44 (2005) 49–74.  
  11. V.A. Il’in and E.I. Moiseev, A boundary control at two ends by a process described by the telegraph equation. Dokl. Akad. Nauk, Ross. Akad. Nauk394 (2004) 154–158(in Russian); English translation : Dokl. Math.69 (2004) 33–37.  
  12. E.H. Moore, On the reciprocal of the general algebraic matrix. Bull. Amer. Math. Soc.26 (1920) 394–395.  
  13. R. Penrose, A generalized inverse for matrices. Proc. Camb. Philos. Soc.51 (1955) 406–413.  
  14. L. Schwartz, Théorie des distributions1, 2. Hermann, Paris (1950–1951).  
  15. G.M. Sklyar and L.V. Fardigola, The Markov power moment problem in problems of controllability and frequency extinguishing for the wave equation on a half-axis. J. Math. Anal. Appl.276 (2002) 109–134.  
  16. G.M. Sklyar and L.V. Fardigola, The Markov trigonometric moment problem in controllability problems for the wave equation on a half-axis. Matem. Fizika, Analiz, Geometriya9 (2002) 233–242.  
  17. J. Vancostenoble and E. Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinder equations with singular potentials. SIAM J. Math. Anal.41 (2009) 1508–1532.  

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