Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-20T21:46:48.389Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  19 September 2011

Larissa V. Fardigola
Larissa V. Fardigola*
Affiliation:
Mathematical Division, Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., 61103 Kharkiv, Ukraine. fardigola@ukr.net
Get access

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.

Keywords

Wave equationhalf-axiscontrollability probleminfluence operatorFourier transformSobolev spaceMoore-Penrose inverse
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 748 - 773
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Belishev, M.I. and Vakulenko, A.F., On a control problem for the wave equation in R3. Zapiski Nauchnykh Seminarov POMI 332 (2006) 19–37 (in Russian); English translation : J. Math. Sci. 142 (2007) 2528–2539. Google Scholar
Erdelyi, I., A generalized inverse for arbitrary operators between Hilbert spaces. Proc. Camb. Philos. Soc. 71 (1972) 43–50. Google Scholar
Fardigola, L.V., On controllability problems for the wave equation on a half-plane. J. Math. Phys. Anal., Geom. 1 (2005) 93–115. Google Scholar
Fardigola, L.V., 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. Google Scholar
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).
Fardigola, L.V. and Khalina, K.S., Controllability problems for the wave equation. Ukr. Mat. Zh. 59 (2007) 939–952 (in Ukrainian), English translation : Ukr. Math. J. 59 (2007) 1040–1058. Google Scholar
S.G. Gindikin and L.R. Volevich, Distributions and convolution equations. Gordon and Breach Sci. Publ., Philadelphia (1992).
Gugat, M., Optimal switching boundary control of a string to rest in finite time. ZAMM Angew. Math. Mech. 88 (2008) 283–305. Google Scholar
Gugat, M. and Leugering, G., L ∞-norm minimal control of the wave equation : on the weakness of the bang-bang principle. ESAIM : COCV 14 (2008) 254–283. Google Scholar
Gugat, M., Leugering, G. and Sklyar, G.M., L p-optimal boundary control for the wave equation. SIAM J. Control Optim. 44 (2005) 49–74. Google Scholar
Il’in, V.A. and Moiseev, E.I., A boundary control at two ends by a process described by the telegraph equation. Dokl. Akad. Nauk, Ross. Akad. Nauk 394 (2004) 154–158 (in Russian); English translation : Dokl. Math. 69 (2004) 33–37. Google Scholar
Moore, E.H., On the reciprocal of the general algebraic matrix. Bull. Amer. Math. Soc. 26 (1920) 394–395. Google Scholar
Penrose, R., A generalized inverse for matrices. Proc. Camb. Philos. Soc. 51 (1955) 406–413. Google Scholar
L. Schwartz, Théorie des distributions 1, 2. Hermann, Paris (1950–1951).
Sklyar, G.M. and Fardigola, L.V., 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. Google Scholar
Sklyar, G.M. and Fardigola, L.V., The Markov trigonometric moment problem in controllability problems for the wave equation on a half-axis. Matem. Fizika, Analiz, Geometriya 9 (2002) 233–242. Google Scholar
Vancostenoble, J. and Zuazua, E., Hardy inequalities, observability, and control for the wave and Schrödinder equations with singular potentials. SIAM J. Math. Anal. 41 (2009) 1508–1532. Google Scholar