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

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

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

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topFardigola, 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

top- 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.
- I. Erdelyi, A generalized inverse for arbitrary operators between Hilbert spaces. Proc. Camb. Philos. Soc.71 (1972) 43–50.
- L.V. Fardigola, On controllability problems for the wave equation on a half-plane. J. Math. Phys. Anal., Geom.1 (2005) 93–115.
- 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.
- 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).
- 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.
- S.G. Gindikin and L.R. Volevich, Distributions and convolution equations. Gordon and Breach Sci. Publ., Philadelphia (1992).
- M. Gugat, Optimal switching boundary control of a string to rest in finite time. ZAMM Angew. Math. Mech.88 (2008) 283–305.
- 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.
- M. Gugat, G. Leugering and G.M. Sklyar, Lp-optimal boundary control for the wave equation. SIAM J. Control Optim.44 (2005) 49–74.
- 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.
- E.H. Moore, On the reciprocal of the general algebraic matrix. Bull. Amer. Math. Soc.26 (1920) 394–395.
- R. Penrose, A generalized inverse for matrices. Proc. Camb. Philos. Soc.51 (1955) 406–413.
- L. Schwartz, Théorie des distributions1, 2. Hermann, Paris (1950–1951).
- 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.
- 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.
- 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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