Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 748-773.

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.

DOI : 10.1051/cocv/2011169
Classification : 93B05, 35B37, 35L05
Mots clés : wave equation, half-axis, controllability problem, influence operator, Fourier transform, Sobolev space, Moore-Penrose inverse
@article{COCV_2012__18_3_748_0,
     author = {Fardigola, Larissa V.},
     title = {Controllability problems for the {1-D} wave equation on a half-axis with the {Dirichlet} boundary control},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {748--773},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {3},
     year = {2012},
     doi = {10.1051/cocv/2011169},
     mrnumber = {3041663},
     zbl = {1252.93023},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2011169/}
}
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
PY  - 2012
SP  - 748
EP  - 773
VL  - 18
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2011169/
DO  - 10.1051/cocv/2011169
LA  - en
ID  - COCV_2012__18_3_748_0
ER  - 
%0 Journal Article
%A Fardigola, Larissa V.
%T Controllability problems for the 1-D wave equation on a half-axis with the Dirichlet boundary control
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 748-773
%V 18
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2011169/
%R 10.1051/cocv/2011169
%G en
%F COCV_2012__18_3_748_0
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, Tome 18 (2012) no. 3, pp. 748-773. doi : 10.1051/cocv/2011169. http://www.numdam.org/articles/10.1051/cocv/2011169/

[1] M.I. Belishev and A.F. Vakulenko, 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. | MR | Zbl

[2] I. Erdelyi, A generalized inverse for arbitrary operators between Hilbert spaces. Proc. Camb. Philos. Soc. 71 (1972) 43-50. | MR | Zbl

[3] L.V. Fardigola, On controllability problems for the wave equation on a half-plane. J. Math. Phys. Anal., Geom. 1 (2005) 93-115. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[7] S.G. Gindikin and L.R. Volevich, Distributions and convolution equations. Gordon and Breach Sci. Publ., Philadelphia (1992). | MR | Zbl

[8] M. Gugat, Optimal switching boundary control of a string to rest in finite time. ZAMM Angew. Math. Mech. 88 (2008) 283-305. | MR | Zbl

[9] M. Gugat and G. Leugering, L∞-norm minimal control of the wave equation : on the weakness of the bang-bang principle. ESAIM : COCV 14 (2008) 254-283. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. Nauk 394 (2004) 154-158 (in Russian); English translation : Dokl. Math. 69 (2004) 33-37. | MR | Zbl

[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. | MR | Zbl

[14] L. Schwartz, Théorie des distributions 1, 2. Hermann, Paris (1950-1951). | MR | Zbl

[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. | MR | Zbl

[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, Geometriya 9 (2002) 233-242. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :