In this paper necessary and sufficient conditions of L^{∞}-controllability and approximate L^{∞}-controllability are obtained for the control system w_{tt} = w_{xx} - q^{2}w, 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 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://archive.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://archive.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://archive.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, Volume 18 (2012) no. 3, pp. 748-773. doi : 10.1051/cocv/2011169. http://archive.numdam.org/articles/10.1051/cocv/2011169/

[1] 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

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

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

,[4] 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] 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] 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

and ,[7] Distributions and convolution equations. Gordon and Breach Sci. Publ., Philadelphia (1992). | MR | Zbl

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

,[9] 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

and ,[10] Lp-optimal boundary control for the wave equation. SIAM J. Control Optim. 44 (2005) 49-74. | MR | Zbl

, and ,[11] 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

and ,[12] On the reciprocal of the general algebraic matrix. Bull. Amer. Math. Soc. 26 (1920) 394-395.

,[13] A generalized inverse for matrices. Proc. Camb. Philos. Soc. 51 (1955) 406-413. | MR | Zbl

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

,[15] 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

and ,[16] 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

and ,[17] Hardy inequalities, observability, and control for the wave and Schrödinder equations with singular potentials. SIAM J. Math. Anal. 41 (2009) 1508-1532. | MR | Zbl

and ,*Cited by Sources: *