March  2015, 5(1): 31-53. doi: 10.3934/mcrf.2015.5.31

Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition

1. 

Mathematical Division, Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., Kharkiv 61103

Received  March 2014 Revised  August 2014 Published  January 2015

In this paper necessary and sufficient conditions of $L^\infty$-controllability and approximate $L^\infty$-controllability are obtained for the control system $ w_{tt}=\frac{1}{\rho} (k w_x)_x+\gamma w$, $w(0,t)=u(t)$, $x>0$, $t\in(0,T)$. Here $\rho$, $k$, and $\gamma$ are given functions on $[0,+\infty)$; $u\in L^\infty(0,\infty)$ is a control; $T>0$ is a constant. These problems are considered in special modified spaces of the Sobolev type introduced and studied in the paper. The growth of distributions from these spaces is associated with the equation data $\rho$ and $k$. Using some transformation operator introduced and studied in the paper, we see that this control system replicates the controllability properties of the auxiliary system $ z_{tt}=z_{xx}-q^2z$, $z(0,t)=v(t)$, $x>0$, $t\in(0,T)$, and vise versa. Here $q\ge0$ is a constant and $v\in L^\infty(0,\infty)$ is a control. Necessary and sufficient conditions of controllability for the main system are obtained from the ones for the auxiliary system.
Citation: Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control & Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31
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show all references

References:
[1]

Elsevier, Amsterdam, 1973.  Google Scholar

[2]

Zapiski Nauchnykh Seminarov POMI, 332 (2006), 19-37; English translation: J. Math. Sci., 142 (2007), 2528-2539. doi: 10.1007/s10958-007-0140-3.  Google Scholar

[3]

ESAIM: Proceedings, 41 (2013), 15-58. doi: 10.1051/proc/201341002.  Google Scholar

[4]

ESAIM: Control, Optim. Calc. Var., 19 (2013), 301-316. doi: 10.1051/cocv/2012009.  Google Scholar

[5]

SIAM J. Control Optim., 15 (1977), 185-220. doi: 10.1137/0315015.  Google Scholar

[6]

J. Math. Phys., Anal., Geom., 1 (2005), 93-115.  Google Scholar

[7]

SIAM J. Control Optim., 47 (2008), 2179-2199. doi: 10.1137/070684057.  Google Scholar

[8]

Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2009), 36-41.  Google Scholar

[9]

ESAIM: Control, Optim. Calc. Var., 18 (2012), 748-773. doi: 10.1051/cocv/2011169.  Google Scholar

[10]

MCRF, 3 (2013), 161-183. doi: 10.3934/mcrf.2013.3.161.  Google Scholar

[11]

SIAM J. Control Optim., 51 (2013), 1781-1801. doi: 10.1137/110858318.  Google Scholar

[12]

Ukr. Mat. Zh., 59 (2007), 939-952, English translation: Ukr. Math. J., 59 (2007), 1040-1058. doi: 10.1007/s11253-007-0068-2.  Google Scholar

[13]

Gordon and Breach Sci. Publ., Philadelphia, 1992.  Google Scholar

[14]

ZAMM Angew. Math. Mech., 89 (2009), 420-444. doi: 10.1002/zamm.200800196.  Google Scholar

[15]

ESAIM: Control Optim. Calc. Var., 14 (2008), 254-283. doi: 10.1051/cocv:2007044.  Google Scholar

[16]

Applicable Analyis, 92 (2013), 2200-2214. doi: 10.1080/00036811.2012.724404.  Google Scholar

[17]

Ann. Inst. H. Poincaré Sect. A (N.S.), 25 (1976), 105-118.  Google Scholar

[18]

J. Math. Phys, 23 (1982), 258-266. doi: 10.1063/1.525347.  Google Scholar

[19]

Dokl. Akad. Nauk, Ross. Akad. Nauk, 446 (2012), 374-377; English translation: Doklady Mathematics, 86 (2012), 657-660. doi: 10.1134/S106456241205016X.  Google Scholar

[20]

Inverse Problems, 6 (1990), 193-204. doi: 10.1088/0266-5611/6/2/004.  Google Scholar

[21]

Ukr. Mat. Zh., 64 (2012), 525-541; English translation: Ukr. Math. J., 64 (2012), 594-615. doi: 10.1007/s11253-012-0666-5.  Google Scholar

[22]

J. Math. Phys., Anal., Geom., 8 (2012), 307-335.  Google Scholar

[23]

Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2012), 24-29. Google Scholar

[24]

Zh. Vychisl. Mat. i Mat. Fiz., 25 (1985), 548-561.  Google Scholar

[25]

C. R. Acad. Sci. Paris. Sér I Math., 302 (1986), 471-475. doi: 10.1007/BFb0007542.  Google Scholar

[26]

Acta Appl. Math., 128 (2013), 181-191. doi: 10.1007/s10440-013-9825-4.  Google Scholar

[27]

American Mathematical Society, Providence, R.I., 2011.  Google Scholar

[28]

Ann. Inst. Poincaré Anal Non Linéaire, 30 (2013), 1097-1126. doi: 10.1016/j.anihpc.2012.11.005.  Google Scholar

[29]

Afr. Mat., 23 (2012), 1-9. doi: 10.1007/s13370-011-0001-6.  Google Scholar

[30]

J. Math. Anal. Appl., 276 (2002), 109-134. doi: 10.1016/S0022-247X(02)00380-3.  Google Scholar

[31]

SIAM J. Math. Anal., 41 (2009), 1508-1532. doi: 10.1137/080731396.  Google Scholar

[32]

in Proceedings of the International Congress of Mathematicians. Vol. IV, Hindustan Book Agency, New Delhi, 2010, 3008-3034. doi: 10.1007/978-0-387-89488-1.  Google Scholar

[33]

in Handbook of Differential Equations: Evolutionary Equations. Vol. III, Elsevier/North-Holland, Amsterdam, 2007, 527-621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

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