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June  2013, 3(2): 161-183. doi: 10.3934/mcrf.2013.3.161

Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control

Larissa V. Fardigola 1,

1. 

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

Received  February 2012 Revised  November 2012 Published  March 2013

In this paper necessary and sufficient conditions of approximate $L^\infty$-controllability at a free time are obtained for the control system $ w_{tt}=w_{xx}-q^2w$, $w_x(0,t)=u(t)$, $x>0$, $t\in(0,T)$, where $q>0$, $T>0$, $u\in L^\infty(0,T)$ is a control. This system is considered in the Sobolev spaces.
Keywords: influence operator, Fourier transform, controllability problem, half-axis, Sobolev space, Wave equation, Moore-Penrose inverse..
Mathematics Subject Classification: Primary: 93B05, 35L05; Secondary: 35B3.
Citation: Larissa V. Fardigola. Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control. Mathematical Control & Related Fields, 2013, 3 (2) : 161-183. doi: 10.3934/mcrf.2013.3.161
References:
[1]

M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbbR^3$, (in Russian) Zapiski Nauchnykh Seminarov POMI, 332 (2006), 19-37; English translation in J. Math. Sci., 142 (2007), 2528-2539. doi: 10.1007/s10958-007-0140-3.  Google Scholar

[2]

I. Erdélyi, A generalized inverse for arbitrary operators between Hilbert spaces, Proc. Camb. Phil. Soc., 71 (1972), 43-50.  Google Scholar

[3]

L. V. Fardigola, On controllability problems for the wave equation on a half-plane, J. Math. Phys. Anal. Geom., 1 (2005), 93-115.  Google Scholar

[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. doi: 10.1137/070684057.  Google Scholar

[5]

L. V.Fardigola, Neumann boundary control problem for the string equation on a half-axis, (in Ukrainian) Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2009), 36-41. Google Scholar

[6]

L. V.Fardigola, Controllability problems for the 1-d wave equation on a half-axis with the Dirichlet boundary control, ESAIM: Control, Optim. Calc. Var., 18 (2012), 748-773. doi: 10.1051/cocv/2011169.  Google Scholar

[7]

L. V. Fardigola and K. S. Khalina, Controllability problems for the string equation, (in Ukrainian) Ukr. Mat. Zh., 59 (2007), 939-952; English translation in Ukr. Math. J., 59 (2007), 1040-1058. doi: 10.1007/s11253-007-0068-2.  Google Scholar

[8]

I. M. Gelfand and G. E. Shilov, "Generalized Functions," Vol. 2, (in Russian) Fismatgiz, Moscow, 1958. Google Scholar

[9]

S. G. Gindikin and L. R. Volevich, "Distributions and Convolution Equations," Gordon and Breach Sci. Publ., Philadelphia, PA, 1992.  Google Scholar

[10]

M. Gugat, A. Keimer and G. Leugering, Optimal distributed control of the wave equation subject to state constraints, ZAMM Angew. Math. Mech., 89 (2009), 420-444. doi: 10.1002/zamm.200800196.  Google Scholar

[11]

M. Gugat and G. Leugering, $L^\infty$-norm minimal control of the wave equation: On the weakness of the bang-bang principle, ESAIM: Control Optim. Calc. Var., 14 (2008), 254-283. doi: 10.1051/cocv:2007044.  Google Scholar

[12]

M. Gugat, G. Leugering and G. M. Sklyar, $L^p$-optimal boundary control for the wave equation, SIAM J. Control Optim., 44 (2005), 49-74. doi: 10.1137/S0363012903419212.  Google Scholar

[13]

V. A. Il'in and E. I. Moiseev, A boundary control at two ends by a process described by the telegraph equation, (in Russian) Dokl. Akad. Nauk, Ross. Akad. Nauk, 394 (2004), 154-158; English translation in Doklady Mathematics, 69 (2004), 33-37.  Google Scholar

[14]

E. H. Moore, On the reciprocal of the general algebraic matrix, Bull. Amer. Math. Soc., 26 (1920), 394-395. Google Scholar

[15]

R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc., 51 (1955), 406-413.  Google Scholar

[16]

L. Schwartz, "Théorie des Distributions," Vol. 1, 2, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966.  Google Scholar

[17]

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. doi: 10.1016/S0022-247X(02)00380-3.  Google Scholar

[18]

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. doi: 10.1137/080731396.  Google Scholar

[19]

X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, in "Proceedings of the International Congress of Mathematicians," Vol. IV, Hindustan Book Agency, New Delhi, (2010), 3008-3034.  Google Scholar

[20]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in "Handbook of Differential Equations: Evolutionary Equations," Vol. 3, Elsevier/North-Holland, Amsterdam, (2006), 527-621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

show all references

References:
[1]

M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbbR^3$, (in Russian) Zapiski Nauchnykh Seminarov POMI, 332 (2006), 19-37; English translation in J. Math. Sci., 142 (2007), 2528-2539. doi: 10.1007/s10958-007-0140-3.  Google Scholar

[2]

I. Erdélyi, A generalized inverse for arbitrary operators between Hilbert spaces, Proc. Camb. Phil. Soc., 71 (1972), 43-50.  Google Scholar

[3]

L. V. Fardigola, On controllability problems for the wave equation on a half-plane, J. Math. Phys. Anal. Geom., 1 (2005), 93-115.  Google Scholar

[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. doi: 10.1137/070684057.  Google Scholar

[5]

L. V.Fardigola, Neumann boundary control problem for the string equation on a half-axis, (in Ukrainian) Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2009), 36-41. Google Scholar

[6]

L. V.Fardigola, Controllability problems for the 1-d wave equation on a half-axis with the Dirichlet boundary control, ESAIM: Control, Optim. Calc. Var., 18 (2012), 748-773. doi: 10.1051/cocv/2011169.  Google Scholar

[7]

L. V. Fardigola and K. S. Khalina, Controllability problems for the string equation, (in Ukrainian) Ukr. Mat. Zh., 59 (2007), 939-952; English translation in Ukr. Math. J., 59 (2007), 1040-1058. doi: 10.1007/s11253-007-0068-2.  Google Scholar

[8]

I. M. Gelfand and G. E. Shilov, "Generalized Functions," Vol. 2, (in Russian) Fismatgiz, Moscow, 1958. Google Scholar

[9]

S. G. Gindikin and L. R. Volevich, "Distributions and Convolution Equations," Gordon and Breach Sci. Publ., Philadelphia, PA, 1992.  Google Scholar

[10]

M. Gugat, A. Keimer and G. Leugering, Optimal distributed control of the wave equation subject to state constraints, ZAMM Angew. Math. Mech., 89 (2009), 420-444. doi: 10.1002/zamm.200800196.  Google Scholar

[11]

M. Gugat and G. Leugering, $L^\infty$-norm minimal control of the wave equation: On the weakness of the bang-bang principle, ESAIM: Control Optim. Calc. Var., 14 (2008), 254-283. doi: 10.1051/cocv:2007044.  Google Scholar

[12]

M. Gugat, G. Leugering and G. M. Sklyar, $L^p$-optimal boundary control for the wave equation, SIAM J. Control Optim., 44 (2005), 49-74. doi: 10.1137/S0363012903419212.  Google Scholar

[13]

V. A. Il'in and E. I. Moiseev, A boundary control at two ends by a process described by the telegraph equation, (in Russian) Dokl. Akad. Nauk, Ross. Akad. Nauk, 394 (2004), 154-158; English translation in Doklady Mathematics, 69 (2004), 33-37.  Google Scholar

[14]

E. H. Moore, On the reciprocal of the general algebraic matrix, Bull. Amer. Math. Soc., 26 (1920), 394-395. Google Scholar

[15]

R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc., 51 (1955), 406-413.  Google Scholar

[16]

L. Schwartz, "Théorie des Distributions," Vol. 1, 2, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966.  Google Scholar

[17]

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. doi: 10.1016/S0022-247X(02)00380-3.  Google Scholar

[18]

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. doi: 10.1137/080731396.  Google Scholar

[19]

X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, in "Proceedings of the International Congress of Mathematicians," Vol. IV, Hindustan Book Agency, New Delhi, (2010), 3008-3034.  Google Scholar

[20]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in "Handbook of Differential Equations: Evolutionary Equations," Vol. 3, Elsevier/North-Holland, Amsterdam, (2006), 527-621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

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