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

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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

Abstract
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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 and 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 $\mathbb{R}^3$, (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.
[2]	I. Erdélyi, A generalized inverse for arbitrary operators between Hilbert spaces, Proc. Camb. Phil. Soc., 71 (1972), 43-50.
[3]	L. V. Fardigola, On controllability problems for the wave equation on a half-plane, J. Math. Phys. Anal. Geom., 1 (2005), 93-115.
[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.
[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.
[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.
[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.
[8]	I. M. Gelfand and G. E. Shilov, "Generalized Functions," Vol. 2, (in Russian) Fismatgiz, Moscow, 1958.
[9]	S. G. Gindikin and L. R. Volevich, "Distributions and Convolution Equations," Gordon and Breach Sci. Publ., Philadelphia, PA, 1992.
[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.
[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.
[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.
[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.
[14]	E. H. Moore, On the reciprocal of the general algebraic matrix, Bull. Amer. Math. Soc., 26 (1920), 394-395.
[15]	R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc., 51 (1955), 406-413.
[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.
[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.
[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.
[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.
[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.

show all references

References:

[1]	M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbb{R}^3$, (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.
[2]	I. Erdélyi, A generalized inverse for arbitrary operators between Hilbert spaces, Proc. Camb. Phil. Soc., 71 (1972), 43-50.
[3]	L. V. Fardigola, On controllability problems for the wave equation on a half-plane, J. Math. Phys. Anal. Geom., 1 (2005), 93-115.
[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.
[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.
[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.
[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.
[8]	I. M. Gelfand and G. E. Shilov, "Generalized Functions," Vol. 2, (in Russian) Fismatgiz, Moscow, 1958.
[9]	S. G. Gindikin and L. R. Volevich, "Distributions and Convolution Equations," Gordon and Breach Sci. Publ., Philadelphia, PA, 1992.
[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.
[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.
[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.
[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.
[14]	E. H. Moore, On the reciprocal of the general algebraic matrix, Bull. Amer. Math. Soc., 26 (1920), 394-395.
[15]	R. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc., 51 (1955), 406-413.
[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.
[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.
[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.
[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.
[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.

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