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

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.
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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