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
References:
[1]

P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions. The Sequential Approach,, Elsevier, (1973).   Google Scholar

[2]

M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbbR^3$ (Russian),, Zapiski Nauchnykh Seminarov POMI, 332 (2006), 19.  doi: 10.1007/s10958-007-0140-3.  Google Scholar

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C. Castro, Exact controllability of the 1-D wave equation from a moving interior point,, ESAIM: Control, 19 (2013), 301.  doi: 10.1051/cocv/2012009.  Google Scholar

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S. Dolecki and D. R. Russel, A general theory of observation and control,, SIAM J. Control Optim., 15 (1977), 185.  doi: 10.1137/0315015.  Google Scholar

[6]

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

[7]

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.  doi: 10.1137/070684057.  Google Scholar

[8]

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

[9]

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

[10]

L. V. Fardigola, Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control,, MCRF, 3 (2013), 161.  doi: 10.3934/mcrf.2013.3.161.  Google Scholar

[11]

L. V. Fardigola, Transformation operators of the Sturm-Liouville problem in controllability problems for the wave equation on a half-axis,, SIAM J. Control Optim., 51 (2013), 1781.  doi: 10.1137/110858318.  Google Scholar

[12]

L. V. Fardigola and K. S. Khalina, Controllability problems for the wave equation (Ukrainian),, Ukr. Mat. Zh., 59 (2007), 939.  doi: 10.1007/s11253-007-0068-2.  Google Scholar

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S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations,, Gordon and Breach Sci. Publ., (1992).   Google Scholar

[14]

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.  doi: 10.1002/zamm.200800196.  Google Scholar

[15]

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.  doi: 10.1051/cocv:2007044.  Google Scholar

[16]

M. Gugat and J. Sokolowski, A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains,, Applicable Analyis, 92 (2013), 2200.  doi: 10.1080/00036811.2012.724404.  Google Scholar

[17]

M. Jaulent and C. Jean, One-dimensional inverse Schrödinger scattering problem with energy-dependent potential, I,, Ann. Inst. H. Poincaré Sect. A (N.S.), 25 (1976), 105.   Google Scholar

[18]

M. Jaulent and C. Jean, Solution of a Schrödinger inverse scattering problem with a polynomial spectral dependence in the potential,, J. Math. Phys, 23 (1982), 258.  doi: 10.1063/1.525347.  Google Scholar

[19]

V. A. Il'in and A. A. Kuleshov, Generalized solutions of the wave equation in the classes $L_p$ and $W_p^1$, $p\ge1$ (Russian),, Dokl. Akad. Nauk, 446 (2012), 374.  doi: 10.1134/S106456241205016X.  Google Scholar

[20]

F. A. Khalilov and E. Ya. Khruslov, Matrix generalisation of the modified Korteweg-de Vries equation,, Inverse Problems, 6 (1990), 193.  doi: 10.1088/0266-5611/6/2/004.  Google Scholar

[21]

K. S. Khalina, Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a half-axis (Ukrainian),, Ukr. Mat. Zh., 64 (2012), 525.  doi: 10.1007/s11253-012-0666-5.  Google Scholar

[22]

K. S. Khalina, On the Neumann boundary controllability for a non-homogeneous string on a half-axis,, J. Math. Phys., 8 (2012), 307.   Google Scholar

[23]

K. S. Khalina, On Dirichlet boundary controllability for a non-homogeneous string on a half-axis (Ukrainian),, Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2012), 24.   Google Scholar

[24]

E. Ya. Khruslov, One-dimensional inverse problems of electrodynamics (Russian),, Zh. Vychisl. Mat. i Mat. Fiz., 25 (1985), 548.   Google Scholar

[25]

J.-L. Lions, Contrôlabilité exacte des systèmes distribués (French) [Exact controllability of distributed systems],, C. R. Acad. Sci. Paris. Sér I Math., 302 (1986), 471.  doi: 10.1007/BFb0007542.  Google Scholar

[26]

Y. Liu, Some sufficient conditions for the controllability of the wave equation with variable coefficients,, Acta Appl. Math., 128 (2013), 181.  doi: 10.1007/s10440-013-9825-4.  Google Scholar

[27]

V. A. Marchenko, Sturm-Liouville Operators and Applications,, American Mathematical Society, (2011).   Google Scholar

[28]

Y. Privat, E. Trélat and E. Zuazua, Optimal location of controllers for the one-dimensional wave equation,, Ann. Inst. Poincaré Anal Non Linéaire, 30 (2013), 1097.  doi: 10.1016/j.anihpc.2012.11.005.  Google Scholar

[29]

Ch. Seck, G. Bayili, A. Séne and M. T. Niane, Contrôlabilité exacte de l'équation des ondes dans des espaces de Sobolev non-réguliers pour un ouvert polygonal (French) [Exact controllability of the wave equation in Sobolev spaces non-regular for an open polygon],, Afr. Mat., 23 (2012), 1.  doi: 10.1007/s13370-011-0001-6.  Google Scholar

[30]

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

[31]

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.  doi: 10.1137/080731396.  Google Scholar

[32]

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, (2010), 3008.  doi: 10.1007/978-0-387-89488-1.  Google Scholar

[33]

E. Zuazua, Controllability and Observability of Partial Differential Equations: Some Results and Open Problems,, in Handbook of Differential Equations: Evolutionary Equations. Vol. III, (2007), 527.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

show all references

References:
[1]

P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions. The Sequential Approach,, Elsevier, (1973).   Google Scholar

[2]

M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbbR^3$ (Russian),, Zapiski Nauchnykh Seminarov POMI, 332 (2006), 19.  doi: 10.1007/s10958-007-0140-3.  Google Scholar

[3]

F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for papabolic problems,, ESAIM: Proceedings, 41 (2013), 15.  doi: 10.1051/proc/201341002.  Google Scholar

[4]

C. Castro, Exact controllability of the 1-D wave equation from a moving interior point,, ESAIM: Control, 19 (2013), 301.  doi: 10.1051/cocv/2012009.  Google Scholar

[5]

S. Dolecki and D. R. Russel, A general theory of observation and control,, SIAM J. Control Optim., 15 (1977), 185.  doi: 10.1137/0315015.  Google Scholar

[6]

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

[7]

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.  doi: 10.1137/070684057.  Google Scholar

[8]

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

[9]

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

[10]

L. V. Fardigola, Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control,, MCRF, 3 (2013), 161.  doi: 10.3934/mcrf.2013.3.161.  Google Scholar

[11]

L. V. Fardigola, Transformation operators of the Sturm-Liouville problem in controllability problems for the wave equation on a half-axis,, SIAM J. Control Optim., 51 (2013), 1781.  doi: 10.1137/110858318.  Google Scholar

[12]

L. V. Fardigola and K. S. Khalina, Controllability problems for the wave equation (Ukrainian),, Ukr. Mat. Zh., 59 (2007), 939.  doi: 10.1007/s11253-007-0068-2.  Google Scholar

[13]

S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations,, Gordon and Breach Sci. Publ., (1992).   Google Scholar

[14]

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.  doi: 10.1002/zamm.200800196.  Google Scholar

[15]

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.  doi: 10.1051/cocv:2007044.  Google Scholar

[16]

M. Gugat and J. Sokolowski, A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains,, Applicable Analyis, 92 (2013), 2200.  doi: 10.1080/00036811.2012.724404.  Google Scholar

[17]

M. Jaulent and C. Jean, One-dimensional inverse Schrödinger scattering problem with energy-dependent potential, I,, Ann. Inst. H. Poincaré Sect. A (N.S.), 25 (1976), 105.   Google Scholar

[18]

M. Jaulent and C. Jean, Solution of a Schrödinger inverse scattering problem with a polynomial spectral dependence in the potential,, J. Math. Phys, 23 (1982), 258.  doi: 10.1063/1.525347.  Google Scholar

[19]

V. A. Il'in and A. A. Kuleshov, Generalized solutions of the wave equation in the classes $L_p$ and $W_p^1$, $p\ge1$ (Russian),, Dokl. Akad. Nauk, 446 (2012), 374.  doi: 10.1134/S106456241205016X.  Google Scholar

[20]

F. A. Khalilov and E. Ya. Khruslov, Matrix generalisation of the modified Korteweg-de Vries equation,, Inverse Problems, 6 (1990), 193.  doi: 10.1088/0266-5611/6/2/004.  Google Scholar

[21]

K. S. Khalina, Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a half-axis (Ukrainian),, Ukr. Mat. Zh., 64 (2012), 525.  doi: 10.1007/s11253-012-0666-5.  Google Scholar

[22]

K. S. Khalina, On the Neumann boundary controllability for a non-homogeneous string on a half-axis,, J. Math. Phys., 8 (2012), 307.   Google Scholar

[23]

K. S. Khalina, On Dirichlet boundary controllability for a non-homogeneous string on a half-axis (Ukrainian),, Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2012), 24.   Google Scholar

[24]

E. Ya. Khruslov, One-dimensional inverse problems of electrodynamics (Russian),, Zh. Vychisl. Mat. i Mat. Fiz., 25 (1985), 548.   Google Scholar

[25]

J.-L. Lions, Contrôlabilité exacte des systèmes distribués (French) [Exact controllability of distributed systems],, C. R. Acad. Sci. Paris. Sér I Math., 302 (1986), 471.  doi: 10.1007/BFb0007542.  Google Scholar

[26]

Y. Liu, Some sufficient conditions for the controllability of the wave equation with variable coefficients,, Acta Appl. Math., 128 (2013), 181.  doi: 10.1007/s10440-013-9825-4.  Google Scholar

[27]

V. A. Marchenko, Sturm-Liouville Operators and Applications,, American Mathematical Society, (2011).   Google Scholar

[28]

Y. Privat, E. Trélat and E. Zuazua, Optimal location of controllers for the one-dimensional wave equation,, Ann. Inst. Poincaré Anal Non Linéaire, 30 (2013), 1097.  doi: 10.1016/j.anihpc.2012.11.005.  Google Scholar

[29]

Ch. Seck, G. Bayili, A. Séne and M. T. Niane, Contrôlabilité exacte de l'équation des ondes dans des espaces de Sobolev non-réguliers pour un ouvert polygonal (French) [Exact controllability of the wave equation in Sobolev spaces non-regular for an open polygon],, Afr. Mat., 23 (2012), 1.  doi: 10.1007/s13370-011-0001-6.  Google Scholar

[30]

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

[31]

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.  doi: 10.1137/080731396.  Google Scholar

[32]

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, (2010), 3008.  doi: 10.1007/978-0-387-89488-1.  Google Scholar

[33]

E. Zuazua, Controllability and Observability of Partial Differential Equations: Some Results and Open Problems,, in Handbook of Differential Equations: Evolutionary Equations. Vol. III, (2007), 527.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

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