American Institute of Mathematical Sciences

Advanced
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

Larissa V. Fardigola 1,

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.
Keywords: transformation operator, controllability problem, Wave equation, modified space of the Sobolev type., half-axis.
Mathematics Subject Classification: Primary: 93B05, 35B30; Secondary: 35L0.
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, Amsterdam, 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), 19-37; English translation: J. Math. Sci., 142 (2007), 2528-2539. 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), 15-58. 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, Optim. Calc. Var., 19 (2013), 301-316. 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), 185-220. doi: 10.1137/0315015.  Google Scholar

[6]

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

[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), 2179-2199. 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), 36-41.  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, Optim. Calc. Var., 18 (2012), 748-773. 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), 161-183. 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), 1781-1801. 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), 939-952, English translation: Ukr. Math. J., 59 (2007), 1040-1058. 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., Philadelphia, 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), 420-444. 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), 254-283. 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), 2200-2214. 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), 105-118.  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), 258-266. 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, Ross. Akad. Nauk, 446 (2012), 374-377; English translation: Doklady Mathematics, 86 (2012), 657-660. 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), 193-204. 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), 525-541; English translation: Ukr. Math. J., 64 (2012), 594-615. 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., Anal., Geom., 8 (2012), 307-335.  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), 24-29. Google Scholar

[24]

E. Ya. Khruslov, One-dimensional inverse problems of electrodynamics (Russian), Zh. Vychisl. Mat. i Mat. Fiz., 25 (1985), 548-561.  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), 471-475. 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), 181-191. doi: 10.1007/s10440-013-9825-4.  Google Scholar

[27]

V. A. Marchenko, Sturm-Liouville Operators and Applications, American Mathematical Society, Providence, R.I., 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), 1097-1126. 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), 1-9. 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), 109-134. 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), 1508-1532. 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, Hindustan Book Agency, New Delhi, 2010, 3008-3034. 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, Elsevier/North-Holland, Amsterdam, 2007, 527-621. 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, Amsterdam, 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), 19-37; English translation: J. Math. Sci., 142 (2007), 2528-2539. 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), 15-58. 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, Optim. Calc. Var., 19 (2013), 301-316. 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), 185-220. doi: 10.1137/0315015.  Google Scholar

[6]

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

[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), 2179-2199. 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), 36-41.  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, Optim. Calc. Var., 18 (2012), 748-773. 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), 161-183. 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), 1781-1801. 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), 939-952, English translation: Ukr. Math. J., 59 (2007), 1040-1058. 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., Philadelphia, 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), 420-444. 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), 254-283. 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), 2200-2214. 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), 105-118.  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), 258-266. 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, Ross. Akad. Nauk, 446 (2012), 374-377; English translation: Doklady Mathematics, 86 (2012), 657-660. 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), 193-204. 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), 525-541; English translation: Ukr. Math. J., 64 (2012), 594-615. 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., Anal., Geom., 8 (2012), 307-335.  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), 24-29. Google Scholar

[24]

E. Ya. Khruslov, One-dimensional inverse problems of electrodynamics (Russian), Zh. Vychisl. Mat. i Mat. Fiz., 25 (1985), 548-561.  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), 471-475. 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), 181-191. doi: 10.1007/s10440-013-9825-4.  Google Scholar

[27]

V. A. Marchenko, Sturm-Liouville Operators and Applications, American Mathematical Society, Providence, R.I., 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), 1097-1126. 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), 1-9. 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), 109-134. 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), 1508-1532. 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, Hindustan Book Agency, New Delhi, 2010, 3008-3034. 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, Elsevier/North-Holland, Amsterdam, 2007, 527-621. doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

[1]

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
[2]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034
[3]

Alexander Zlotnik. The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis. Kinetic & Related Models, 2015, 8 (3) : 587-613. doi: 10.3934/krm.2015.8.587
[4]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[5]

Nakao Hayashi, Pavel I. Naumkin. Modified wave operator for Schrodinger type equations with subcritical dissipative nonlinearities. Inverse Problems & Imaging, 2007, 1 (2) : 391-398. doi: 10.3934/ipi.2007.1.391
[6]

Kenji Nakanishi. Modified wave operators for the Hartree equation with data, image and convergence in the same space. Communications on Pure & Applied Analysis, 2002, 1 (2) : 237-252. doi: 10.3934/cpaa.2002.1.237
[7]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061
[8]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215
[9]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016
[10]

Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097
[11]

Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261
[12]

Muslim Malik, Anjali Rose, Anil Kumar. Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition. Discrete & Continuous Dynamical Systems - S, 2022, 15 (2) : 387-407. doi: 10.3934/dcdss.2021068
[13]

Gael Diebou Yomgne. On a nonlinear Laplace equation related to the boundary Yamabe problem in the upper-half space. Communications on Pure & Applied Analysis, 2022, 21 (2) : 517-539. doi: 10.3934/cpaa.2021186
[14]

Linglong Du. Characteristic half space problem for the Broadwell model. Networks & Heterogeneous Media, 2014, 9 (1) : 97-110. doi: 10.3934/nhm.2014.9.97
[15]

Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková. Sharp Sobolev type embeddings on the entire Euclidean space. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2011-2037. doi: 10.3934/cpaa.2018096
[16]

Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure & Applied Analysis, 2015, 14 (2) : 527-548. doi: 10.3934/cpaa.2015.14.527
[17]

Linglong Du, Haitao Wang. Pointwise wave behavior of the Navier-Stokes equations in half space. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1349-1363. doi: 10.3934/dcds.2018055
[18]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511
[19]

Jinrong Wang, Michal Fečkan, Yong Zhou. Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 471-486. doi: 10.3934/eect.2017024
[20]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2021, 11 (4) : 857-883. doi: 10.3934/mcrf.2020049

2020 Impact Factor: 1.284

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy