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On the influence of the coupling on the dynamics of single-observed cascade systems of PDE's
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 |
References:
[1] |
P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier, Amsterdam, 1973. |
[2] |
M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbb{R}^3$ (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. |
[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. |
[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. |
[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. |
[6] |
L. V. Fardigola, On controllability problems for the wave equation on a half-plane, J. Math. Phys., Anal., Geom., 1 (2005), 93-115. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations, Gordon and Breach Sci. Publ., Philadelphia, 1992. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
E. Ya. Khruslov, One-dimensional inverse problems of electrodynamics (Russian), Zh. Vychisl. Mat. i Mat. Fiz., 25 (1985), 548-561. |
[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. |
[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. |
[27] |
V. A. Marchenko, Sturm-Liouville Operators and Applications, American Mathematical Society, Providence, R.I., 2011. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier, Amsterdam, 1973. |
[2] |
M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbb{R}^3$ (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. |
[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. |
[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. |
[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. |
[6] |
L. V. Fardigola, On controllability problems for the wave equation on a half-plane, J. Math. Phys., Anal., Geom., 1 (2005), 93-115. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations, Gordon and Breach Sci. Publ., Philadelphia, 1992. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
E. Ya. Khruslov, One-dimensional inverse problems of electrodynamics (Russian), Zh. Vychisl. Mat. i Mat. Fiz., 25 (1985), 548-561. |
[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. |
[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. |
[27] |
V. A. Marchenko, Sturm-Liouville Operators and Applications, American Mathematical Society, Providence, R.I., 2011. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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