-
Previous Article
Zubov's equation for state-constrained perturbed nonlinear systems
- MCRF Home
- This Issue
-
Next Article
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, (1973).
|
[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. |
[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. |
[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. |
[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. |
[6] |
L. V. Fardigola, On controllability problems for the wave equation on a half-plane,, J. Math. Phys., 1 (2005), 93.
|
[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. |
[8] |
L. V. Fardigola, Neumann boundary control problem for the string equation on a half-axis (Ukrainian),, Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2009), 36.
|
[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. |
[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. |
[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. |
[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. |
[13] |
S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations,, Gordon and Breach Sci. Publ., (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.
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.
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[22] |
K. S. Khalina, On the Neumann boundary controllability for a non-homogeneous string on a half-axis,, J. Math. Phys., 8 (2012), 307.
|
[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.
|
[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. |
[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. |
[27] |
V. A. Marchenko, Sturm-Liouville Operators and Applications,, American Mathematical Society, (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.
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.
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.
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.
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, (2010), 3008.
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, (2007), 527.
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, (1973).
|
[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. |
[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. |
[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. |
[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. |
[6] |
L. V. Fardigola, On controllability problems for the wave equation on a half-plane,, J. Math. Phys., 1 (2005), 93.
|
[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. |
[8] |
L. V. Fardigola, Neumann boundary control problem for the string equation on a half-axis (Ukrainian),, Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2009), 36.
|
[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. |
[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. |
[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. |
[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. |
[13] |
S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations,, Gordon and Breach Sci. Publ., (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.
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.
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.
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.
|
[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. |
[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. |
[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. |
[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. |
[22] |
K. S. Khalina, On the Neumann boundary controllability for a non-homogeneous string on a half-axis,, J. Math. Phys., 8 (2012), 307.
|
[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.
|
[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. |
[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. |
[27] |
V. A. Marchenko, Sturm-Liouville Operators and Applications,, American Mathematical Society, (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.
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.
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.
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.
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, (2010), 3008.
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, (2007), 527.
doi: 10.1016/S1874-5717(07)80010-7. |
[1] |
Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 |
[2] |
Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363 |
[3] |
Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027 |
[4] |
Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201 |
[5] |
Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 |
[6] |
Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 |
[7] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[8] |
Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379 |
[9] |
Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817 |
[10] |
Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269 |
[11] |
Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024 |
[12] |
Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020210 |
[13] |
Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 |
[14] |
Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 |
[15] |
Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190 |
[16] |
Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067 |
[17] |
Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 |
[18] |
Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
[19] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[20] |
Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]