메뉴 건너뛰기




Volumn 94, Issue 5, 2010, Pages 520-554

Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control

Author keywords

Bilinear control; Control of partial differential equations; Schr dinger equation; Wave equation

Indexed keywords


EID: 77957800526     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2010.04.001     Document Type: Article
Times cited : (148)

References (64)
  • 1
    • 33847209264 scopus 로고    scopus 로고
    • Controllability of the Schroedinger equation via intersection of eigenvalues, in: Proceedings of the 44rd IEEE Conference on Decision and Control, December 12-15, 2005, Seville (Spain). Also in: 'Control Systems: Theory, Numerics and Applications, Roma, Italy, 30 March-1 April 2005, POS, Proceeding of Science
    • R. Adami, U. Boscain, Controllability of the Schroedinger equation via intersection of eigenvalues, in: Proceedings of the 44rd IEEE Conference on Decision and Control, December 12-15, 2005, Seville (Spain). Also in: 'Control Systems: Theory, Numerics and Applications, Roma, Italy, 30 March-1 April 2005, POS, Proceeding of Science.
    • Adami, R.1    Boscain, U.2
  • 2
    • 33745808312 scopus 로고    scopus 로고
    • An estimation of the controllability time for single-input systems on compact Lie groups
    • Agrachev A., Chambrion T. An estimation of the controllability time for single-input systems on compact Lie groups. ESAIM Control Optim. Calc. Var. 2006, 12(3):409-441.
    • (2006) ESAIM Control Optim. Calc. Var. , vol.12 , Issue.3 , pp. 409-441
    • Agrachev, A.1    Chambrion, T.2
  • 3
    • 28444476416 scopus 로고    scopus 로고
    • Control theory from the geometric viewpoint
    • Springer-Verlag, Berlin, Control Theory and Optimization, II
    • Agrachev A., Sachkov Y.L. Control theory from the geometric viewpoint. Encyclopaedia of Mathematical Sciences 2004, vol. 87. Springer-Verlag, Berlin.
    • (2004) Encyclopaedia of Mathematical Sciences , vol.87
    • Agrachev, A.1    Sachkov, Y.L.2
  • 4
    • 15944401206 scopus 로고    scopus 로고
    • Navier-Stokes equations: Controllability by means of low modes forcing
    • Agrachev A., Sarychev A.V. Navier-Stokes equations: Controllability by means of low modes forcing. J. Math. Fluid Mech. 2005, 7(1):108-152.
    • (2005) J. Math. Fluid Mech. , vol.7 , Issue.1 , pp. 108-152
    • Agrachev, A.1    Sarychev, A.V.2
  • 5
    • 0042883845 scopus 로고    scopus 로고
    • Notions of controllability for bilinear multilevel quantum systems
    • Albertini F., D'Alessandro D. Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Automat. Control 2003, 48(8):1399-1403.
    • (2003) IEEE Trans. Automat. Control , vol.48 , Issue.8 , pp. 1399-1403
    • Albertini, F.1    D'Alessandro, D.2
  • 6
    • 0004157168 scopus 로고
    • Opérateurs Pseudo-différentiels et Théorème de Nash-Moser
    • Intereditions, Paris
    • Alinhac S., Gérard P. Opérateurs Pseudo-différentiels et Théorème de Nash-Moser. Collection Savoirs Actuels 1991, Intereditions, Paris.
    • (1991) Collection Savoirs Actuels
    • Alinhac, S.1    Gérard, P.2
  • 7
    • 0035981789 scopus 로고    scopus 로고
    • Controllability of quantum mechanical systems by root space decomposition of su(n)
    • Altafini C. Controllability of quantum mechanical systems by root space decomposition of su(n). J. Math. Phys. 2002, 43(5):2051-2062.
    • (2002) J. Math. Phys. , vol.43 , Issue.5 , pp. 2051-2062
    • Altafini, C.1
  • 8
    • 54049119716 scopus 로고    scopus 로고
    • Cubic nonlinear Schrödinger equation on three-dimensional balls with radial data
    • Anton R. Cubic nonlinear Schrödinger equation on three-dimensional balls with radial data. Comm. Partial Differential Equations 2008, 33(10-12):1862-1889.
    • (2008) Comm. Partial Differential Equations , vol.33 , Issue.10-12 , pp. 1862-1889
    • Anton, R.1
  • 11
    • 38149014790 scopus 로고    scopus 로고
    • A bilinear optimal control problem applied to a time dependent Hartree-Fock equation coupled with classical nuclear dynamics
    • Baudouin L. A bilinear optimal control problem applied to a time dependent Hartree-Fock equation coupled with classical nuclear dynamics. Port. Math. (N.S.) 2006, 63(3):293-325.
    • (2006) Port. Math. (N.S.) , vol.63 , Issue.3 , pp. 293-325
    • Baudouin, L.1
  • 12
    • 23644461902 scopus 로고    scopus 로고
    • Regularity for a Schrödinger equation with singular potential and application to bilinear optimal control
    • Baudouin L., Kavian O., Puel J.-P. Regularity for a Schrödinger equation with singular potential and application to bilinear optimal control. J. Differential Equations 2005, 216:188-222.
    • (2005) J. Differential Equations , vol.216 , pp. 188-222
    • Baudouin, L.1    Kavian, O.2    Puel, J.-P.3
  • 13
    • 41949123114 scopus 로고    scopus 로고
    • Constructive solutions of a bilinear control problem for a Schrödinger equation
    • Baudouin L., Salomon J. Constructive solutions of a bilinear control problem for a Schrödinger equation. Systems Control Lett. 2008, 57(6):453-464.
    • (2008) Systems Control Lett. , vol.57 , Issue.6 , pp. 453-464
    • Baudouin, L.1    Salomon, J.2
  • 14
    • 20044377637 scopus 로고    scopus 로고
    • Local controllability of a 1-D Schrödinger equation
    • J
    • Beauchard K. Local controllability of a 1-D Schrödinger equation. J. Math. Pures Appl. July 2005, 84:851-956.
    • (2005) J. Math. Pures Appl. , vol.84 , pp. 851-956
    • Beauchard, K.1
  • 15
    • 38149044222 scopus 로고    scopus 로고
    • Controllability of a quantum particle in a 1D variable domain
    • Beauchard K. Controllability of a quantum particle in a 1D variable domain. ESAIM Control Optim. Calc. Var. 2008, 14(1):105-147.
    • (2008) ESAIM Control Optim. Calc. Var. , vol.14 , Issue.1 , pp. 105-147
    • Beauchard, K.1
  • 16
    • 55549119623 scopus 로고    scopus 로고
    • Local controllability of a 1-D beam equation
    • Beauchard K. Local controllability of a 1-D beam equation. SIAM J. Control Optim. 2008, 47(3):1219-1273.
    • (2008) SIAM J. Control Optim. , vol.47 , Issue.3 , pp. 1219-1273
    • Beauchard, K.1
  • 17
    • 31744450577 scopus 로고    scopus 로고
    • Controllability of a quantum particle in a moving potential well
    • Beauchard K., Coron J.-M. Controllability of a quantum particle in a moving potential well. J. Funct. Anal. 2006, 232:328-389.
    • (2006) J. Funct. Anal. , vol.232 , pp. 328-389
    • Beauchard, K.1    Coron, J.-M.2
  • 18
  • 19
    • 77950961578 scopus 로고    scopus 로고
    • Practical stabilization of a quantum particle in a one-dimensional infinite square potential well
    • Beauchard K., Mirrahimi M. Practical stabilization of a quantum particle in a one-dimensional infinite square potential well. SIAM J. Control Optim. 2009, 48(2):1179-1205.
    • (2009) SIAM J. Control Optim. , vol.48 , Issue.2 , pp. 1179-1205
    • Beauchard, K.1    Mirrahimi, M.2
  • 21
    • 12844252459 scopus 로고    scopus 로고
    • Resonance of minimizers for n-level quantum systems with an arbitrary cost
    • Boscain U., Charlot G. Resonance of minimizers for n-level quantum systems with an arbitrary cost. ESAIM Control Optim. Calc. Var. 2004, 10(4):593-614.
    • (2004) ESAIM Control Optim. Calc. Var. , vol.10 , Issue.4 , pp. 593-614
    • Boscain, U.1    Charlot, G.2
  • 22
    • 29344442575 scopus 로고    scopus 로고
    • Optimal control of the Schrödinger equation with two or three levels
    • Springer, Berlin, Nonlinear and Adaptive Control
    • Boscain U., Charlot G., Gauthier J.-P. Optimal control of the Schrödinger equation with two or three levels. Lecture Notes in Control and Inform. Sci. 2003, vol. 280. Springer, Berlin.
    • (2003) Lecture Notes in Control and Inform. Sci. , vol.280
    • Boscain, U.1    Charlot, G.2    Gauthier, J.-P.3
  • 23
    • 0035981790 scopus 로고    scopus 로고
    • Optimal control in laser-induced population transfer for two- and three-level quantum systems
    • Boscain U., Charlot G., Gauthier J.-P., Guérin S., Jauslin H.-R. Optimal control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys. 2002, 43(5).
    • (2002) J. Math. Phys. , vol.43 , Issue.5
    • Boscain, U.1    Charlot, G.2    Gauthier, J.-P.3    Guérin, S.4    Jauslin, H.-R.5
  • 24
    • 0035981790 scopus 로고    scopus 로고
    • Optimal control in laser-induced population transfer for two- and three-level quantum systems
    • Boscain U., Charlot G., Gauthier J.-P., Guérin S., Jauslin H.-R. Optimal control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys. 2002, 43:2107-2132.
    • (2002) J. Math. Phys. , vol.43 , pp. 2107-2132
    • Boscain, U.1    Charlot, G.2    Gauthier, J.-P.3    Guérin, S.4    Jauslin, H.-R.5
  • 25
    • 33745683248 scopus 로고    scopus 로고
    • Time minimal trajectories for a spin 1/2 particle in a magnetic field
    • Boscain U., Mason P. Time minimal trajectories for a spin 1/2 particle in a magnetic field. J. Math. Phys. 2006, 47(6):62101-62129.
    • (2006) J. Math. Phys. , vol.47 , Issue.6 , pp. 62101-62129
    • Boscain, U.1    Mason, P.2
  • 26
    • 0000871390 scopus 로고
    • Lie theory and control systems defined on spheres
    • Brockett R. Lie theory and control systems defined on spheres. SIAM J. Appl. Math. 1973, 25(2):213-225.
    • (1973) SIAM J. Appl. Math. , vol.25 , Issue.2 , pp. 213-225
    • Brockett, R.1
  • 27
    • 0008723973 scopus 로고
    • Contrôle de l'équation des plaques en présence d'obstacles strictement convexes
    • Burq N. Contrôle de l'équation des plaques en présence d'obstacles strictement convexes. Mem. Soc. Math. Fr. 1993, 55.
    • (1993) Mem. Soc. Math. Fr. , vol.55
    • Burq, N.1
  • 28
    • 61849117399 scopus 로고    scopus 로고
    • Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain
    • Cerpa E., Crépeau E. Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain. Ann. Inst. H. Poincaré Anal. Non Linéaire 2009, 26(2):457-475.
    • (2009) Ann. Inst. H. Poincaré Anal. Non Linéaire , vol.26 , Issue.2 , pp. 457-475
    • Cerpa, E.1    Crépeau, E.2
  • 30
    • 2442663960 scopus 로고    scopus 로고
    • Local controllability of a 1-D Tank containing a fluid modeled by the shallow water equations
    • J
    • Coron J.-M. Local controllability of a 1-D Tank containing a fluid modeled by the shallow water equations. ESAIM Control Optim. Calc. Var. June 2002, 8:513-554.
    • (2002) ESAIM Control Optim. Calc. Var. , vol.8 , pp. 513-554
    • Coron, J.-M.1
  • 31
    • 29944433591 scopus 로고    scopus 로고
    • On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well
    • Coron J.-M. On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well. C. R. Acad. Sci. Paris, Ser. I 2006, 342:103-108.
    • (2006) C. R. Acad. Sci. Paris, Ser. I , vol.342 , pp. 103-108
    • Coron, J.-M.1
  • 33
    • 33749642834 scopus 로고    scopus 로고
    • Stabilization and control for the nonlinear Schrödinger equation on a compact surface
    • D
    • Dehman B., Gérard P., Lebeau G. Stabilization and control for the nonlinear Schrödinger equation on a compact surface. Math. Z. December 2006, 254(4):729-749.
    • (2006) Math. Z. , vol.254 , Issue.4 , pp. 729-749
    • Dehman, B.1    Gérard, P.2    Lebeau, G.3
  • 34
    • 0034165527 scopus 로고    scopus 로고
    • Contrôle optimal bilinéaire d'une équation de Schrödinger
    • Cancès E., Le Bris C., Pilot M. Contrôle optimal bilinéaire d'une équation de Schrödinger. C. R. Acad. Sci. Paris 2000, 330:567-571.
    • (2000) C. R. Acad. Sci. Paris , vol.330 , pp. 567-571
    • Cancès, E.1    Le Bris, C.2    Pilot, M.3
  • 35
    • 71849105789 scopus 로고    scopus 로고
    • Approximate controllability for a system of Schrödinger equations modeling a single trapped ion
    • Ervedoza S., Puel J.-P. Approximate controllability for a system of Schrödinger equations modeling a single trapped ion. Ann. Inst. H. Poincaré Anal. Non Linéaire 2009, 26(6):2111-2136.
    • (2009) Ann. Inst. H. Poincaré Anal. Non Linéaire , vol.26 , Issue.6 , pp. 2111-2136
    • Ervedoza, S.1    Puel, J.-P.2
  • 36
    • 84974774417 scopus 로고
    • Results on exact internal controllability for the Schrödinger equation and their asymptotic limits: Application to some vibrating-plate equations
    • (in French)
    • Fabre C. Résultats de contrôlabilité exacte interne pour l'équation de Schrödinger et leurs limites asymptotiques: application à certaines équations de plaques vibrantes. Asymptot. Anal. 1992, 5(4):343-379. (in French).
    • (1992) Asymptot. Anal. , vol.5 , Issue.4 , pp. 343-379
    • Fabre, C.1
  • 37
    • 0000089068 scopus 로고
    • Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire
    • Haraux A. Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl. 1989, 68:457-465.
    • (1989) J. Math. Pures Appl. , vol.68 , pp. 457-465
    • Haraux, A.1
  • 38
    • 0000522384 scopus 로고
    • On the Nash-Moser implicit function theorem
    • Hörmander
    • Hörmander On the Nash-Moser implicit function theorem. Ann. Acad. Sci. Fenn. 1985, 255-259.
    • (1985) Ann. Acad. Sci. Fenn. , pp. 255-259
  • 39
  • 40
    • 84983715460 scopus 로고    scopus 로고
    • Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer
    • Khaneja N., Glaser S.J., Brockett R. Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer. Phys. Rev. A (3) 2002, 65(11):032301.
    • (2002) Phys. Rev. A (3) , vol.65 , Issue.11 , pp. 032301
    • Khaneja, N.1    Glaser, S.J.2    Brockett, R.3
  • 41
    • 34547640709 scopus 로고    scopus 로고
    • Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state
    • Lange H., Teismann H. Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state. Math. Methods Appl. Sci. 2007, 30(13):1483-1505.
    • (2007) Math. Methods Appl. Sci. , vol.30 , Issue.13 , pp. 1483-1505
    • Lange, H.1    Teismann, H.2
  • 42
    • 84972514083 scopus 로고
    • Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet controls
    • Lasiecka I., Triggiani R. Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet controls. Differential Integral Equations 1992, 5:535-571.
    • (1992) Differential Integral Equations , vol.5 , pp. 535-571
    • Lasiecka, I.1    Triggiani, R.2
  • 43
    • 2942683404 scopus 로고    scopus 로고
    • Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates
    • Lasiecka I., Triggiani R., Zhang X. Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. J. Inverse Ill-Posed Probl. 2004, 12:183-231.
    • (2004) J. Inverse Ill-Posed Probl. , vol.12 , pp. 183-231
    • Lasiecka, I.1    Triggiani, R.2    Zhang, X.3
  • 44
    • 77952035855 scopus 로고    scopus 로고
    • Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifold of dimension 3
    • Laurent C. Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifold of dimension 3. SIAM Journal on Mathematical Analysis 2010, 42(2):785-832.
    • (2010) SIAM Journal on Mathematical Analysis , vol.42 , Issue.2 , pp. 785-832
    • Laurent, C.1
  • 45
    • 77951182367 scopus 로고    scopus 로고
    • Global controllability and stabilization for the nonlinear Schrödinger equation on an interval
    • Laurent C. Global controllability and stabilization for the nonlinear Schrödinger equation on an interval. ESAIM Control Optim. Calc. Var. 2010, 16(2):359-379.
    • (2010) ESAIM Control Optim. Calc. Var. , vol.16 , Issue.2 , pp. 359-379
    • Laurent, C.1
  • 46
    • 0000977450 scopus 로고
    • Contrôle de l'équation de Schrödinger
    • Lebeau G. Contrôle de l'équation de Schrödinger. J. Math. Pures Appl. 1992, 71:267-291.
    • (1992) J. Math. Pures Appl. , vol.71 , pp. 267-291
    • Lebeau, G.1
  • 47
    • 0028257122 scopus 로고
    • Exact controllability for the Schrödinger equation
    • Machtyngier
    • Machtyngier Exact controllability for the Schrödinger equation. SIAM J. Control Optim. 1994, 32:24-34.
    • (1994) SIAM J. Control Optim. , vol.32 , pp. 24-34
  • 48
    • 69949096628 scopus 로고    scopus 로고
    • Lyapunov control of a quantum particle in a decaying potential
    • Mirrahimi M. Lyapunov control of a quantum particle in a decaying potential. Ann. Inst. H. Poincaré (c) Nonlinéar Anal. 2009, 26:1743-1765.
    • (2009) Ann. Inst. H. Poincaré (c) Nonlinéar Anal. , vol.26 , pp. 1743-1765
    • Mirrahimi, M.1
  • 49
    • 2942557186 scopus 로고    scopus 로고
    • Controllability of quantum harmonic oscillators
    • Mirrahimi M., Rouchon P. Controllability of quantum harmonic oscillators. IEEE Trans. Automat. Control 2004, 49(5):745-747.
    • (2004) IEEE Trans. Automat. Control , vol.49 , Issue.5 , pp. 745-747
    • Mirrahimi, M.1    Rouchon, P.2
  • 50
    • 24944551258 scopus 로고    scopus 로고
    • Lyapunov control of bilinear Schrödinger equations
    • Mirrahimi M., Rouchon P., Turinici G. Lyapunov control of bilinear Schrödinger equations. Automatica 2005, 41:1987-1994.
    • (2005) Automatica , vol.41 , pp. 1987-1994
    • Mirrahimi, M.1    Rouchon, P.2    Turinici, G.3
  • 51
    • 77957768068 scopus 로고    scopus 로고
    • Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, preprint
    • V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, preprint, 2009.
    • (2009)
    • Nersesyan, V.1
  • 52
    • 70349678394 scopus 로고    scopus 로고
    • Growth of Sobolev norms and controllability of Schrödinger equation
    • Nersesyan V. Growth of Sobolev norms and controllability of Schrödinger equation. Comm. Math. Phys. 2009, 290(1):371-387.
    • (2009) Comm. Math. Phys. , vol.290 , Issue.1 , pp. 371-387
    • Nersesyan, V.1
  • 54
    • 23344440954 scopus 로고    scopus 로고
    • A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator
    • Ramdani K., Takahashi T., Tenenbaum G., Tucsnak M. A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator. J. Funct. Anal. 2005, 226:193-229.
    • (2005) J. Funct. Anal. , vol.226 , pp. 193-229
    • Ramdani, K.1    Takahashi, T.2    Tenenbaum, G.3    Tucsnak, M.4
  • 55
    • 84996141528 scopus 로고    scopus 로고
    • Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain
    • Rosier L. Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM Control Optim. Calc. Var. 1997, 2:33-55.
    • (1997) ESAIM Control Optim. Calc. Var. , vol.2 , pp. 33-55
    • Rosier, L.1
  • 56
    • 77950847498 scopus 로고    scopus 로고
    • Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval
    • Rosier L., Zhang B.-Y. Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval. SIAM J. Control Optim. 2009, 48(2):972-992.
    • (2009) SIAM J. Control Optim. , vol.48 , Issue.2 , pp. 972-992
    • Rosier, L.1    Zhang, B.-Y.2
  • 57
    • 77957773900 scopus 로고    scopus 로고
    • Control of a quantum particle in a moving potential well, in: 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville
    • P. Rouchon, Control of a quantum particle in a moving potential well, in: 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville, 2003.
    • (2003)
    • Rouchon, P.1
  • 58
    • 21444446596 scopus 로고    scopus 로고
    • Exact controllability and stabilizability of the Korteweg-de Vries equation
    • Russel D., Zhang B.-Y. Exact controllability and stabilizability of the Korteweg-de Vries equation. Trans. Amer. Math. Soc. 1996, 348(9):3643-3672.
    • (1996) Trans. Amer. Math. Soc. , vol.348 , Issue.9 , pp. 3643-3672
    • Russel, D.1    Zhang, B.-Y.2
  • 59
    • 33745613246 scopus 로고    scopus 로고
    • Approximate controllability of three-dimensional Navier-Stokes equations
    • Shirikyan A. Approximate controllability of three-dimensional Navier-Stokes equations. Comm. Math. Phys. 2006, 266(1):123-151.
    • (2006) Comm. Math. Phys. , vol.266 , Issue.1 , pp. 123-151
    • Shirikyan, A.1
  • 61
    • 0004976311 scopus 로고    scopus 로고
    • On the controllability of bilinear quantum systems
    • Springer, C. Le Bris, M. Defranceschi (Eds.) Mathematical Models and Methods for Ab Initio Quantum Chemistry
    • Turinici G. On the controllability of bilinear quantum systems. Lecture Notes in Chemistry 2000, vol. 74. Springer. C. Le Bris, M. Defranceschi (Eds.).
    • (2000) Lecture Notes in Chemistry , vol.74
    • Turinici, G.1
  • 62
    • 0035371423 scopus 로고    scopus 로고
    • Quantum wave function controllability
    • Turinici G., Rabitz H. Quantum wave function controllability. Chem. Phys. 2001, 267:1-9.
    • (2001) Chem. Phys. , vol.267 , pp. 1-9
    • Turinici, G.1    Rabitz, H.2
  • 63
    • 0032201986 scopus 로고    scopus 로고
    • Exact boundary controllability of the Korteweg-de Vries equation
    • Zhang B.-Y. Exact boundary controllability of the Korteweg-de Vries equation. SIAM J. Control Optim. 1999, 37(2):543-565.
    • (1999) SIAM J. Control Optim. , vol.37 , Issue.2 , pp. 543-565
    • Zhang, B.-Y.1
  • 64
    • 85012981810 scopus 로고
    • Exact controllability for semilinear wave equations in one space dimension
    • Zuazua E. Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire 1993, 10(1):109-129.
    • (1993) Ann. Inst. H. Poincaré Anal. Non Linéaire , vol.10 , Issue.1 , pp. 109-129
    • Zuazua, E.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.