Exact controllability for the fourth order Schrödinger equation

Abstract

The boundary controllability of the fourth order Schrödinger equation in a bounded domain is studied. By means of an L ²-Neumann boundary control, the authors prove that the solution is exactly controllable in H ⁻²(Ω) for an arbitrarily small time. The method of proof combines both the HUM (Hilbert Uniqueness Method) and multiplier techniques.

Keywords

Fourth order Schrödinger equation / HUM method / Controllability / Multiplier

References

[1] Bardos C., Lebeau G., Rauch J.. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim., 1992, 30: 1024-1065

[2] Burq N., Zworski M.. Geometric control in the presence of a black box. J. Amer. Math. Soc., 2004, 17: 443-471

[3] Chou H., Guo Y.. Null boundary controllability for a fourth order semilinear equation. Taiwanese J. Math., 2006, 10: 251-263

[4] Díaz J. I., Ramos A. M.. On the approximate controllability for higher order parabolic nonlinear equations of Cahn-Hilliard type. Internat. Ser. Numer. Math., 1998, 126: 111-127

[5] Fu X., Yong J., Zhang X.. Controllability and observability of a heat equation with hyperbolic memory kernel. J. Differential Equations, 2009, 247: 2395-2439

[6] Fursikov A., Imanuvilov O.. Controllability of Evolution Equations, 1996, Seoul: Seoul National University

[7] Hao C., Hsiao L., Wang B.. Wellposedness for the fourth order nonlinear Schrödinger equations. J. Math. Anal. Appl., 2006, 320: 246-265

[8] Hao C., Hsiao L., Wang B.. Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces. J. Math. Anal. Appl., 2007, 328: 58-83

[9] Hörmander L.. Linear Partial Differential Operators, 1976, Berlin: Springer-Verlag

[10] Lin, P. and Zhou, Z., Observability estimate for a one-dimensional fourth order parabolic equation, Proceedings of the 29th Chinese Control Conference, China, 2010, 830–832.

[11] Lions J. L.. Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, 1988, Paris: Masson 1-68

[12] Lions J. L.. Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev., 1988, 30: 1-68

[13] Lions J. L., Magenes E.. Problèmes aux Limites Non Homogènes et Applications, 1968, Paris: Dunod

[14] Machtyngier E.. Exact controllability for the Schrödinger equation. SIAM J. Control Optim., 1994, 32: 24-34

[15] Miller L.. Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal., 2005, 218: 425-444

[16] Pausader B.. Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn. Partial Differ. Equ., 2007, 4: 197-225

[17] Pausader B.. The cubic fourth-order Schrödinger equation. J. Funct. Anal., 2009, 256: 2473-2517

[18] 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

[19] Simon J.. Compact sets in the space L _p(0, T;B). Ann. Mat. Pura Appl., 1987, 146(4): 65-96

[20] Zhang X.. Exact controllability of semilinear plate equations. Asympt. Anal., 2001, 27: 95-125