Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity

Micu, Sorin; Rovenţa, Ionel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 277-293.

This article considers the linear 1-d Schrödinger equation in (0,π) perturbed by a vanishing viscosity term depending on a small parameter ε > 0. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls v_ε as ε goes to zero. It is shown that, for any time T sufficiently large but independent of ε and for each initial datum in H^-1(0,π), there exists a uniformly bounded family of controls (v_ε)_ε in L²(0, T) acting on the extremity x = π. Any weak limit of this family is a control for the Schrödinger equation.

[1] O.M. Aamo, A. Smyshlyaev and M. Krstić, Boundary control of the linearized Ginzburg-Landau model of vortex shedding. SIAM J. Control Optim. 43 (2005) 1953-1971. | MR | Zbl

[2] S.A. Avdonin and S.A. Ivanov, Families of exponentials. The method of moments in controllability problems for distributed parameter systems. Cambridge University Press (1995). | MR | Zbl

[3] J. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semi-linear control systems. Commun. Pure Appl. Math. XXXII (1979) 555-587. | MR | Zbl

[4] M. Bartuccelli, P. Constantin, C.R. Doering, J.D. Gibbon and M. Gisselfält, On the possibility of soft and hard turbulence in the complex Ginzburg Landau equation. Physica D 44 (1990) 421-444. | MR | Zbl

[5] L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Probl. 18 (2002) 1537-1554. | MR | Zbl

[6] J.-M. Coron, Control and nonlinearity, Mathematical Surveys and Monographs 136. Am. Math. Soc., Providence (2007). | MR | Zbl

[7] J.-M. Coron and S. Guerrero, Singular optimal control : a linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237-257. | MR | Zbl

[8] R.J. Diperna, Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82 (1983) 27-70. | MR | Zbl

[9] X. Fu, A weighted identity for partial differential operators of second order and its applications. C. R. Acad. Sci. Paris, Sér. I 342 (2006) 579-584. | MR

[10] X. Fu, Null controllability for the parabolic equation with a complex principal part. J. Funct. Anal. 257 (2009) 1333-1354. | MR | Zbl

[11] A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, Lect. Notes Ser. 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR | Zbl

[12] O. Glass, A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit. J. Funct. Anal. 258 (2010) 852-868. | MR | Zbl

[13] L. Ignat and E. Zuazua, Dispersive Properties of Numerical Schemes for Nonlinear Schrödinger Equations, in Foundations of Computational Mathematics, Santander 2005, London Math. Soc. Lect. Notes 331, L.M. Pardo, A. Pinkus, E. Suli and M.J. Todd Eds., Cambridge University Press (2006) 181-207. | MR | Zbl

[14] L. Ignat and E. Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 47 (2009) 1366-1390. | MR | Zbl

[15] A.E. Ingham, A note on Fourier transform. J. London Math. Soc. 9 (1934) 29-32. | MR | Zbl

[16] A.E. Ingham, Some trigonometric inequalities with applications to the theory of series. Math. Zeits. 41 (1936) 367-379. | MR | Zbl

[17] J.P. Kahane, Pseudo-Périodicité et Séries de Fourier Lacunaires. Ann. Scient. Ec. Norm. Sup. 37 (1962) 93-95. | Numdam | MR | Zbl

[18] V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer-Verlag, New York (2005). | MR | Zbl

[19] M. Léautaud, Uniform controllability of scalar conservation laws in the vanishing viscosity limit. Preprint (2010). | Zbl

[20] G. Lebeau, Contrôle de l'équation de Schrödinger. J. Math. Pures Appl. 71 (1992) 267-291. | Zbl

[21] C.D. Levermore and M. Oliver, The complex Ginzburg Landau equation as a model problem, in Dynamical Systems and Probabilistic Methods in Partial Differential Equations, in Lect. Appl. Math. 31, Am. Math. Soc., Providence (1996) 141-190. | MR | Zbl

[22] A. López, X. Zhang and E. Zuazua, Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations. J. Math. Pures Appl. 79 (2000) 741-808. | MR | Zbl

[23] E. Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Control Optim. 32 (1994) 24-34. | MR | Zbl

[24] A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights. Inverse Probl. 24 (2008) 150-170. | MR | Zbl

[25] S. Micu and L. De Teresa, A spectral study of the boundary controllability of the linear 2-D wave equation in a rectangle. Asymptot. Anal. 66 (2010) 139-160. | MR | Zbl

[26] R.E.A.C. Paley and N. Wiener, Fourier Transforms in Complex Domains, AMS Colloq. Publ. 19. Am. Math. Soc., New York (1934). | MR | Zbl

[27] R.M. Redheffer, Completeness of sets of complex exponentials. Adv. Math. 24 (1977) 1-62. | MR | Zbl

[28] L. Rosier and B.-Y. Zhang, Null controllability of the complex Ginzburg Landau equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 649-673. | Numdam | MR | Zbl

[29] M. Salerno, B.A. Malomed and V.V. Konotop, Shock wave dynamics in a discrete nonlinear Schrödinger equation with internal losses. Phys. Rev. 62 (2000) 8651-8656. | MR

[30] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts, Springer, Basel (2009). | MR | Zbl

[31] R. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980). | MR | Zbl

[32] J. Zabczyk, Mathematical Control Theory : An Introduction. Birkhäuser, Basel (1992). | MR | Zbl

[33] X. Zhang, A remark on null exact controllability of the heat equation. SIAM J. Control Optim. 40 (2001) 39-53. | MR | Zbl