Internal control of the Schrödinger equation
نویسنده
چکیده
In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schrödinger equation. After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension 1 using some propagation results. We then discuss how to obtain some similar results on a compact manifold where the zone of control satisfies the Geometric Control Condition. We also discuss some known results and open questions when this condition is not satisfied. Then, we present the links between the controllability and some resolvent estimates. Finally, we discuss the new difficulties when we consider the Nonlinear Schrödinger equation.
منابع مشابه
Control and Stabilization of the Nonlinear Schrödinger Equation on Rectangles
This paper studies the local exact controllability and the local stabilization of the semilinear Schrödinger equation posed on a product of n intervals (n ≥ 1). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as fa...
متن کاملGlobal controllability and stabilization for the nonlinear Schrödinger equation on an interval
We prove global internal controllability in large time for the nonlinear Schrödinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L 2. We also get a regularity result about the control if the data are assumed smoother.
متن کاملBoundary constrained control of delayed nonlinear Schrödinger equation
This paper studies regional boundary stabilization of nonlinear Schrödinger equation with state delay and bounded internal disturbance. The boundary constrained control law is designed by using the backstepping method. Regional input-tostate stability (ISS) of the perturbed system with time-delay is established by a Lyapunov function and a generalized Halanay’s inequality. Estimates on the set ...
متن کاملControllability of the bilinear Schrödinger equation with several controls and application to a 3D molecule
We show the approximate rotational controllability of a polar linear molecule by means of three nonresonant linear polarized laser fields. The result is based on a general approximate controllability result for the bilinear Schrödinger equation, with wavefunction varying in the unit sphere of an infinite-dimensional Hilbert space and with several control potentials, under the assumption that th...
متن کاملStabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback
In this paper, we investigate the effect of time delays in boundary or internal feedback stabilization of the multidimensional Schrödinger equation. In both cases, under suitable assumptions, we establish sufficient conditions on the delay term that guarantee the exponential stability of the solution. These results are obtained by using suitable energy functionals and some observability estimates.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2017