Rigorous numerics for NLS: Bound states, spectra, and controllability

In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schrödinger equation (NLS); specifically, to determining bound-state solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof (Beauchard et al., 2015) of the local exact controllability of NLS

Local controllability of 1D linear and nonlinear Schr\"odinger equations with bilinear control

We consider a linear Schr\"odinger equation, on a bounded interval, with bilinear control, that represents a quantum particle in an electric field (the control). We prove the controllability of this system, in any positive time, locally around the ground state. Similar results were proved for particular models (by the first author and with J.M. Coron), in non optimal spaces, in long time and the proof relied on the Nash-Moser implicit function theorem in order to deal with an a priori loss of regularity. In this article, the model is more general, the spaces are optimal, there is no restriction on the time and the proof relies on the classical inverse mapping theorem. A hidden regularizing effect is emphasized, showing there is actually no loss of regularity. Then, the same strategy is applied to nonlinear Schr\"odinger equations and nonlinear wave equations, showing that the method works for a wide range of bilinear control systems

Controllability of the 1D Schrodinger equation by the flatness approach

We derive in a straightforward way the exact controllability of the 1-D Schrodinger equation with a Dirichlet boundary control. We use the so-called flatness approach, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control input achieving the exact controllability in the energy space. As an application, we derive an explicit pair of control inputs achieving the exact steering to zero for a simply-supported beam

Explicit approximate controllability of the Schr\"odinger equation with a polarizability term

We consider a controlled Schr\"odinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts non linearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schr\"odinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system

Simultaneous local exact controllability of 1D bilinear Schr\"odinger equations

We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schr\"odinger equations on a bounded interval. This is a bilinear control system in which the state is the N-tuple of wave functions. The control is the real amplitude of the laser field. For N=1, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time if N is greater or equal than 2. Still, for N=2, we prove using Coron's return method that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. We also prove that for N greater or equal than 3, local controllability does not hold in small time even up to a global phase. Finally, for N=3, we prove that local controllability holds up to a global phase and a global delay

Rigorous numerics for NLS: bound states, spectra, and controllability

In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schr\"odinger equation (NLS); specifically, to determining bound--state solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof [6] of the local exact controllability of NLS.Comment: 30 pages, 2 figure

Local controllability of 1D Schr\"odinger equations with bilinear control and minimal time

We consider a linear Schr\"odinger equation, on a bounded interval, with bilinear control. Beauchard and Laurent proved that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. Coron proved that a positive minimal time is required for this controllability, on a particular degenerate example. In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution