[Paper Review] Controllability of the cubic Schroedinger equation via a low-dimensional source term

Andrey Sarychev|arXiv (Cornell University)|May 9, 2011

Advanced Mathematical Physics Problems21 references19 citations

TL;DR

This paper establishes that the 2D defocusing cubic Schrödinger equation on the torus is approximately controllable and controllable in finite-dimensional projections using only four controlled modes via an additive source term. It proves that controlling just four complex exponentials in the source term suffices for approximate controllability in $ H^{1+\sigma}(\mathbb{T}^2) $ and controllability in any finite-dimensional subspace, while also showing that exact controllability in the full space is impossible with finite-dimensional control.

ABSTRACT

We study controllability of $d$-dimensional defocusing cubic Schroedinger equation under periodic boundary conditions. The control is applied additively, via a source term, which is a linear combination of few complex exponentials (modes) with time-variant coefficients - controls. We manage to prove that controlling at most $2^d$ modes one can achieve controllability of the equation in any finite-dimensional projection of the evolution space $H^{s}(\\mathbb{T}^d), \\ s>d/2$, as well as approximate controllability in $H^{s}(\\mathbb{T}^d)$. We also present negative result regarding exact controllability of cubic Schroedinger equation via a finite-dimensional source term.

Motivation & Objective

• To investigate the controllability of the 2D defocusing cubic Schrödinger equation under periodic boundary conditions with additive control via a low-dimensional source term.
• To determine whether controlling only a finite number of Fourier modes (specifically four) can achieve controllability in finite-dimensional projections or approximate controllability in the full Sobolev space $ H^{1+\sigma}(\mathbb{T}^2) $.
• To establish a negative result: exact controllability in $ H^{1+\sigma}(\mathbb{T}^2) $ is impossible when the source term has finite-dimensional range.
• To apply geometric control theory and Lie algebraic methods to nonlinear PDEs, extending techniques from fluid dynamics to nonlinear Schrödinger systems.

Proposed method

• The study models the cubic Schrödinger equation as an evolution in the Sobolev space $ H^{1+\sigma}(\mathbb{T}^2) $ with $ \sigma > 0 $, ensuring sufficient regularity to isolate controllability issues from analytic complications.
• Control is applied via an additive source term $ F(t,x) = \sum_{k \in \hat{\mathcal{K}}} v_k(t) e^{ik \cdot x} $, where $ \hat{\mathcal{K}} \subset \mathbb{Z}^2 $ is a finite set of wave vectors and $ v_k(t) \in L^\infty[0,T] $ are time-varying control coefficients.
• The analysis employs a Lie algebraic approach from geometric control theory, focusing on the span of iterated Lie brackets of the vector fields associated with the controlled modes to assess reachability.
• The paper uses a partitioning argument and compactness criteria in Hilbert space to prove the equicontinuity of the control operator, relying on the boundedness of the nonlinear term and the isometric property of the free propagator $ e^{-i\tau\Delta} $.
• It establishes a key estimate on the integral of the control influence via the resolvent norm $ \|\cdot\|^{rx} $, showing that small $ L^1 $-norm of the control input leads to small net effect on the state, which is essential for proving approximate controllability.
• A contradiction argument is used to prove that exact controllability in the full space $ H^{1+\sigma} $ is impossible with finite-dimensional control, based on the structure of the nonlinear term and the finite-dimensional range of the control space.

Experimental results

Research questions

• RQ1 Can the 2D defocusing cubic Schrödinger equation be approximately controlled in $ H^{1+\sigma}(\mathbb{T}^2) $ using only four controlled Fourier modes in the source term?
• RQ2 Is it possible to achieve controllability in any finite-dimensional projection of $ H^{1+\sigma}(\mathbb{T}^2) $ by controlling just four modes?
• RQ3 What is the minimal number of modes required to achieve approximate controllability or finite-dimensional projection controllability in this system?
• RQ4 Can exact controllability in the full $ H^{1+\sigma}(\mathbb{T}^2) $ space be achieved with a finite-dimensional source term?
• RQ5 How does the structure of the nonlinear term $ |u|^2 u $ interact with finite-dimensional control to limit or enable controllability?

Key findings

• Controlling just four modes via an additive source term is sufficient to achieve approximate controllability in $ H^{1+\sigma}(\mathbb{T}^2) $ for any $ \sigma > 0 $.
• Controllability in any finite-dimensional projection of $ H^{1+\sigma}(\mathbb{T}^2) $ can be achieved using only four controlled modes.
• The paper identifies a universal family of four-mode control sets that suffice for both approximate controllability and finite-dimensional projection controllability.
• Exact controllability in $ H^{1+\sigma}(\mathbb{T}^2) $ is impossible when the source term has finite-dimensional range, regardless of the choice of controlled modes.
• The proof relies on a compactness argument involving the resolvent norm $ \|\cdot\|^{rx} $ and the equicontinuity of the control operator, showing that small control inputs yield small state changes.
• The nonlinear term $ |u|^2 u $, while bounded in $ H^{1+\sigma} $, prevents the control operator from being surjective onto the full space, which blocks exact controllability.