Small-time local control of a Schrödinger equation: a negative and a positive quadratic result

Technical Summary: Small-time local control of a Schrödinger equation: a negative and a positive quadratic result

Problem Statement
This paper investigates the small-time local controllability (STLC) of a bilinear Schrödinger equation with Neumann boundary conditions, linearized around its ground state $\phi_0$. The system is governed by:
$$\begin{cases} i\partial_t\psi(t, x) = -\partial^2_x\psi(t, x) - u(t)\mu(x)\psi(t, x), & t \in (0, T), x \in (0, 1), \\ \partial_x\psi(t, 0) = \partial_x\psi(t, 1) = 0, & t \in (0, T), \end{cases}$$
where $\mu \in H^2((0, 1); \mathbb{R})$ is a fixed spatial function and $u(t)$ is a real-valued scalar control.

The study focuses on the degenerate case where the linearized system is not controllable. Specifically, the authors consider situations where the coupling coefficient $\langle \mu, \phi_k \rangle = 0$ for some mode $k \in \mathbb{N}$. In such scenarios, linear control theory is insufficient to determine controllability, necessitating a second-order (quadratic) analysis. The central question is whether the system remains STLC when the linearized system fails, or if a "quadratic obstruction" prevents controllability.

Methodology
The authors employ a power-series expansion of the state $\psi = \phi_0 + \psi_1 + \psi_2 + \dots$ with respect to the control $u$.

1. Quadratic Form Analysis: The core of the analysis involves the second-order term $\psi_2(T)$, specifically its projection onto the uncontrollable mode $\phi_k$. This projection is expressed as a bilinear form $Q(u, u)$ involving an integral kernel $K(t-s)$.
2. Fourier-Based Approach: A critical methodological innovation is the avoidance of standard integration-by-parts arguments. The integral kernel $K$ associated with the Schrödinger equation lacks the regularity (specifically, its derivative is not a function but a distribution involving Dirac masses and principal values) required to extract the drift term via classical integration by parts. Instead, the authors analyze the quadratic form in the Fourier frequency domain. They derive an explicit expression for the Fourier transform of the kernel, decomposing it into Dirac masses, a principal value part, and a regular part.
3. Drift Quantification:
   • Negative Result: When a specific coefficient $a_k$ (derived from the kernel's singularity) is non-zero, the authors prove a coercivity estimate showing that the quadratic motion is constrained to a half-space, preventing reachability of certain targets. This corresponds to a non-vanishing Lie bracket obstruction.
   • Positive Result: When $a_k = 0$, the authors construct specific controls using high-frequency oscillations. They exploit the sign-changing properties of the imaginary part of the Fourier-transformed kernel at high frequencies to generate motion in the directions $\pm i\phi_k$.
4. Kernel Modulation Tools: The authors develop general tools (Section 5) to transfer estimates from a reference quadratic form to one modulated by a regular function, allowing them to handle the general case $k \neq 0$ based on the reference case $k=0$.
5. Remainder Estimates: To close the argument, they derive precise energy estimates for the nonlinear remainder terms using an auxiliary system with inhomogeneous Neumann boundary conditions, ensuring the higher-order terms do not dominate the quadratic effects.

Key Results
The paper establishes a dichotomy based on the coefficient $a_k$:
$$a_k := \sum_{j \in \mathbb{N}} \left( \lambda_j - \frac{\lambda_k}{2} \right) \langle \mu, \phi_j \rangle \langle \phi_j, \mu \phi_k \rangle$$

1. Negative Result (Theorem 1.7): If $a_k \neq 0$, the system is not STLC.

   • There exists a "drift" in the direction $-i a_k \phi_k$ quantified by the squared $H^{-1}$ norm of the control primitive ($\|u_1\|_{L^2}^2$).
   • This provides a new instance of Sussmann's classical quadratic obstruction for PDEs, corresponding to a non-vanishing Lie bracket $[\mu, [\mu, A]]\phi_0$.
   • Specifically, targets of the form $\psi^*_\epsilon = \sqrt{1-\epsilon^2}\phi_0 + i \text{sgn}(a_k)\epsilon \phi_k$ are unreachable for small $\epsilon$.

2. Positive Result (Theorem 1.9): If $a_k = 0$ and the remaining coupling coefficients satisfy a generic non-degeneracy condition ($|\langle \mu, \phi_j \rangle| \geq c \langle j \rangle^{-2}$ for $j \neq k$), the system is STLC.

   • The authors construct controls that achieve motion in the directions $\pm i \phi_k$ with an amplitude scaling like $T \|u_1\|_{L^2}^2$.
   • Using a tangent vector argument and time concatenation, they extend this to motion in any direction $z \phi_k$ for $z \in \mathbb{C}$.
   • The cost of control scales as $\|u\|_{L^2} \lesssim |\langle \psi^*, \phi_k \rangle|^{1/2}$, which is characteristic of quadratic-order controllability.

Significance and Claims
The paper claims the following contributions to control theory:

• New PDE Instance of Sussmann's Obstruction: The negative result (Theorem 1.7) identifies a specific PDE instance where the classical quadratic obstruction (linked to Lie brackets of length 3) prevents STLC, extending previous results known for ODEs and other PDEs (like Dirichlet Schrödinger or KdV systems).
• First Positive Quadratic STLC for a Physical System: The positive result (Theorem 1.9) is claimed to be the first to establish small-time local controllability at the quadratic order for a "reasonable physical system" (the bilinear Schrödinger equation with Neumann conditions) driven by a single scalar control. Previous positive quadratic results (e.g., in [4]) relied on artificially constructed ("cooked-up") nonlinearities.
• Methodological Advancement: The authors highlight their Fourier-based approach as a crucial tool for handling kernels with low regularity (where standard integration by parts fails). They also provide a general framework for multiplying kernels by modulation functions, which they anticipate will be reusable for other systems.
• Technical Rigor: The work includes detailed energy estimates for the Schrödinger equation with inhomogeneous Neumann boundary conditions and rigorous proofs of the convergence of the series defining the drift coefficients.

The authors remain modest regarding the scope, noting that their results apply to the specific Neumann setting and that the positive result relies on the specific structure of the Schrödinger nonlinearity, which happens to satisfy the necessary sign-changing conditions in the frequency domain. They do not propose experimental applications but frame the work as a theoretical development in the methodology for analyzing quadratic obstructions in infinite-dimensional control systems.