[Paper Review] Local controllability of 1D Schrödinger equations with bilinear control and minimal time

Karine Beauchard, Morgan Morancey|arXiv (Cornell University)|Aug 27, 2012

Stability and Controllability of Differential Equations22 references4 citations

TL;DR

This paper establishes a general framework for local controllability of 1D Schrödinger equations with bilinear control, proving that controllability is possible only when the time exceeds a positive minimal threshold. The minimal time arises from the second-order term in the power series expansion of the solution, and the authors identify conditions on the dipolar moment $\mu$ under which controllability fails for arbitrarily small times, extending prior results on minimal time requirements in degenerate cases.

ABSTRACT

We consider a linear Schrödinger 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.

Motivation & Objective

• To identify a general condition on the dipolar moment $\mu$ under which local controllability of the 1D Schrödinger equation with bilinear control fails for arbitrarily small times.
• To characterize the existence of a positive minimal time for local controllability, extending previous results on degenerate cases.
• To link the minimal time requirement to the behavior of the second-order term in the power series expansion of the solution.
• To provide a framework where controllability holds in large time but not in small time, based on spectral and regularity properties of $\mu$.

Proposed method

• Analyzes the power series expansion of the solution to the Schrödinger equation, focusing on the second-order term to determine minimal time constraints.
• Uses the linear test and inverse mapping theorem to establish controllability in large time under non-degeneracy assumptions on $\mu$.
• Applies the Ingham inequality and compactness arguments to show that vanishing of the second-order term leads to contradiction unless the control is zero.
• Employs moment problems and spectral analysis to prove that the minimal time threshold arises from the structure of the operator $K$ associated with the system's dynamics.
• Introduces the functional $\mathcal{Q}_T(S)$ to quantify controllability and uses variational methods to analyze its supremum over control functions.
• Uses the operator $L_T$ to project controls onto the span of eigenfunctions, enabling estimation of $L^2$-norms and perturbation bounds.

Experimental results

Research questions

• RQ1 Under what conditions on the dipolar moment $\mu$ does the 1D Schrödinger equation with bilinear control fail to be locally controllable in arbitrarily small time?
• RQ2 How is the minimal time for controllability related to the second-order term in the power series expansion of the solution?
• RQ3 Can a general framework be established to distinguish between systems that require a positive minimal time and those that are controllable in arbitrary time?
• RQ4 What role does the spectral behavior of $\mu$ at the boundaries play in determining the minimal time for controllability?
• RQ5 How do the regularity and decay properties of $\langle \mu \varphi_1, \varphi_k \rangle$ influence the existence of a minimal time?

Key findings

• A positive minimal time is required for local controllability when the second-order term in the power series expansion of the solution does not vanish, which occurs generically for certain $\mu$.
• The minimal time threshold is linked to the non-vanishing of the functional $\mathcal{Q}_T(S)$, which is shown to be negative for small $T$ under the given assumptions.
• For $T < \tilde{T}_{\text{min}}^1$, the functional $\mathcal{Q}_T(S)$ is uniformly bounded above by a negative constant, implying failure of controllability in small time.
• The proof shows that assuming $\lambda(T) = 0$ leads to a contradiction via iterative differentiation and compactness, forcing the control to be zero.
• The minimal time threshold depends on the spectral properties of $\mu$, particularly the decay rate of $|\langle \mu \varphi_1, \varphi_k \rangle|$ as $k \to \infty$.
• The framework generalizes previous results, such as Coron's example with $\mu(x) = x - 1/2$, and provides a unified condition for minimal time existence.