Small-time estimates for a real moment problem with two-term Weyl spectral law

1. Problem Statement

The paper addresses the solvability of moment problems involving real exponentials and their application to the null-controllability and local exact controllability of linear and bilinear parabolic equations.

• The Moment Problem: Given a sequence of distinct positive real numbers $(\lambda_k)_{k \in \mathbb{N}^*}$ and a target sequence $(x_k) \subset \mathbb{R}$, the goal is to find a control function $u \in L^2(0, T)$ such that:
  $$\int_0^T e^{\lambda_k t} u(t) \, dt = x_k, \quad \forall k \in \mathbb{N}^*.$$
  The authors focus on estimating the control cost (the norm $\|u\|_{L^2}$) as the time horizon $T \to 0^+$.

• The Spectral Assumptions: The sequence $(\lambda_k)$ is not required to satisfy the classical uniform gap condition (i.e., $\inf |\lambda_{k+1} - \lambda_k| > 0$). Instead, it satisfies a two-term Weyl asymptotic law:
  $$c_{W,1}\Gamma^a - c_{W,2}\Gamma^b \leq N_\lambda(\Gamma) \leq c_{W,1}\Gamma^a + c_{W,2}\Gamma^b,$$
  where $N_\lambda(\Gamma)$ is the counting function, $1/2 \leq a < 5/8$, and $0 < b < 1/2$.
  Additionally, the sequence satisfies a weak spectral gap condition:
  $$\lambda_{k+1} - \lambda_k \geq \exp(-c_w \lambda_k^{1/2}).$$

• The Challenge: Existing results often require uniform block spacing or specific spectral gaps to derive short-time estimates. This paper tackles cases where eigenvalues may cluster significantly (unbounded block sizes) while still satisfying the Weyl law, a scenario common in higher-dimensional domains (e.g., rectangles with irrational aspect ratios).

2. Methodology

The authors employ a combination of biorthogonal family construction and the Lebeau-Robbiano method (spectral truncation).

1. Biorthogonal Families:

   • The core of the moment problem solution relies on constructing a biorthogonal family $(\sigma_k)$ to the exponentials $(e^{\lambda_k t})$.
   • The authors utilize and refine a result from [26] (Proposition 2.1), which provides explicit estimates for the $L^2$-norm of biorthogonal functions based on the spectral density and gap properties.
   • They introduce an auxiliary spectral sequence $(\nu_k)$ that modifies the original sequence $(\mu_k)$ above a cutoff frequency $\Lambda$. This modification allows them to handle the clustering of eigenvalues by artificially creating gaps for high frequencies while preserving the low-frequency structure.

2. Lebeau-Robbiano Strategy:

   • The proof of null-controllability for the associated linear system uses a spectral truncation approach.
   • Step 1: Control the low-frequency modes (below a cutoff $\Lambda$) using the moment method. The control cost for this part depends on the biorthogonal estimates.
   • Step 2: Exploit the inherent dissipation of the parabolic operator (heat equation) to drive the high-frequency modes to zero without additional control effort, provided the time is sufficient for the dissipation to dominate.
   • By optimizing the cutoff parameter $\Lambda$ as a function of time $T$, the authors derive the total control cost.

3. Bilinear Controllability:

   • For the bilinear case, the authors use a linearization argument. They show that if the linearized system around an eigenstate is null-controllable with a specific cost estimate (exponential in $1/T^\gamma$), then the nonlinear bilinear system is locally exactly controllable to that eigenstate.

3. Key Contributions and Results

A. Solvability of the Moment Problem (Theorem 1.1)

The authors prove that for sequences satisfying the two-term Weyl law ($1/2 \leq a < 5/8$) and the weak gap condition, the moment problem is solvable.

• Result: There exists a control $u$ such that $\|u\|_{L^2(0,T)} \leq C \exp(C T^{-\alpha_{\varepsilon, \delta}}) \|x\|_{\ell^2}$, where $\alpha_{\varepsilon, \delta}$ is a constant depending on the spectral parameters.
• Significance: This extends previous results by removing the requirement for uniform block spacing. It handles sequences with unbounded block sizes (e.g., eigenvalues of the Laplacian on a square where multiplicities grow).

B. Null-Controllability of Linear Systems (Theorem 1.3)

For the abstract linear control problem $\xi' + A\xi + Bv = 0$, the system is null-controllable in arbitrarily small time $T$.

• Result: The control cost satisfies the same exponential estimate as the moment problem.
• Significance: This generalizes controllability results to higher-dimensional domains where spectral gaps are not uniform, provided the control operator $B$ satisfies a non-degeneracy condition (exponential lower bound on Fourier coefficients).

C. Local Exact Controllability to Eigensolutions (Theorem 1.8)

The paper establishes small-time local exact controllability for bilinear parabolic equations of the form $\psi' + A\psi + vB\psi = 0$.

• Result: For any initial state close to an eigenstate $\phi_j$, there exists a control steering the system exactly to the eigenstate trajectory $\Phi_j(T) = e^{-\mu_j T}\phi_j$.
• Significance: This overcomes the "Ball-Marsden-Slemrod" obstruction (which suggests bilinear systems are not exactly controllable in general) by restricting the target to eigenstates and utilizing the specific spectral properties of the operator.

D. Application to Fractional/Higher-Dimensional Equations

• The authors provide a concrete example: The Bi-Laplace operator on a rectangle $\Omega = (0, a) \times (0, b)$ where $a^2/b^2$ is an algebraic irrational number.
• In this setting, the eigenvalues satisfy the required Weyl law and gap conditions, but fail the uniform block spacing condition. This demonstrates the applicability of their theory to complex geometric domains where classical methods fail.

4. Significance and Impact

1. Relaxation of Spectral Assumptions: The paper breaks the barrier of requiring uniform spectral gaps or bounded block multiplicities. It shows that the two-term Weyl law is sufficient to derive short-time controllability estimates, even in the presence of significant eigenvalue clustering.
2. Explicit Cost Estimates: Unlike many qualitative controllability results, this work provides explicit exponential estimates for the control cost as $T \to 0$. This is crucial for numerical implementations and understanding the energy required for fast control.
3. Bilinear Control in Higher Dimensions: It resolves the local exact controllability to eigenstates for bilinear heat equations in dimensions $d \geq 2$ under realistic geometric conditions (irrational aspect ratios), extending previous 1D results.
4. Methodological Advancement: The construction of the auxiliary sequence $\nu_k$ to handle the "weak gap" property via the Lebeau-Robbiano method offers a robust template for analyzing other control problems with non-uniform spectra.

5. Limitations and Future Directions

• Parameter Range: The main theorems are stated for $1/2 \leq a < 5/8$. The authors note in the Appendix that the case $0 < a < 1/2$ can be treated, but the resulting control cost estimates appear non-optimal due to the method used.
• Optimality: The authors acknowledge that the derived control cost exponents might not be optimal, particularly for the $a < 1/2$ case, suggesting a need for refined techniques in future work.

In summary, Buffe and Duca provide a rigorous framework for controlling parabolic systems with clustered spectra, bridging the gap between abstract moment problem theory and concrete applications in higher-dimensional PDEs with complex geometries.