Cost of controllability of the Burgers' equation linearized at a steady shock in the vanishing viscosity limit

We consider the one-dimensional Burgers’ equation linearized at a stationary shock, and investigate its null-controllability cost with a control at the left endpoint. We give an upper and a lower bound on the control time required for this cost to remain bounded in the vanishing viscosity limit, and construct an admissible control with an explicit limit behavior. We also provide an extension of the analysis to the case where the control acts on both endpoints. The proof relies on complex analysis and adapts methods previously used to tackle the same issue with a constant transport term.

💡 Research Summary

The paper investigates the null‑controllability of the one‑dimensional Burgers equation linearized around a stationary viscous shock, focusing on the behavior of the control cost as the viscosity ε tends to zero. The authors consider the system on a finite interval (−L, L) with a Dirichlet control applied at the left endpoint (and later also at the right endpoint). For a given control time T, the null‑controllability cost C(T,L,σ,ε) is defined as the supremum over unit‑norm initial data of the minimal L²‑norm of the control that drives the solution to zero at time T. The central question is to determine the minimal time T_unif such that C(T,L,σ,ε) stays bounded uniformly in ε as ε→0.

Limit system (ε=0).
When ε is formally set to zero, the PDE reduces to a pure transport equation with a discontinuous velocity field sgn(x−σ). By explicit characteristic formulas, the solution can be written in closed form, revealing that the system is null‑controllable only if T>L+|σ|. Moreover, the optimal control for the limit system is a simple “average‑cancelling” function: h₀(t)=−(2/(T−L−σ))∫_{−L}^{L}u₀(x)dx for t<T−L−σ, and zero afterwards. The associated cost is C(T,0)=√(2L)√(T−L−σ).

Spectral analysis of the linearized operator.
The linearized operator L_{ε,σ}u = ∂ₓ(U_{ε,σ}u) − ε∂ₓₓu acts on H₀¹∩H². Its adjoint L_{ε,σ}* = −ε∂ₓₓ − U_{ε,σ}∂ₓ has a discrete spectrum. The authors prove:

• The smallest eigenvalue λ_{ε,σ,0} is exponentially small, λ_{ε,σ,0}=O(ε e^{−L²/ε}), reflecting the translational invariance of the shock.
• For k≥1, eigenvalues satisfy ¼ε + (kπ)²/(4L²ε) < λ_{ε,σ,k} < ¼ε + ((k+1)π)²/(4L²ε).
• Gaps satisfy |λ_{ε,σ,k}−λ_{ε,σ,j}| ≥ |k²−j²| eπ²/(4L²).
• Boundary derivatives of eigenfunctions obey precise estimates: |εψ’_{ε,σ,0}(−L)| ≤ C, while for k≥1 they grow like (4Lkπ)/√ε (σ≥0) or (4Lkπ e^{2|σ|/ε})/√ε (σ<0).

These spectral facts are crucial for applying the method of moments, which requires a uniform lower bound on eigenvalue gaps and control of the boundary traces of eigenfunctions.

Upper bound on T_unif (controllability for large times).
The authors construct a two‑step control strategy:

1. Killing the slow mode. Using the first eigenfunction, they design a control that cancels the mean of the solution in a short interval of length ≈T−T* (with T* defined below). This step yields a term proportional to √(T−T*).
2. Exploiting diffusion for higher modes. The remaining modes decay rapidly because of the 1/(4ε) term in the eigenvalues. By choosing T larger than a critical value T*, the diffusion is sufficient to drive all higher modes to zero within the remaining time.

The critical time T* is given explicitly:

• If σ≥0, T* = 4√3 L.
• If σ<0, T* = 4(2|σ| + √(4σ² + 3L²)).

For any T > T*, they prove the existence of a control h_{ε} such that ‖h_{ε}‖{L²(0,T)} ≤ 4√L √(T−T*) + C e^{−c/ε}, and h{ε} converges in L²(0,T) to the optimal limit control h₀ as ε→0, with an error O(ε√(T−T*)). Consequently, the null‑controllability cost remains bounded for all T > T*, establishing an upper bound on T_unif.

Lower bound on T_unif (obstruction for short times).
Using complex‑analytic techniques reminiscent of those in Lissy (2015) and Dardé‑Ervedoza (2015), the authors show that if T < (4√2−2)L, then the control cost necessarily blows up as ε→0. The argument hinges on the fact that the first eigenfunction’s boundary trace cannot be sufficiently damped in such a short time, due to its O(1/√ε) magnitude. This yields the lower bound T_unif ≥ (4√2−2)L, which is slightly larger than the minimal time L+|σ| required for the limit transport system.

Two‑boundary control extension.
When controls are allowed at both x=−L and x=+L, the authors repeat the spectral analysis and moment construction. The presence of the second control eliminates the “short‑time obstruction” entirely: for any T > L+|σ| the system becomes uniformly null‑controllable. The upper bound on T_unif improves accordingly, matching the best known results for constant‑velocity transport‑diffusion equations.

Conclusions and perspectives.
The paper provides a complete quantitative description of how the vanishing viscosity limit influences controllability for a linearized Burgers shock. The key contributions are:

• Precise eigenvalue/eigenfunction asymptotics that capture the metastable slow mode.
• Sharp upper and lower bounds on the uniform controllability time, with explicit constants.
• Construction of admissible controls that converge to the optimal limit control.
• Extension to the two‑boundary setting, showing the benefit of additional actuation.

These results bridge the gap between the well‑studied constant‑velocity transport‑diffusion models and the more intricate variable‑velocity case arising from shock linearization. The techniques (spectral reduction to a Schrödinger operator, moment method with explicit gap estimates, complex analysis) are likely adaptable to other conservation laws with internal layers or to nonlinear control problems where one first drives the system close to a traveling wave before applying linearized controls.