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

This paper investigates the null-controllability cost of the one-dimensional Burgers' equation linearized at a stationary shock in the vanishing viscosity limit, establishing bounds on the required control time and constructing explicit admissible controls using complex analysis methods.

The Gist

The Big Picture: Taming a Wild River

Imagine a river flowing through a narrow canyon. This river represents the Burgers' equation, a mathematical model used to describe how fluids (like air or water) move.

In this specific story, the river has a shockwave. Think of this as a sudden, sharp wall of water where the flow speed drops instantly from fast to slow. It's like a traffic jam on a highway that suddenly forms and stays in one spot.

Now, imagine the water isn't perfectly smooth; it has a tiny bit of "stickiness" or viscosity (like honey mixed in). The paper asks a very specific question: If we want to stop this river completely (make the water level zero everywhere) by only controlling the water at the left edge of the canyon, how much effort (energy) does it take?

The tricky part is that the authors are looking at the limit where the "stickiness" (viscosity) disappears. They want to know: As the water gets thinner and thinner (approaching a perfect, frictionless fluid), does the effort required to stop the river stay reasonable, or does it explode to infinity?

The Cast of Characters

1. The River (The Equation): A flow that naturally wants to form a shockwave (a stationary wall of water).
2. The Shock (The Obstacle): A permanent bump in the river that doesn't move. It's the "steady state" the system is stuck in.
3. The Controller (The Hand): A valve or gate at the left end of the canyon. You can open or close it to push or pull water.
4. The Viscosity (The Honey): A tiny amount of friction. The paper studies what happens when this honey is almost gone.
5. The Cost (The Energy Bill): The amount of energy you need to spend on the valve to stop the river.

The Main Problem: The "Metastable" Ghost

The authors discovered something strange about this river. Because of the shockwave, the system has a "ghost" mode.

Imagine trying to push a heavy boulder that is stuck in a deep, narrow valley. If you push it just a tiny bit, it rolls back. But if you push it exactly right, it might stay there for a very long time before moving. In math terms, this is called metastability.

There is one specific "mode" (a way the water can wiggle) that is incredibly hard to kill. It's like a ghost that refuses to leave the room.

• The Good News: If you wait long enough, the natural friction (viscosity) will eventually kill this ghost.
• The Bad News: As the friction gets smaller (approaching zero), it takes much longer for the ghost to die naturally.

The Two-Step Strategy

The paper proposes a clever two-step plan to stop the river efficiently, even when the friction is tiny.

Step 1: The "Surgical Strike" (Killing the Ghost)

First, you need to get rid of that stubborn "ghost" mode immediately. You can't wait for friction to do it because it's too slow.

• The Analogy: Imagine the river has a specific, annoying vibration. You apply a very precise, short burst of energy at the left gate to cancel out that specific vibration instantly.
• The Result: The "ghost" is gone. The river is now in a state where the rest of the water is easy to control.

Step 2: The "Squeeze" (Letting Friction Do the Rest)

Once the ghost is gone, the rest of the river behaves nicely. It's like a hot cup of coffee cooling down.

• The Analogy: You just let the system sit there. The natural friction (even if it's tiny) will quickly dampen the remaining ripples.
• The Result: The river stops completely.

The Big Discovery: How Long Do We Need?

The authors calculated the minimum time required for this plan to work without the energy bill becoming astronomical.

• If the shock is in the middle: You need a specific amount of time (roughly proportional to the length of the canyon). If you try to stop the river faster than this, the energy required to kill the "ghost" explodes to infinity.
• If the shock is off-center: The math gets more complex, but the rule is similar: You need enough time for the "ghost" to be killed before the friction takes over.

The "Magic Number": The paper proves that if you give the system enough time (specifically, a time related to the length of the canyon multiplied by a constant like $4\sqrt{3}$), you can stop the river with a reasonable amount of energy, even if the water is almost frictionless.

The Twist: Two Gates vs. One Gate

The authors also asked: What if we have control gates at both ends of the canyon (left and right)?

• One Gate: The water has to travel all the way across the canyon to be stopped. The "bad" side of the river (where the flow goes against your control) makes it hard.
• Two Gates: You can push from the left and pull from the right simultaneously.
• The Result: This is a game-changer! With two gates, you can stop the river twice as fast with the same energy. The "ghost" mode is easier to kill because you can attack it from both sides. The paper shows that with two gates, the minimum time required is cut in half.

Why Does This Matter?

This isn't just about math puzzles. It helps engineers understand:

1. Supersonic Flight: How to control shockwaves in air (like in jet engines).
2. Traffic Flow: How to clear traffic jams efficiently.
3. Safety: Knowing the "minimum time" tells us how fast we can react to a disaster before the system becomes uncontrollable.

Summary in a Sentence

The paper proves that while stopping a fluid with a shockwave is incredibly difficult when there is almost no friction, you can still do it with reasonable effort if you act quickly enough to kill the "stubborn ghost" of the system, and if you have control on both sides of the river, you can do it twice as fast.

Technical Summary

1. Problem Statement

The paper investigates the null-controllability cost of the one-dimensional viscous Burgers' equation linearized around a stationary shock profile as the viscosity parameter $\varepsilon$ tends to zero (the vanishing viscosity limit).

• The System: The study focuses on the linearized equation:
  $$\partial_t u_\varepsilon + \partial_x(U_\varepsilon u_\varepsilon) = \varepsilon \partial_{xx} u_\varepsilon$$
  on the interval $(-L, L)$, where $U_\varepsilon$ is a stationary viscous shock profile transitioning from $1$ to $-1$ (centered at $\sigma$).
• Control Configuration: The control acts via Dirichlet boundary conditions. The primary focus is on one-sided control (at the left endpoint $x=-L$), with an extension to two-sided control (at both $x=-L$ and $x=L$).
• The Core Question: For a fixed control time $T$, does the cost of driving the system to zero remain bounded as $\varepsilon \to 0$? If not, what is the minimal uniform control time $T_{\text{unif}}$ required to ensure the cost remains bounded?
• Context: This problem is a nonlinear analogue of the "Coron-Guerrero" problem (transport-diffusion with constant velocity), but the presence of a shock introduces a metastable mode (an eigenvalue exponentially close to zero) which complicates the control strategy.

2. Methodology

The author employs a combination of spectral analysis, complex analysis, and the method of moments.

A. Spectral Analysis of the Linearized Operator

The linearized operator is $L_\varepsilon^\sigma u = \partial_x(U_\varepsilon^\sigma u) - \varepsilon \partial_{xx} u$.

• Transformation: The author reduces the non-self-adjoint eigenvalue problem to a Schrödinger-type operator with a constant potential using a conjugation technique involving hyperbolic functions ($\text{sech}$ and $\cosh$).
• Eigenvalue Distribution:
  • First Eigenvalue ($\lambda_{\varepsilon, 0}$): It is shown to be exponentially small ($O(e^{-L/\varepsilon})$), corresponding to the metastability of the shock position.
  • Higher Eigenvalues ($\lambda_{\varepsilon, k}, k \ge 1$): They are distributed similarly to the Laplacian with a shift, roughly $\lambda_{\varepsilon, k} \approx \frac{1}{4\varepsilon} + \frac{k^2 \pi^2}{4L^2 \varepsilon}$.
• Eigenfunctions: Precise estimates are derived for the eigenfunctions and their derivatives at the boundary, which are crucial for the moment method. The behavior differs significantly depending on whether the shock is centered ($\sigma=0$) or shifted ($\sigma \neq 0$).

B. Control Strategy (Two-Step Approach)

To handle the exponentially small eigenvalue, the control time $T$ is split into two intervals $(0, \tau)$ and $(\tau, T)$:

1. Step 1 (Killing the Metastable Mode): A control is applied in time $(0, \tau)$ specifically to cancel the projection of the initial state onto the first eigenfunction ($\psi_{\varepsilon, 0}$). This step requires careful handling to avoid re-exciting other modes.
2. Step 2 (Controlling the Rest): Once the first mode is eliminated, the remaining system is controlled using the method of moments (constructing a bi-orthogonal family to the exponentials $e^{\lambda_{\varepsilon, k} t}$). The strong dissipation of the higher modes ($\lambda_{\varepsilon, k} \gg 1$) ensures that the cost of this second step is exponentially small ($O(e^{-C/\varepsilon})$) provided $T - \tau$ is sufficiently large.

C. Lower Bound via Complex Analysis

To prove the necessity of a minimal time, the author uses complex analysis techniques (Hadamard factorization and properties of entire functions of exponential type).

• By assuming a control exists with bounded cost, the author constructs an entire function related to the Fourier transform of the control.
• Using the distribution of eigenvalues and the specific values the function must take (to satisfy the null-controllability condition), a contradiction is derived if $T$ is too small, proving that the cost must blow up.

3. Key Contributions and Results

A. One-Sided Control (Left Endpoint Only)

The paper establishes sharp bounds for the minimal uniform control time $T_{\text{unif}}$:

• Case $\sigma \ge 0$ (Shock at or right of center):
  $$(4\sqrt{2} - 2)L \le T_{\text{unif}} \le 4\sqrt{3}L$$
• Case $\sigma < 0$ (Shock left of center):
  $$(4\sqrt{2} - 2)L + 8|\sigma| \le T_{\text{unif}} \le 4(2|\sigma| + \sqrt{4\sigma^2 + 3L^2})$$
• Significance: The lower bound is strictly greater than the time required for the inviscid limit system ($T > L + |\sigma|$). This confirms the existence of a "short-time obstruction" where the viscous system is controllable, but the cost explodes as $\varepsilon \to 0$ if $T$ is too small. The obstruction is caused by the difficulty of controlling the metastable shock position against the transport term.

B. Two-Sided Control (Both Endpoints)

When control is available at both $x=-L$ and $x=L$:

• Result: The minimal uniform time is significantly reduced:
  $$T_{\text{unif}}^{\text{TS}} \in [L, 2\sqrt{3}L]$$
• Mechanism: The symmetry of the problem allows the control to be decoupled into even and odd modes. The "bad" metastable mode (even) can be killed efficiently, and the transport term aids control on both halves of the domain simultaneously.
• Significance: The lower bound obstruction disappears (or is reduced to the trivial limit system constraint $T > L$). Controlling from both sides eliminates the cost explosion associated with the shock's position.

C. Explicit Control Construction

The paper provides a constructive proof of the upper bounds. It explicitly constructs an admissible control $h_\varepsilon$ that:

1. Converges in $L^2$ to the optimal control of the limit inviscid system as $\varepsilon \to 0$.
2. Satisfies the cost estimate:
   $$\|h_\varepsilon\|_{L^2} \le \left( \frac{C}{\sqrt{T - T^*}} + C e^{-C/\varepsilon} \right) \|u_0\|_{L^2}$$
   where $T^*$ is the derived minimal time.

4. Significance and Implications

• Resolution of Metastability: The paper rigorously quantifies how the "metastability" of viscous shocks (the tendency to stay in a quasi-stationary state) impacts controllability costs. It shows that while the inviscid system is controllable in time $L+|\sigma|$, the viscous system requires a longer time to control the shock's position without infinite energy.
• Extension of Coron-Guerrero Results: It extends the well-known results on transport-diffusion equations with constant velocity to the case of variable velocity generated by a shock profile.
• Nonlinear Relevance: While the analysis is linear, the results provide critical insights for the local controllability of the full nonlinear Burgers' equation near a shock. The linearized cost bounds often dictate the feasibility of local control strategies for the nonlinear problem.
• Methodological Advancement: The adaptation of the bi-orthogonal family construction to handle an isolated, exponentially small eigenvalue without re-exciting it is a technical novelty that could apply to other singular perturbation problems in PDE control.

In summary, Laheurte demonstrates that vanishing viscosity controllability is not uniform for all times in the presence of a shock. There is a critical time threshold below which the energy required to control the shock position becomes infinite as viscosity vanishes, a phenomenon that can be mitigated by using controls at both boundaries.