Exact controllability of two-dimensional hydroelastic waves

1. Problem Statement

The paper addresses the exact controllability of the two-dimensional hydroelastic wave system in a periodic setting. This system models an irrotational, inviscid, and incompressible fluid with a free upper boundary in contact with a frictionless thin elastic sheet.

• Governing Equations: The system is described by the Euler equations for the fluid interior and dynamic boundary conditions on the free surface $\Gamma_t$. The dynamic condition includes a nonlinear elastic restoring force (proportional to the flexural rigidity $\sigma$) and an exterior pressure control $P_{ext}$.
• Formulation: The problem is reformulated using the Zakharov-Craig-Sulem variables: the surface elevation $\eta(t, x)$ and the trace of the velocity potential $\psi(t, x)$ on the free surface. The system is a quasi-linear evolution equation of order $5/2$.
• Control Objective: Given initial data $(\eta_{in}, \psi_{in})$ and final data $(\eta_{end}, \psi_{end})$ in a sufficiently small neighborhood of zero in a high-regularity Sobolev space, the goal is to find an exterior pressure control $P_{ext}(t, x)$ supported on an arbitrary non-empty open set $\omega \subset \mathbb{T}$ and a time interval $[0, T]$ such that the system evolves from the initial state to the final state in arbitrary short time $T > 0$.

2. Methodology

The authors employ a direct Nash-Moser-Hörmander approach, adapted from the work of Baldi, Haus, and Montalto on Hamiltonian nonlinear Schrödinger equations. The proof strategy involves several sophisticated steps:

A. Linearization and Reduction

The core difficulty lies in the quasi-linear nature of the system, which prevents standard linear control techniques from being directly applicable. The authors perform a rigorous diagonalization and symmetrization of the linearized operator $P'(u)$ around a solution $u = (\eta, \psi)$. This is achieved through a sequence of conjugations:

1. Good Unknowns: Transforming variables to the Alinhac "good unknown" $(\eta, \psi - T_B \eta)$ to handle the Dirichlet-Neumann operator's shape derivative.
2. Coordinate Changes: Applying time-dependent changes of space variables and time reparametrization to eliminate variable coefficients in the transport terms.
3. Matrix Conjugations: Using a series of matrix transformations ($Z, B, A, P, Q, S, M$) to reduce the principal part of the operator to a diagonal matrix with constant coefficients.
4. Symmetrization: Symmetrizing the leading order ($5/2$) and eliminating sub-leading terms (order $3/2$) via specific conjugations involving Fourier multipliers and pseudo-differential operators.
5. Fourier Integral Operators (FIOs): Constructing FIOs to eliminate lower-order terms (orders $1$ and $1/2$), reducing the system to a diagonal system of constant-coefficient dispersive equations plus a smoothing remainder.

B. Observability and Linear Controllability

Once the linearized system is reduced to a diagonal form with constant coefficients:

• Well-posedness: The well-posedness of the linearized Cauchy problem is established using energy estimates derived from the reduction steps.
• Observability: The authors prove an observability inequality for the adjoint linearized system. This relies on an Ingham-type inequality (Lemma 1.1) applied to the eigenvalues of the reduced constant-coefficient operator. The inequality ensures that the energy of the solution can be bounded from below by its observation on the control domain $\omega$.
• HUM: Using the Hilbert Uniqueness Method (HUM), the observability of the adjoint system implies the exact controllability of the linearized system.

C. Nash-Moser Iteration

To extend the result from the linearized system to the full nonlinear system:

• The authors apply the Nash-Moser-Hörmander Theorem (Theorem A.6).
• This theorem is necessary because the linearized operator loses derivatives (a common feature in quasi-linear problems), making the standard Banach fixed-point theorem inapplicable.
• The "tame" estimates for the linearized inverse (Proposition 8.1) are crucial. The smallness assumption on the data is required to ensure the nonlinear terms remain perturbative within the iteration scheme.

3. Key Contributions

1. First Exact Controllability Result: This is the first proof of exact controllability for the full 2D hydroelastic wave system in the periodic setting with arbitrary short time.
2. Handling Quasi-linearity: The paper successfully adapts the Nash-Moser-Hörmander framework to a hydroelastic context, which is more complex than standard water waves due to the higher-order elastic terms ($\partial_x^4$).
3. Technical Reduction: The authors provide a detailed, step-by-step reduction of the linearized operator to a diagonal constant-coefficient form. This involves eliminating sub-leading terms of order $3/2$ and $1/2$, which is technically more involved than in gravity-capillary water waves.
4. Arbitrary Control Domain: The result holds for any non-empty open set $\omega$, demonstrating that the control can be localized.

4. Main Results

Theorem 1.2 (Main Theorem):
Let $T > 0$ be an arbitrary time and $\omega \subset \mathbb{T}$ be a non-empty open set. There exist constants $s^*$ (regularity index) and $\delta^*$ (smallness threshold) such that if the initial and final data $(\eta_{in}, \psi_{in})$ and $(\eta_{end}, \psi_{end})$ satisfy:
$$\|( \eta_{in}, \psi_{in} )\|_{H^{s^*}} + \|( \eta_{end}, \psi_{end} )\|_{H^{s^*}} < \delta^*$$
Then there exists an exterior pressure control $P_{ext} \in C([0, T]; H^{s^*})$ supported on $[0, T] \times \omega$ such that:

1. The hydroelastic wave system admits a unique solution $(\eta, \psi)$ in the space $C([0, T]; H^{s^*}) \cap C^1([0, T]; H^{s^*-5/2}) \cap C^2([0, T]; H^{s^*-5})$.
2. The solution satisfies the boundary conditions: $(\eta, \psi)|_{t=0} = (\eta_{in}, \psi_{in})$ and $(\eta, \psi)|_{t=T} = (\eta_{end}, \psi_{end})$.
3. The control and solution satisfy tame estimates bounded by the size of the initial and final data.

5. Significance and Implications

• Theoretical Advancement: The work bridges the gap between control theory for linear dispersive equations and fully nonlinear quasi-linear hydroelastic systems. It demonstrates that the complex interplay between fluid dynamics and elastic structures can be controlled using external pressure.
• Methodological Robustness: The reduction technique developed here (symmetrization and diagonalization of quasi-linear operators) is applicable to other quasi-linear dispersive PDEs, such as the Muskat problem with elastic interfaces or higher-order water wave models.
• Limitations and Future Work: The result is currently restricted to small data due to the quasi-linear nature of the problem. The authors note that extending this to large data would likely require a stabilization result (damping) for the hydroelastic system, similar to arguments used for gravity water waves. Additionally, the regularity required ($s^*$) is high, which is typical for Nash-Moser approaches but may be reducible in future work.

In summary, this paper provides a rigorous mathematical foundation for controlling hydroelastic waves, utilizing advanced microlocal analysis and iterative fixed-point methods to overcome the challenges posed by the system's nonlinearity and high-order derivatives.