Bilinear control to trajectories of 1D degenerate parabolic equations in moving domains

Technical Summary: Bilinear Control to Trajectories of 1D Degenerate Parabolic Equations in Moving Domains

Problem Statement
This paper investigates the local controllability properties of one-dimensional semilinear degenerate parabolic equations defined on bounded domains that evolve over time. The specific system under consideration is governed by a bilinear control acting through the reaction term coefficient. The domain $\Omega_t$ is defined as $(0, \ell(t))$, where the boundary $\ell(t)$ is a continuously differentiable function of time, creating a non-cylindrical space-time domain $\hat{Q}$. The diffusion coefficient $a(x)$ degenerates at the boundary $x=0$ (e.g., $a(x) = x^\alpha$ for $\alpha \in (0,1)$).

The primary objective is to establish local exact controllability to a positive trajectory. Specifically, given a positive trajectory $\tilde{u}$ satisfying the uncontrolled equation, the authors seek to determine if, for initial data sufficiently close to the trajectory's initial state, there exists a control function $h$ supported in an arbitrarily small open set $\hat{\omega} \times (0, T)$ such that the controlled state $u$ reaches the trajectory state $\tilde{u}$ at time $T$.

Methodology
The authors employ a multi-step strategy combining geometric transformations, linearization, and functional analysis:

1. Domain Transformation: To handle the moving boundary, the authors construct a diffeomorphism $\tau_t$ that maps the non-cylindrical domain $\hat{Q}$ onto a fixed cylindrical domain $Q = (0, 1) \times (0, T)$. This transformation rescales the spatial variable, converting the original problem into a parabolic equation with time-dependent coefficients and additional lower-order terms on a fixed domain.
2. Linearization and Adjoint Analysis: The controllability of the nonlinear system is approached via the linearization of the transformed system around the zero state (after shifting by the target trajectory). The authors analyze the associated linearized system, which includes a degenerate diffusion term and a bilinear control term.
3. Carleman Estimates: A critical component of the proof is the derivation of global Carleman estimates for the linearized adjoint system. These estimates utilize specific weight functions that depend on time and space, designed to handle the degeneracy at $x=0$ and the time-dependence of the domain. These estimates lead to an observability inequality.
4. Null Controllability of the Linear System: Using the observability inequality and the Hilbert Uniqueness Method (HUM) adapted to weighted spaces, the authors prove the global null controllability of the linearized system. They establish the existence of a control that drives the solution to zero at time $T$, along with specific regularity estimates for the control and the state.
5. Inverse Function Theorem: Finally, the authors apply the Liusternik Inverse Function Theorem in Banach spaces. By verifying that the mapping defined by the nonlinear system is $C^1$ and that its linearization at the origin is surjective (based on the linear controllability results), they deduce the local exact controllability of the original nonlinear system to the positive trajectory.

Key Contributions and Results

• Main Theorem: The paper proves that the nonlinear degenerate parabolic system (1) is locally exactly controllable to any positive trajectory $\tilde{u}$ at any time $T > 0$, provided the initial data is sufficiently close to the trajectory's initial data in the $H^1_a$ norm.
• Handling Moving Domains: The work extends controllability results from fixed domains to non-cylindrical, moving domains for degenerate equations, a setting that introduces significant complexity due to the time-varying geometry and the degeneracy of the diffusion coefficient.
• Bilinear Control Mechanism: The study focuses on bilinear controls (multiplicative controls) acting on the reaction term, which modulate growth or decay rates proportional to the state. This is distinct from additive controls and is shown to be effective for trajectory tracking in this context.
• Regularity Estimates: The authors derive detailed weighted estimates for the control and the state of the linearized system, ensuring the control possesses sufficient regularity (specifically in weighted $L^2$ and $H^1$ spaces) to handle the nonlinear terms in the fixed-point argument.

Significance and Claims
The authors position this work as a contribution to the theory of control for degenerate partial differential equations (PDEs) in non-standard geometries. They note that while controllability for non-degenerate and fixed-domain degenerate equations is well-developed, the combination of degeneracy, nonlinearity, and moving boundaries presents a more complex challenge requiring specific techniques.

The paper claims to provide the first results on the bilinear controllability of such degenerate systems in moving domains. The significance lies in demonstrating that the system can be steered to a desired positive trajectory, which corresponds to regulating complex dynamical systems (such as those in phase transitions or population dynamics) under realistic constraints where diffusion vanishes at boundaries and the domain evolves. The authors explicitly state that their results are local (requiring small initial deviations) and rely on the Liusternik Inverse Function Theorem, a standard but rigorous approach for establishing local controllability in infinite-dimensional systems.

The paper concludes by suggesting potential extensions to systems with nonlocal nonlinearities and one-phase Stefan-like problems, indicating that the techniques developed here could serve as a foundation for these more complex scenarios.