Global Control in Cahn-Hilliard Equation

• The paper demonstrates that finite-dimensional Fourier mode controls can approximate trajectories of the nonlinear Cahn–Hilliard equation in arbitrarily small time.
• It employs a two-phase strategy that first uses global approximate controllability and then switches to localized null controllability to steer phase separation.
• Methodological innovations integrate geometric control theory, unique continuation via spectral inequalities, and fixed-point techniques to handle fourth-order dynamics.

The global control properties of the Cahn-Hilliard equation address the extent to which the system’s macroscopic evolution can be externally manipulated via internal forces or controls. This subject is critical for understanding phase separation processes and for designing strategies to steer these nonlinear, fourth-order parabolic systems toward desired configurations, either exactly or approximately, in finite or infinitesimal time. The Cahn-Hilliard framework is also a testbed for high-dimensional controllability techniques, nontrivially extending linear and second-order parabolic theory to the nonlinear and higher-order regime.

1. Functional Setting for the Cahn-Hilliard Equation

The canonical controlled Cahn-Hilliard equation on the $d$-dimensional flat torus, $\mathbb{T}^d = \mathbb{R}^d / (2\pi \mathbb{Z}^d)$, is given by

$$\partial_t u + \Delta^2 u + \Delta u = \Delta(u^3) + \eta(t,x), \quad (t, x) \in (0, T) \times \mathbb{T}^d,$$

with periodic boundary conditions and initial datum $u(0, x) = u_0(x) \in H^k(\mathbb{T}^d)$. The control $\eta(t,x)$ acts as an internal source. Analysis is conducted in Sobolev spaces, with the standard norm structure. The uncontrolled system is globally well-posed in $X_T^0$ via energy estimates and parabolic semigroup theory, with the nonlinearity managed through fixed point arguments.

2. Small-Time Global Approximate Controllability

Global $H^k$-approximate controllability in arbitrarily small time is achieved by restricting controls to a finite-dimensional space $\mathcal{H}_0$ spanned by Fourier modes: $$\mathcal{H}_0 = \mathrm{span}\{1, \sin(x \cdot e_i), \cos(x \cdot e_i): i=1,\dots,d\}.$$ For any $u_0, u_1 \in H^k(\mathbb{T}^d)$ and any $\varepsilon > 0$, there exists $\eta \in L^\infty(0,T; \mathcal{H}_0)$ such that $\|u(T) - u_1\|_{H^k} < \varepsilon$, uniformly for any $T > 0$ (even as $T \to 0$). The central mechanism involves:

• Stability in $H^k$ with respect to control and initial data,
• An asymptotic reachability property leveraging shifts by large Fourier-mode controls,
• A geometric control "saturation" argument: repeated application expands the reachable set, ultimately densing in $H^k$ as the union of iterated images.

Hence, for every target, any trajectory can be steered arbitrarily near any other in arbitrarily small time by concatenated projected controls in $\mathcal{H}_0$, a result rooted in the infinite-dimensional Agrachev–Sarychev framework (Hernández-Santamaría et al., 14 Dec 2025).

3. Null Controllability for the Linearized System with Localized Control

Linearization about a trajectory $U(t, x)$ yields

$$w_t + \Delta^2 w + \Delta w = v(t,x)\,\chi_\omega(x),$$

where the control $v$ is localized in $\omega \subset \mathbb{T}^d$. Null controllability is established for all positive measure subsets $\omega$ by leveraging:

• Quantitative unique continuation (“propagation of smallness”),
• Lebeau–Robbiano-type spectral inequalities, enabling construction of a control $v$ supported in $\omega \times (0,T)$ such that $w(T) = 0$. Control cost admits a bound $\|v\|_{L^\infty} \leq C \exp(A/T)\|w_0\|_{L^2}$, with explicit control cost scaling as $T \to 0$ (Hernández-Santamaría et al., 14 Dec 2025).

4. Local Null Controllability for the Nonlinear System

For the nonlinear equation with a spatially localized forcing,

$$u_t + \Delta^2 u + \Delta u = \Delta(u^3) + v(t,x)\,\chi_\omega(x), \quad u(0) = u_0,$$

controllability is recast as a fixed-point problem: the nonlinearity is treated as a perturbation to the controllable linear system, yielding local null controllability for sufficiently small data. For $d=1,2$, this is realized in $L^2$; for $d=3$, the analysis is upgraded to $H^2$ based on the algebra property of the space. The key is the ability to invert the nonlinear map by a contraction argument, critically hinging on the null controllability of the linearized system (Hernández-Santamaría et al., 14 Dec 2025).

5. Two-Phase Strategy for Global Null Controllability

Global null controllability—driving any initial data to zero in finite time—is obtained by combining the small-time approximate controllability with local null controllability in a two-phase process:

• Phase I (Fourier-mode control): Use $\mathcal{H}_0$-valued controls on $(0, \delta)$ to drive the system arbitrarily close to zero.
• Phase II (Localized control): Switch to an $\omega$-supported control on $[\delta, T]$ to drive to zero exactly, exploiting local null controllability.

Thus, for $d = 1,2,3$ and any open $\omega \subset \mathbb{T}^d$, every initial datum in $L^2(\mathbb{T}^d)$ can be null-controlled at an arbitrary final time via a composite control strategy. The precise control sequence is

$$\eta(t,x) = \begin{cases} \eta_1(t,x)\in \mathcal{H}_0, & t \in (0, \epsilon), \ 0, & t \in (\epsilon, \delta), \ v(t,x)\,\chi_\omega(x), & t \in (\delta, T), \end{cases}$$

with $0 < \epsilon < \delta < T$ (Hernández-Santamaría et al., 14 Dec 2025).

6. Methodological Innovations and Open Problems

The established results leverage an overview of geometric control theory (Agrachev–Sarychev iterative commutator expansion), propagation-of-smallness techniques from unique continuation, and fixed-point arguments in appropriate weighted spaces. Control on finite Fourier-bandwidth spaces interfaces with localized control via spectral inequalities.

Open problems include:

• Extension to singular (e.g., logarithmic) potentials,
• Boundary (as opposed to internal) control in bounded domains,
• Mass-conserving and obstacle-constrained regimes,
• Analysis for multi-component and higher-order nonlinearities,
• Optimality and robustness of control cost as $T \to 0$, and issues posed by discretization (Hernández-Santamaría et al., 14 Dec 2025).

7. Relation to Broader Control and Cahn-Hilliard Theory

The established global control framework for the cubic Cahn-Hilliard equation fundamentally extends the reach of fourth-order parabolic control theory. It connects with variational and gradient-flow methods developed for aggregation-diffusion systems (Carrillo et al., 2023), optimal control of coupled Cahn-Hilliard-Darcy models with singular potentials (Abatangelo et al., 2023), as well as fractional and generalized Cahn-Hilliard systems (Colli et al., 2018). Particularly, the approach described here inspires control strategies for systems with complex free-energy landscapes, as in multicomponent and pattern-forming settings (Kagawa et al., 29 Jun 2025).

In summary, the global control properties of the Cahn-Hilliard equation enable small-time approximate steering and global null controllability using a combination of finite-dimensional mode controls and localized strategies. These methodologies have established the rigorous basis for practical phase control in a broad class of nonlinear, high-dimensional parabolic PDEs (Hernández-Santamaría et al., 14 Dec 2025).