Local controllability of the Cahn-Hilliard-Burgers' equation around certain steady states

In this article we study the local controllability of the one-dimensional Cahn-Hilliard-Navier-Stokes equation, that is Cahn-Hilliard-Burgers’ equation, around a certain steady state using a localized interior control acting only in the concentration equation. To do it, we first linearize the nonlinear equation around the steady state. The linearized system turns out to be a system coupled between second order and fourth order parabolic equations and the control acts in the fourth order parabolic equation. The null controllability of the linearized system is obtained by a duality argument proving an observability inequality. To prove the observability inequality, a new Carleman inequality for the coupled system is derived. Next, using the source term method, it is shown that the null controllability of the linearized system with non-homogeneous terms persists provided the non-homogeneous terms satisfy certain estimates in a suitable weighted space. Finally, using a Banach fixed point theorem in a suitable weighted space, the local controllability of the nonlinear system is obtained.

💡 Research Summary

The paper investigates the local null‑controllability of a one‑dimensional Cahn‑Hilliard‑Burgers system, which can be viewed as a simplified Cahn‑Hilliard‑Navier‑Stokes model. The authors focus on a steady state consisting of a constant concentration φ̄ (different from the pure phases 0, ±1) and a velocity field ū that solves a small‑forcing Burgers‑type equation. By introducing perturbations w = u − ū and ψ = φ − φ̄, the original nonlinear problem is rewritten as a coupled system of a second‑order parabolic equation (for w) and a fourth‑order Cahn‑Hilliard equation (for ψ). The control enters only the fourth‑order equation as an interior source term supported in an arbitrary open sub‑interval O⊂(0,1).

The analysis proceeds in several stages. First, the system is linearized around the steady state, yielding a coupled linear system where the second‑order equation is driven by the spatial derivative of ψ, while ψ is directly actuated by the control. The main controllability result for this linear system (Theorem 1.3) states that for any initial data in L²(0,1)² and any time horizon T>0, there exists a control h∈L²(0,T;L²(O)) that drives the state to zero at time T. The associated control cost satisfies an explicit exponential estimate of the form ‖h‖ ≤ M exp(M(T+1/T^m))‖(w₀,ψ₀)‖, with m>3.

The cornerstone of the proof is a new Carleman inequality adapted to the coupled second‑ and fourth‑order system with mixed Dirichlet–Neumann–third‑order boundary conditions. Existing Carleman estimates apply separately to heat‑type or Kuramoto‑Sivashinsky‑type equations, but they cannot handle the present coupling. By constructing a single weight function that simultaneously controls both equations, the authors derive a unified Carleman estimate, which yields an observability inequality for the adjoint system. Duality then provides the null‑controllability of the linearized system.

Next, the source‑term method is employed to extend the controllability result to the linear system with non‑homogeneous terms. By working in weighted Hilbert spaces where the weight blows up as t→T⁻, the authors show that the same observability inequality holds, guaranteeing that the presence of suitable source terms does not destroy controllability.

Finally, the nonlinear system is tackled using a Banach fixed‑point argument. The nonlinear terms N₁(w,ψ) and N₂(w,ψ) are quadratic and satisfy Lipschitz estimates on a small ball around the origin. Defining a mapping that takes a pair (w,ψ) to the solution of the nonlinear system with the control obtained from the linear theory, the authors prove that this mapping is a contraction provided the initial perturbation is sufficiently small (‖(w₀,ψ₀)‖ ≤ μ). Consequently, a unique fixed point exists, delivering a control that steers the original nonlinear system to the chosen steady state at time T (Theorem 1.4).

The paper’s contributions are threefold: (i) the derivation of a novel Carleman inequality for a coupled second‑order/ fourth‑order parabolic system with nonstandard boundary conditions; (ii) the demonstration that a localized interior control acting only on the fourth‑order equation suffices to control the whole coupled system; (iii) the combination of Carleman‑based observability, source‑term techniques, and fixed‑point theory to achieve local null‑controllability of the nonlinear Cahn‑Hilliard‑Burgers model. The results open the way for further investigations in higher dimensions, more general boundary conditions, and multi‑control configurations.