Decay of solutions and bilinear control to trajectories of a 1D degenerate parabolic system

Technical Summary: Decay of Solutions and Bilinear Control to Trajectories of a 1D Degenerate Parabolic System

Problem Statement
This paper investigates the controllability properties of a class of one-dimensional nonlinear parabolic systems characterized by weakly degenerate diffusion coefficients. The system, denoted as (1), consists of two coupled equations for state variables $y$ and $z$ defined on the domain $(0, 1) \times (0, T)$. The diffusion term involves a coefficient $a(x)$ that degenerates at the boundary $x=0$ (specifically, $a(0)=0$), modeling heterogeneous media where transport mechanisms weaken near critical regions. The system is non-autonomous, with time-dependent coefficients $b(t)$ and lower-order terms $B_i(x,t)$.

The primary control mechanism is bilinear, acting through the reaction term as $h(x,t)\omega(x)y$, where $h$ is the control function supported on a non-empty open subset $\omega \subset (0, 1)$. Unlike additive controls, this bilinear structure modulates the growth or decay rates proportionally to the current state. The paper addresses two main objectives:

1. Local Exact Controllability to Trajectories: Determining if, for any initial state close to a reference trajectory $(\bar{y}, \bar{z})$ (solution of the uncontrolled system), there exists a control $h$ that steers the system to match the trajectory at time $T$.
2. Global Exact Controllability to Trajectories: Determining if this steering is possible for any initial state, provided the time $T$ is sufficiently large.

Methodology
The authors employ a combination of functional analysis techniques, energy estimates, and Carleman inequalities tailored to the degenerate setting.

1. Reformulation and Linearization:
   To study controllability to a trajectory $(\bar{y}, \bar{z})$, the problem is reformulated by introducing the error variables $w_1 = y - \bar{y}$ and $w_2 = z - \bar{z}$. This transforms the exact controllability problem into a null controllability problem for the error system (6). The nonlinear system is then decomposed into a linear part and a nonlinear remainder.

2. Decay of Solutions (Global Controllability Strategy):
   To achieve global controllability, the authors first analyze the uncontrolled system ($h \equiv 0$). Under specific structural assumptions on the nonlinear terms $F_i$ (specifically a monotonicity condition), they prove that solutions to the error system exhibit exponential decay in time. This decay ensures that for any initial data, the system state enters a small neighborhood of the origin after a sufficiently large time $\tilde{T}$.

3. Local Inversion via Liusternik's Theorem:
   The core of the local controllability proof relies on Liusternik's Inverse Function Theorem in Banach spaces. The authors define a mapping $\mathcal{A}$ from a space of states and controls ($V$) to a space of residuals and initial data ($W$). To apply the theorem, they must verify:

   • $\mathcal{A}$ is continuously differentiable ($C^1$).
   • The Fréchet derivative $\mathcal{A}'(0)$ is surjective (onto).

4. Linear Null Controllability and Carleman Estimates:
   The surjectivity of the linearized operator is established by proving the null controllability of the associated linearized system (14). This is achieved using Carleman inequalities (Proposition 1), which provide observability estimates for the adjoint system. These estimates allow the construction of a control $h$ that drives the linear system to zero. The authors utilize weighted energy spaces with specific weight functions $\rho_j$ (dependent on time and space) to handle the degeneracy and the bilinear nature of the control.

5. Regularity and Estimates:
   The proof involves deriving detailed a priori estimates for the linearized system and the control function. Specifically, Corollary 3 establishes the regularity of the control $h$ in weighted Sobolev spaces, which is crucial for handling the multiplicative term $h \omega y$ in the nonlinear analysis.

Key Contributions and Results

• Theorem 1 (Local Exact Controllability): The paper establishes that the nonlinear degenerate system (1) is locally exactly controllable to any trajectory $(\bar{y}, \bar{z})$ at any time $T > 0$. This holds for initial data sufficiently close to the trajectory in the $H^1_a(0,1)$ norm. The proof relies on the successful application of Liusternik's theorem, contingent on the solvability of the linearized null controllability problem.
• Theorem 2 (Global Exact Controllability): The authors prove that the system is globally exactly controllable to trajectories for any initial data, provided the time horizon $T$ is sufficiently large. This result is a direct consequence of the exponential decay of the uncontrolled system (Theorem 2 in Section 1.A) combined with the local controllability result. The decay allows the system to be brought into the "local" regime where Theorem 1 applies.
• Decay Estimates: A significant technical contribution is the proof of exponential decay for solutions of the nonlinear degenerate system without control (Theorem 2), under the assumption of a specific sign condition on the nonlinearities.
• Extension of Previous Work: The results improve upon previous studies (specifically [28]) by incorporating bilinear controls (multiplicative) rather than just additive controls and by including semilinear terms $F_i(x,t,y,z)$. The paper also addresses the specific challenges of non-autonomous coefficients and weak degeneracy.

Significance and Claims
The paper claims to provide a rigorous framework for controlling degenerate parabolic systems where the diffusion vanishes at the boundary and the control acts multiplicatively on the state. The authors position their work as an advancement in the control of non-autonomous degenerate coupled systems, which model real-world phenomena such as thermal processes in nonhomogeneous materials, sedimentation, and diffusion in anisotropic media.

The significance lies in the combination of:

1. Bilinear Control: Addressing the realistic mechanism where control modulates reaction rates (e.g., catalysts in chemical reactions or reproduction rates in ecology) rather than simply adding a source term.
2. Degeneracy Handling: Successfully adapting Carleman inequalities and functional analytic tools to the weakly degenerate setting ($a(x) \sim x^\alpha$).
3. Global Results: Bridging the gap between local controllability (valid for small data) and global controllability (valid for arbitrary data) through the analysis of solution decay.

The authors note that these results can be extended to systems with nonlocal terms (e.g., dependence on $\int y$) and free boundary problems (Stefan-like problems), though these extensions are reserved for future work. The study is supported by standard assumptions on the regularity and boundedness of coefficients and the non-degeneracy of the control region $\omega$.