Memory-Type Null Controllability for Non-Autonomous Degenerate Parabolic Equations with Boundary Degeneracy

Abstract: This paper studies the memory-type null controllability of a class of one-dimensional non-autonomous degenerate parabolic equations with Volterra-type memory terms. The diffusion operator is considered in both divergence and non-divergence forms and may exhibit weak or strong degeneracy at the boundary, while the diffusion coefficient depends explicitly on time. Due to the presence of memory effects, classical null controllability is insufficient, and a stronger notion requiring the vanishing of both the state and the accumulated memory is introduced. To address this problem, we establish new Carleman estimates adapted to non-autonomous degenerate operators in weighted spaces. The memory term is handled as a lower-order perturbation within the Carleman framework. These estimates yield suitable observability inequalities, which allow us to prove memory-type null controllability under appropriate structural conditions. Extensions to cases with double boundary degeneracy and moving control regions are also discussed.

• The paper establishes a framework proving that memory-type null controllability is achievable for degenerate parabolic equations with both weak and strong boundary degeneracy.
• It employs advanced Carleman estimates to derive observability inequalities and explicit control bounds by utilizing moving control regions.
• The study offers practical insights for actuator placement and numerical schemes in real-world models featuring time delays and spatial degeneracy.

Memory-Type Null Controllability for Non-Autonomous Degenerate Parabolic Equations with Boundary Degeneracy

Introduction and Problem Setting

The study addresses the memory-type null controllability for a broad class of one-dimensional non-autonomous degenerate parabolic equations with Volterra-type memory effects and boundary degeneracy. Specifically, the analysis includes both divergence and non-divergence forms of the main operator, allowing for both weak and strong degeneracy at the domain boundaries, with explicit time dependence in the diffusion coefficients. Such problems encapsulate significant physical phenomena, encompassing climate models with delayed feedback, viscoelasticity, and population genetics, where degeneracy and memory are prevalent. The memory term introduces a nonlocal-in-time structure, which demands a strengthened notion of controllability: both the state and the accumulated memory must vanish at the terminal time, a concept known as memory-type null controllability.

A salient structural challenge is the interplay between degeneracy (in which the diffusion vanishes at the boundary), non-autonomous (time-dependent) coefficients, and the nonlocal memory term. Classical tools for parabolic controllability (e.g., Carleman estimates, observability inequalities) are insufficient or must be rigorously extended. Moreover, the introduction of moving control regions is necessary, as fixed interior controls cannot, in general, counteract the memory effects—an obstruction that is intrinsic rather than technical, as established for the heat equation with memory (2604.02916).

Main Contributions and Technical Methods

Definitions of Degeneracy and Functional Framework

Two classes of degeneracy are isolated:

• Weak Degeneracy (WD): Characterized by the weight $a(x)$ vanishing at the boundary with a degeneracy rate parameter $K \in [0,1)$ and satisfying a specified monotonicity property.
• Strong Degeneracy (SD): $a(x)$ vanishes at the boundary with $K \in [1,2)$ and more rapid "degenerate" vanishing.

Both cases require weighted Hilbert and Sobolev spaces, as standard $L^2$ or $H^1$ spaces are inadequate for meaningful formulations and estimates.

Formulation with Volterra-Type Memory and Moving Controls

The model problem is:

$$u_t - \mathcal{A}(t)u + \int_0^t M(t,s)u(s)\,ds = f(t,x)\chi_{\omega(t)}(x), \quad (t,x)\in (0,T)\times(0,1),$$

subject to appropriate homogeneous (possibly degenerate) boundary conditions. The operator $\mathcal{A}(t)$ is either in divergence or non-divergence form with $a(x)$ as degeneracy and $b(t)$ as time-dependent diffusion. The control region $K \in [0,1)$0 moves in time and must dynamically cover the whole spatial domain, following geometric conditions well-established for memory-affected PDEs [Chaves-Silva/Rosier/Zuazua 2014].

Carleman Estimates for Non-Autonomous Degenerate Parabolic Operators

A fundamental tool is the derivation of global Carleman estimates for the adjoint systems. This requires a delicate construction of weight functions adapted to both boundary degeneracy (via integrals involving $K \in [0,1)$1 and powers of the distance to degeneracy points) and non-autonomous coefficients (encoding specific time-singularity profiles). For the non-divergence case, the estimate takes the form:

$K \in [0,1)$2

where $K \in [0,1)$3, $K \in [0,1)$4 and $K \in [0,1)$5 are weight functions reflecting both time and the spatial degeneracy geometry.

A delicate point is that the memory term is controlled as a lower-order perturbation: under the suitable choice of Carleman weights, the Volterra term does not interfere with the leading-order Carleman structure and can be absorbed into the main energy estimates.

Observability Inequalities and Hilbert Uniqueness Method (HUM)

The Carleman estimates yield observability inequalities for the adjoint system, now posed in weighted spaces adapted to the degeneracy structure, for moving controls:

$K \in [0,1)$6

This observability result, established for both non-divergence and divergence cases, is equivalent to memory-type null controllability for the original forward problem: for every initial datum, one can find a distributed control $K \in [0,1)$7 (supported in the moving region) that drives both the state and its accumulated memory to zero.

Main Results and Strong/Critical Claims

• Simultaneous Handling of Degeneracy, Memory, and Non-Autonomy: The analysis proves that it is possible, even in the presence of strong/weak boundary degeneracy, non-autonomous (time-dependent) principal part, and nonlocal (memory) terms, to obtain memory-type null controllability provided the control moves and structural conditions ($K \in [0,1)$8 on $K \in [0,1)$9) are satisfied.
• Sharp Geometric Condition for Control: Fixed controls cannot achieve null controllability except in trivial cases. The necessity of moving controls is structural and follows explicitly from the model's Volterra/Oscillator decomposition.
• Carleman Estimate Improvements: The Carleman estimates and observability inequalities derived for the non-divergence case require weaker assumptions than previous results and address cases with double degeneracy at both boundaries.
• Quantitative Control Bounds: The paper establishes explicit bounds of the control in terms of the initial energy in the weighted space, ensuring practical relevance for engineering and applied science modellers.

Implications, Limitations, and Future Directions

The results clarify the limits of controllability in degenerate, non-autonomous, memory-affected PDEs. On the theoretical side, they establish a comprehensive framework for deducing controllability based on Carleman techniques, geometric conditions, and the handling of nonlocal terms as perturbations. These results can be expected to be broadly applicable to related models in viscoelasticity, biology, and climate science where time delays and spatial heterogeneity are intrinsic.

Practically, the necessity of moving controls suggests that actuator placement for such systems must exploit both spatial and temporal flexibility. For numerics and optimal control, the explicit Carleman weights and observability constants provide guidance for constructing minimal-norm controls and for designing robust sensor-actuator trajectories.

Open directions include extension to multi-dimensional domains, more general forms of nonlocality (e.g., multi-term memory kernels, delay equations with non-convolution structure), and to nonlinear settings with degenerate principle parts. There is also potential for hybrid results combining stochastic effects and boundary control with memory, inspired by contemporary work on Carleman estimates for SPDEs.

Conclusion

This work establishes memory-type null controllability for a substantial class of non-autonomous degenerate parabolic PDEs with boundary degeneracy and Volterra-type memory. The framework unifies and extends both theoretical and technical methodologies (notably advanced Carleman estimates in weighted spaces and the necessity of moving controls) to cover structurally challenging open problems. The results yield both sharp theoretical insight and practical guidance for the control of degenerate systems with memory (2604.02916).