Exact boundary controllability of a singular/degenerate wave equation via singular Sturm-Liouville theory

The paper proves exact boundary controllability for a class of one-dimensional singular/degenerate wave equations of Sturm–Liouville type:

utt − (xα ux)x − βxα−1 ux − µxα−2 u = 0, x ∈ (0, 1),

with a Dirichlet condition at the regular endpoint and a boundary control acting at the singular endpoint x = 0. The analysis is carried out in the fractional energy space X = Hν+1/2 × Hν−1/2, associated with the corresponding singular Sturm–Liouville operator. Using the spectral decomposition induced by Bessel functions, the paper establishes admissibility of the boundary observation operator and derives precise lower estimates for the observation coefficients.

Exact observability is obtained through Ingham-type inequalities together with an abstract observability result for unitary groups. The proof treats simultaneously the subcritical, critical logarithmic, and limit-point regimes by means of singular Sturm–Liouville theory and boundary traces identified through the Lagrange bracket of the singular Sturm–Liouville expression.

The main result, Theorem 1, states: "Let 0 ≤ α < 2, β, µ ∈ R with (1 − α − β)2 ≥ 4µ. Then system (1) is exactly controllable in any time T > 4/(2−α). More precisely, for every U0, UT ∈ X there exists a control f ∈ L2 ((0, T); C) such that the corresponding weak solution of the controlled system (1) satisfies (u(T), ut (T)) = UT."

A distinctive feature is that the observation operator is not postulated a priori but emerges naturally from the singular Green identity and the asymptotic structure of principal/non-principal solutions at the singular endpoint. This provides a unified framework covering both limit-circle and limit-point regimes and recovering the classical wave equation as a particular case when α = β = µ = 0.

The paper is organized as follows: Section 2 introduces the singular Sturm–Liouville framework, self-adjoint realizations, and spectral decomposition. Section 3 constructs fractional spaces and the abstract wave dynamics. Section 4 establishes admissibility, exact observability, and exact boundary controllability.

Improvements to AI Systems:

1. Physics-Informed Spectral Controllability Module: Enhance AI systems (e.g., PDE solvers or control agents) with a built-in module that automatically detects singular/degenerate coefficients (like x alpha) and selects the correct fractional energy space H nu+1/2 times H nu-1/2 for state representation, rather than assuming standard Sobolev spaces. This allows AI to handle boundary control problems for wave equations with singular endpoints without manual regularization.

2. Unified Regime Handling via Bessel-Based Feature Extraction: Train AI to decompose solutions using Bessel-function spectral bases (as in the paper) instead of generic Fourier or polynomial bases. This enables the AI to simultaneously treat subcritical, critical logarithmic, and limit-point regimes in one framework, improving robustness when parameters (alpha, beta, mu) vary across critical thresholds.

3. Automatic Observation Operator Discovery: Implement an AI component that derives boundary observation operators from the singular Green identity and Lagrange brackets (as done in the paper) rather than assuming a fixed observation. This lets the AI generate physically meaningful control signals for systems where the natural observation point is singular, and where naive pointwise observations fail.

4. Ingham-Type Time-Optimal Control Predictor: Use the proven controllability time T > 4/(2-alpha) as a hard constraint in AI-based optimal control solvers. The AI can now predict minimal control horizons for singular wave equations, avoiding trial-and-error and ensuring exact state transfer (U 0 to U T) in any time above this threshold, with L squared control inputs.

5. Admissibility-Aware Reinforcement Learning for Boundary Control: In reinforcement learning environments simulating wave equations, add a reward shaping term that penalizes control signals violating the admissibility conditions derived from the observation coefficients. This improves sample efficiency and stability when learning controllers for singular endpoints, as the AI avoids unstable or non-physical control actions.

6. Cross-Regime Transfer Learning: Pre-train a neural network surrogate on the classical wave equation (alpha= beta= mu=0) and fine-tune it for singular cases using the paper’s unified framework. The AI can then transfer controllability knowledge across regimes, reducing training data requirements for new singular parameter sets.

What the Improved AI System Can Do:

• Solve exact boundary control problems for singular/degenerate wave equations (e.g., in plasma physics, acoustics with tapered media, or quantum graphs) in real time, with guaranteed controllability for any T > 4/(2-alpha).

• Automatically identify the correct energy space and observation functional for a given singular Sturm–Liouville operator, eliminating the need for human expert tuning.

• **Generate control inputs f(t) ** that are provably admissible and achieve exact state transfer, even in limit-point regimes where classical methods fail.

• Predict controllability times and design time-optimal control strategies for systems with fractional-order singularities, improving efficiency in engineering applications like flexible structures or waveguides with singular cross-sections.

• Adapt to new singular parameter sets (e.g., changing alpha, beta, mu) without retraining from scratch, thanks to the unified spectral framework and transfer learning capabilities.

We prove exact boundary controllability for a class of one-dimensional singular/degenerate wave equations of Sturm--Liouville type, u tt-(x alpha u x) x-beta x alpha-1u x-mu x alpha-2u=0, x in(0,1), with a Dirichlet condition at the regular endpoint and a boundary control acting at the singular endpoint x=0. The analysis is carried out in the fractional energy space % X= H nu+1/2 times H nu-1/2, associated with the corresponding singular Sturm--Liouville operator. Using the spectral decomposition induced by Bessel functions, we establish admissibility of the boundary observation operator and derive precise lower estimates for the observation coefficients. Exact observability is obtained through Ingham-type inequalities together with an abstract observability result for unitary groups. The proof treats simultaneously the subcritical, critical logarithmic, and limit-point regimes by means of singular Sturm--Liouville theory and boundary traces identified through the Lagrange bracket of the singular Sturm--Liouville expression.