TITLE:
Toward a Unified Controllability Theory for Coupled Degenerate Elliptic-Parabolic and Hyperbolic Systems
AUTHORS:
Thierno Sarr
KEYWORDS:
Renormalized Solutions, Exact Boundary Controllability, Hilbert Uniqueness Method, Degenerate Elliptic-Parabolic Equations, Coupled Wave Systems, Poroelasticity
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.10,
October
21,
2026
ABSTRACT: The conclusion of the thesis sets as a long-term objective the unification of its two parts—the theory of renormalized solutions for a degenerate elliptic-parabolic equation (Part I) and the Hilbert Uniqueness Method (HUM) for exact boundary controllability of a system of coupled wave equations (Part II)—within a single methodological framework. This note proposes a poroelastic-type coupled system linking the two problems, precisely identifies the central mathematical obstruction preventing a direct fusion of the two theories—a regularity mismatch between HUM’s finite-energy solutions and the L1 framework of renormalized solutions—and shows that an appropriate choice of coupling terms, built on objects already present in both parts of the thesis (the truncations
T
k
(
u
)
on one side, finite-energy weak solutions on the other), makes it possible to bypass this obstruction at a fixed truncation level. A well-posedness result for the truncated system is established via a fixed-point argument, resting on a precise stability statement for renormalized solutions under right-hand side perturbations (Proposition 3') whose hypotheses and convergence topology are made explicit. An HUM control-cost estimate for the wave system forced by a fixed interior source is proved (Proposition 4'), giving an explicit dependence of the control norm on that source and an explicit, k-linear a priori bound relevant to the passage to the limit. Coupled controllability of the truncated system is formulated as a conjecture drawing on Zuazua’s techniques for semilinear systems, with the missing ingredient for a complete proof identified explicitly. A dedicated section discusses the physical scope and the limits of the chosen coupling relative to the classical Biot equations. The passage to the limit
k→+∞
—the true heart of the long-term objective—is identified as a second open problem, distinct from and more difficult than the first. Every numbered statement in this note carries an explicit status tag—Established, Established in this note, Conjectured by analogy, or Open—so that the boundary between what is proved and what remains a research objective is visible at each step, not left to be inferred from the surrounding prose.