Structural Compatibility and Uniform Stability of Temporally Degenerate Parabolic Systems

Abstract: Modern feedback design for distributed parameter systems presupposes that the closed-loop dynamics define a well-posed evolution problem. This presupposition becomes nontrivial for temporally degenerate parabolic systems, where temporal degeneracy affects not only the analytical properties of the evolution equation but also the mathematical formulation of the feedback interconnection itself. It is shown that admissible feedback interconnections for temporally degenerate parabolic systems are completely characterized by an operator compatibility condition linking the singular reaction operator with the actuator and observation operators. This characterization removes the singular component of the closed-loop dynamics and reduces the degenerate evolution equation to a regular evolution equation. Building upon this regularized formulation, a critical--residual decomposition yields a uniform exponential stability certificate, which is subsequently extended to the original infinite-dimensional evolution through a finite-to-infinite lifting theorem. A constructive static output feedback synthesis is finally obtained as a consequence of these results. Numerical experiments illustrate the regularization mechanism, validate the stability certificate, and confirm the finite-to-infinite lifting.