Structural Inheritance in Sensitivity Systems of Differential-Algebraic Equations

We prove that adding sensitivity equations to a parametrised differential algebraic equation (DAE) preserves its structural properties identically. Under the standing hypotheses of real analyticity and locally constant structural data, the Σ-matrix (signature matrix), canonical offsets, and system Jacobian of every sen sitivity equation equal those of the original DAE. The complete sensitivity system (CSS) Jacobian is block lower-triangular with N(q) diagonal blocks, each equal to the original system Jacobian J, giving determinant (det J) N(q) . This result—the structural inheritance theorem—implies that whenever Pryce’s structural analysis succeeds on the original DAE, it succeeds on the complete sensitivity system for any finite index and any requested finite sensitivity order, without further structural derivation. The proof proceeds in stages: existence of sensitivities via the implicit function theorem, dependence ordering via operator decomposition, preservation of the Σ-matrix and offsets for first-order and higher-order equations, preservation of the system Jacobian, extension to the block lower-triangular CSS Jacobian, and consistent-point inheritance. Numerical confirmation on test systems spanning indices 0 through 5 verifies the predicted block structure, offset repetition, and Jacobian determinant identity, with errors at the level of floating-point roundoff (≤ 10−14).