Constructing Adjoint DAEs via Structural Index Reduction and Offset-Based Lifting

We address the problem of constructing the adjoint system for a general differential algebraic equation (DAE) when only the forward DAE and a cost functional are given. The classical route—form the continuous Lagrangian and take variations—yields the adjoint DAE directly but the resulting system is often of high index and numerically unstable. We propose a three-stage alternative: first reduce the DAE to an underlying ODE using Pryce’s structural analysis, compute the adjoint ODE by standard optimal control theory, then lift the adjoint ODE back to an adjoint DAE by reintroducing algebraic constraints and Lagrange multipliers. The key insight is that the lifting step is governed by dual offsets cˆj = ν − dj , d ˆ i = ν − ci derived from the forward Pryce offsets. We prove that the resulting adjoint DAE retains the offset level ν of the original, with its algebraic part in index-1 form. At regular solution points, the transformation J Tµ recovers a local augmented adjoint form that generalises the augmented system of Cao, Li, and Petzold (2003) to the Pryce-structured setting (Theorem 5.1). We construct explicit adjoint DAEs for index-1, index-2, and index-3 benchmark problems and validate them numerically: the index-1 adjoint gradient is computed by the combined DAE pipeline and validated against central finite differences (relative error < 10−6 ), the index-2 GGL pendulum adjoint gradient is validated via a two-pass Python BDF solver and central finite differences (relative error 3.1 × 10−6 ), and the index-3 combined DAE pipeline compiles and runs against the DAETS framework. An open-source C++ library automates the full pipeline: structural analysis, type-level time reversal, combined forward-adjoint DAE execution, and gradient extraction once the problem-specific combined DAE has been supplied.