A Unified Framework for Error Analysis of Optimal Control Problems Governed by Saddle‐Point Systems

This work develops an abstract framework for conducting both a priori and a posteriori error analysis for a class of optimal control problems (OCPs) governed by saddle point systems. The approach employs variational discretization for the control variable and relies on minimal assumptions imposed on the underlying saddle point system at both the continuous and discrete levels. Within this setting, an abstract error equivalence result is first established through the introduction of suitably constructed auxiliary problems. Since the assumptions considered are mild and can be verified for a broad class of saddle point systems, the proposed framework is widely applicable. As an application, the framework is employed to derive an error equivalence result for OCPs governed by the weakly symmetric linear elasticity problem. This, in turn, leads to the development of a priori and a posteriori error estimates that are robust with respect to the Lamé parameter , thereby ensuring applicability in the nearly incompressible regime. Additionally, the mixed formulation of the Poisson problem is also investigated as part of the applications.