Functional-analytic validation of geometric optimal control assumptions

Establish a functional-analytic treatment for concrete classes of geometric PDE-constrained optimal control problems and verify the abstract differentiability assumptions used in the Lagrange-Newton framework for appropriate problem classes.

Background

The paper develops an abstract Lagrange-Newton framework for optimal control problems whose state equations are geometric variational equations on manifolds. Its derivation assumes regularity properties such as differentiability of manifold-valued mappings, vector-bundle sections, and the Lagrangian derivatives, but the paper does not verify these assumptions for specific infinite-dimensional problem classes. The authors identify the rigorous functional-analytic analysis and validation of these assumptions as an unresolved theoretical issue.

References

Certainly, a number of important theoretical and algorithmic issues are still open and subject to future research. First of all, a functional analytic treatment of concrete classes of problems needs to be done. In particular, our abstract assumptions, e.g. on differentiability, need to be verified for appropriate classes of problems.

— A Lagrange-Newton method for solving optimal control problems on manifolds  (2610.06310 - Weigl et al., 5 Oct 2026) in Section Conclusion