Optimal control for the thin-film equation: Convergence of a multi-parameter approach to track state constraints avoiding degeneracies

We consider an optimal control problem subject to the thin-film equation which is deduced from the Navier--Stokes equation. The PDE constraint lacks well-posedness for general right-hand sides due to possible degeneracies; state constraints are used to circumvent this problematic issue and to ensure well-posedness, and the rigorous derivation of necessary optimality conditions for the optimal control problem is performed. A multi-parameter regularization is considered which addresses both, the possibly degenerate term in the equation and the state constraint, and convergence is shown for vanishing regularization parameters by decoupling both effects. The fully regularized optimal control problem allows for practical simulations which are provided, including the control of a dewetting scenario, to evidence the need of the state constraint, and to motivate proper scalings of involved regularization and numerical parameters