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

On the use of state constraints in optimal control of singular PDEs

We consider optimal control of nonlinear partial differential equations involving potentially singular solution-dependent terms. Singularity can be prevented by either restricting controls to a closed admissible set for which well-posedness of the equation can be guaranteed, or by explicitly enforcing pointwise bounds on the state. By means of an elliptic model problem, we contrast the requirements for deriving the existence of solutions and first order optimality conditions for both the control-constrained and the state-constrained formulation. Our analysis as well as numerical tests illustrate that control constraints lead to severe restrictions on the attainable states, which is not the case for state constraints