Further Results on the Optimal Control of Fronts Generated by Level Set Methods

The control of level sets generated by partial differential equations is still a challenge because of its complexity both from the theoretical and computational points of view. Specifically, we focus on the space-dependent optimal control problem of a moving front and search for an approximate solution method that is computationally feasible. We formulate the problem in an Eulerian setting and develop an efficient approximation scheme based on the extended Ritz method. Such a method consists in adopting a control law with fixed structure that depends nonlinearly from a number of parameters to be suitably chosen by using a gradient-based technique. Toward this end, we derive the adjoint equations for optimal control problems involving the normal and mean curvature flow partial differential equations. The adjoint equations allow to compute the gradient of the cost with respect to the vector of parameters of the control law. Numerical results are reported to show the effectiveness of the proposed approach in some 2D and 3D examples.