Feedback control on the velocity field and source term of a normal flow equation

The normal flow equation is a nonlinear partial differential equation that is quite popular in numerous research fields related to the so-called level set methods. Specifically, we have investigated the feedback control of such an equation by proposing two different regulators. The first approach consists in considering the velocity field of the equation as a control action; in such a case a simple proportional regulator is proved to be stable. In the second case, the control acts on the source term, and it relies on a Luenberger observer that provides an estimate of the norm of the gradient involved in the normal flow equation. Also this controller is proved to be stable by using Lyapunov arguments. Simulation results are presented to show the effectiveness of the proposed approaches.