Practical Observers for Velocity Field Estimation of Normal Flow Equations

We propose a new approach for the estimation of the velocity field of normal flow partial differential equations by means of practical observers. In more detail, we suppose to know the evolution over time in each point of the domain of the solution of a normal flow equation subject to an unknown velocity field, which is assumed to be smooth but time-varying in general. Numerical results are presented to showcase the effectiveness of the proposed approach in one-and two-dimensional examples. In particular, the performance of the practical observer has been tested also to estimate the rate of spread of a fire propagation model based on the normal flow equation. Predicting wildfire evolution presents a multitude of challenges due to the variable nature of wildfires (influenced by a range of factors, from weather conditions and vegetation type to landscape topography), thus motivating the need to have at disposal an accurate estimate of the velocity vector to predict the evolution of the fire front.