The minimum energy estimator for a cubic wave equation

The minimum energy estimator - also called Mortensen observer - was originally designed for the reconstruction of the state of nonlinear nite-dimensional dynamical systems subject to deterministic disturbances based on partial and awed measurements. In this presentation we propose a generalization to systems governed by hyperbolic PDEs. Using the example of a nonlinear defocusing wave equation we formulate the underlying optimal control problem and formally derive the associated observer.

After discussing theoretical results on well-posedness we introduce a spatial discretization of the wave equation inspired by spectral methods. This allows the numerical realization of the observer based on a polynomial approximation of the value function. We conclude with a comparison of the obtained state reconstruction to the well-known extended Kalman filter.