Local well-posedness of the Mortensen observer

Abstract: The analytical background of nonlinear observers based on minimal energy estimation is discussed. It is shown that locally the derivation of the observer equation based on a trajectory with pointwise minimal energy can be done rigorously. The result is obtained by a local sensitivity analysis of the value function based on Pontryagin's maximum principle and the Hamilton-Jacobi-Bellman equation. The consideration of a differential Riccati equation reveals that locally the second derivative of the value function is a positive definite matrix. The local convexity ensures existence of a trajectory minimizing the energy, which is then shown to satisfy the observer equation.