Local well-posedness of the minimum energy estimator for a defocusing cubic wave equation

Abstract

This work is concerned with the minimum energy estimator for a nonlinear hyperbolic partial differential equation. The Mortensen observer – originally introduced for the energy-optimal reconstruction of the state of nonlinear finite-dimensional systems – is formulated for a disturbed cubic wave equation and the associated observer equation is derived. An in depth study of the associated optimal control problem and sensitivity analysis of the corresponding value function reveals that the energy optimal state estimator is well-defined. Deploying a classical fixed point argument we proceed to show that the observer equation is locally well-posed.