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

TL;DR

This paper extends the Mortensen observer framework to a nonlinear cubic wave equation, demonstrating that the energy-optimal state estimator is well-defined and locally well-posed through fixed point analysis.

Contribution

It formulates a minimum energy estimator for a nonlinear hyperbolic PDE and proves its local well-posedness, a novel extension of the Mortensen observer to infinite-dimensional systems.

Findings

The energy optimal state estimator is well-defined.

The observer equation is locally well-posed.

Sensitivity analysis supports the estimator's robustness.

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.