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

Jesper Schröder

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.

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.

Paper Structure

This paper contains 21 sections, 31 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Motivation and available literature
  3. The Mortensen observer for a nonlinear wave equation
  4. Contributions of this work
  5. Notation
  6. Analytical framework
  7. The forward equation
  8. The backward state equation
  9. Optimal control problem
  10. The data-to-state operator
  11. Existence of a minimizer
  12. Optimality conditions
  13. Regularity of the value function
  14. Properties of the second derivative
  15. Local coercivity
  16. ...and 6 more sections

Key Result

Lemma 2.1

The product of three functions $w_1,w_2,w_3 \in H^1_0(\Omega)$ lies in $L^2(\Omega)$ and it holds

Theorems & Definitions (68)

  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • Proposition 2.7
  • ...and 58 more