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Semiglobal oblique projection exponential dynamical observers for nonautonomous semilinear parabolic-like equations

Sérgio S. Rodrigues

TL;DR

This work develops a semiglobal exponential observer for nonautonomous semilinear parabolic-like PDEs using a Luenberger-type dynamical observer with an explicit oblique-projection-based output injection. By leveraging a finite-dimensional output from averaged measurements and carefully chosen sensor/auxiliary-function subspaces, the authors prove that, for any prescribed decay rate and initial error bound, there exist sufficiently many sensors and a large enough gain to guarantee $|z(t)|_V \ frac{\le}{ } \\varrho e^{-\\mu(t-s)} |z(s)|_V$, with $z=\,\widehat{y}-y$. The analysis hinges on a decomposition of the error via oblique projections, nonlinear term bounds, and a Poincaré-type condition that strengthens observability as sensor count grows, with concrete verification in rectangular domains. Numerical simulations on the unit square corroborate the exponential stability and illustrate the need for ample sensors and gain to achieve stabilization, while offering practical insights into parameter choices and discretization effects. Overall, the paper advances continuous data assimilation for challenging nonautonomous PDE systems by delivering a tangible, rigorously justified observer framework with explicit injection design and semiglobal guarantees.

Abstract

The estimation of the full state of a nonautonomous semilinear parabolic equation is achieved by a Luenberger type dynamical observer. The estimation is derived from an output given by a finite number of average measurements of the state on small regions. The state estimate given by the observer converges exponentially to the real state, as time increases. The result is semiglobal in the sense that the error dynamics can be made stable for an arbitrary given initial condition, provided a large enough number of measurements, depending on the norm of the initial condition, is taken. The output injection operator is explicit and involves a suitable oblique projection. The results of numerical simulations are presented showing the exponential stability of the error dynamics.

Semiglobal oblique projection exponential dynamical observers for nonautonomous semilinear parabolic-like equations

TL;DR

This work develops a semiglobal exponential observer for nonautonomous semilinear parabolic-like PDEs using a Luenberger-type dynamical observer with an explicit oblique-projection-based output injection. By leveraging a finite-dimensional output from averaged measurements and carefully chosen sensor/auxiliary-function subspaces, the authors prove that, for any prescribed decay rate and initial error bound, there exist sufficiently many sensors and a large enough gain to guarantee , with . The analysis hinges on a decomposition of the error via oblique projections, nonlinear term bounds, and a Poincaré-type condition that strengthens observability as sensor count grows, with concrete verification in rectangular domains. Numerical simulations on the unit square corroborate the exponential stability and illustrate the need for ample sensors and gain to achieve stabilization, while offering practical insights into parameter choices and discretization effects. Overall, the paper advances continuous data assimilation for challenging nonautonomous PDE systems by delivering a tangible, rigorously justified observer framework with explicit injection design and semiglobal guarantees.

Abstract

The estimation of the full state of a nonautonomous semilinear parabolic equation is achieved by a Luenberger type dynamical observer. The estimation is derived from an output given by a finite number of average measurements of the state on small regions. The state estimate given by the observer converges exponentially to the real state, as time increases. The result is semiglobal in the sense that the error dynamics can be made stable for an arbitrary given initial condition, provided a large enough number of measurements, depending on the norm of the initial condition, is taken. The output injection operator is explicit and involves a suitable oblique projection. The results of numerical simulations are presented showing the exponential stability of the error dynamics.

Paper Structure

This paper contains 33 sections, 14 theorems, 194 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The main result
  3. Motivation
  4. On previous related works in literature
  5. Illustrating example. Scalar parabolic equations
  6. Contents and notation
  7. Assumptions
  8. Exponential stability of the error dynamics
  9. Auxiliary results
  10. Proof of the main Theorem \ref{['T:main']}
  11. Boundedness of the output injection operator
  12. On the existence and uniqueness of solutions for the error
  13. On the existence and uniqueness of solutions for systems \ref{['sys-y-o']} and \ref{['sys-haty-o']}
  14. Parabolic equations evolving in rectangular domains
  15. Previous related work.
  16. ...and 18 more sections

Key Result

Theorem 3.1

Let Assumptions A:A0sp--A:DS hold true and let us be given $\ell\in[0,2]$, $R>0$, $\varrho>1$, and $\mu>0$. Then there exists a pair $(S^*,\lambda^*)\in{\mathbb N}_0\times{\mathbb R}_0$ such that: for all pairs $(S,\lambda)$ satisfying $S\ge S^*$ and $\lambda\ge\lambda^*(S)$, the error dynamical sys

Figures (6)

  • Figure 1: The sensor supports as in \ref{['mxe-hd_suppsensors']}. Case $\Omega^\times\subset{\mathbb R}^d$, $d=2$.
  • Figure 2: The sensor supports as in \ref{['mxe-hd_suppsensors']}. Case $\Omega^\times\subset{\mathbb R}^d$, $d=2$.
  • Figure 3: Locations of sensors in cases $S_\sigma\in\{4,9,16\}$.
  • Figure 4: The free dynamics and the case of $4$ sensors.
  • Figure 5: The case of $9$ and $16$ sensors.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 2.9
  • Theorem 3.1
  • Remark 3.2
  • ...and 18 more