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Exponential Decay for a Boundary-Controlled Nonlinear Parabolic Reactor Model

Yevgeniia Yevgenieva, Alexander Zuyev, Christophe Prieur, Peter Benner

Abstract

We study an axial dispersion tubular reactor model governed by a nonlinear parabolic equation with Robin-type boundary conditions and boundary feedback control. We derive sufficient conditions for the exponential stability of the steady-state solution of the closed-loop system and provide an explicit estimate of the decay rate. In addition, numerical simulations are presented to illustrate the sharpness of the obtained decay rate for different choices of the feedback gain parameter.

Exponential Decay for a Boundary-Controlled Nonlinear Parabolic Reactor Model

Abstract

We study an axial dispersion tubular reactor model governed by a nonlinear parabolic equation with Robin-type boundary conditions and boundary feedback control. We derive sufficient conditions for the exponential stability of the steady-state solution of the closed-loop system and provide an explicit estimate of the decay rate. In addition, numerical simulations are presented to illustrate the sharpness of the obtained decay rate for different choices of the feedback gain parameter.
Paper Structure (10 sections, 8 theorems, 48 equations, 2 figures, 1 table)

This paper contains 10 sections, 8 theorems, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Exponential stability of nonlinear Sturm--Liouville systems
  3. Axial Dispersion Tubular Reactor Model
  4. Proofs of the main results
  5. Proof of Theorem \ref{['th1']}
  6. Proof of Theorem \ref{['th2']}
  7. System in deviations
  8. Auxiliary results
  9. Proof of Proposition \ref{['prop1']}
  10. Simulation results

Key Result

Theorem 2.1

Let system eq2.1 satisfy the following conditions: Then the equilibrium $\xi=0$ of system eq2.1 is exponentially stable in $X_0$. Moreover, there is a constant $C\ge 1$ such that, for any initial state $\xi_0\in X_0$, the corresponding mild solution of eq2.1, IC satisfies In particular, if $X_0=X$, then the equilibrium $\xi=0$ is globally exponentially stable.

Figures (2)

  • Figure 1: Norms of the solutions for different values of $\alpha$.
  • Figure 2: Decay rate depending on $\alpha$.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Remark 2.1
  • Theorem 3.1
  • Proposition 1
  • Lemma 4.1
  • proof
  • Corollary 1
  • Lemma 4.2
  • ...and 5 more