Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability and decay rate estimates for a nonlinear dispersed flow reactor model with boundary control

Yevgeniia Yevgenieva, Alexander Zuyev, Peter Benner

TL;DR

This work analyzes a nonlinear parabolic PDE model of a dispersed-flow tubular reactor with boundary control applied at the inlet. By casting the problem into an abstract evolution framework and applying $C_0$-semigroup theory along with a Lipschitz analysis of the nonlinearity, the authors prove existence and uniqueness of mild solutions and design a boundary feedback to achieve exponential stability of the steady state. A Lyapunov-based energy in a weighted $L^2$ space yields a global decay-rate lower bound $\lambda_T=\frac{v^2}{16D_{ax}}$, illustrating robust stabilization that is, in principle, independent of the reaction order $n$. Numerical simulations across different $n$ and feedback gains $\alpha$ validate the exponential decay, reveal how decay rates depend on parameters, and highlight the conservativeness of the analytical bound. The results provide a rigorous stability framework for boundary-controlled nonlinear DFTR models with practical implications for reactor operation and control.

Abstract

We investigate a nonlinear parabolic partial differential equation whose boundary conditions contain a single control input. This model describes a chemical reaction of the type ``$A \to $ product'', occurring in a dispersed flow tubular reactor. The existence and uniqueness of solutions to the nonlinear Cauchy problem under consideration are established by applying the theory of strongly continuous semigroups of operators. We also prove the stability of the equilibrium of the closed-loop system with a proposed feedback law. Additionally, using Lyapunov's direct method, we evaluate the exponential decay rate of the solutions.

Stability and decay rate estimates for a nonlinear dispersed flow reactor model with boundary control

TL;DR

This work analyzes a nonlinear parabolic PDE model of a dispersed-flow tubular reactor with boundary control applied at the inlet. By casting the problem into an abstract evolution framework and applying -semigroup theory along with a Lipschitz analysis of the nonlinearity, the authors prove existence and uniqueness of mild solutions and design a boundary feedback to achieve exponential stability of the steady state. A Lyapunov-based energy in a weighted space yields a global decay-rate lower bound , illustrating robust stabilization that is, in principle, independent of the reaction order . Numerical simulations across different and feedback gains validate the exponential decay, reveal how decay rates depend on parameters, and highlight the conservativeness of the analytical bound. The results provide a rigorous stability framework for boundary-controlled nonlinear DFTR models with practical implications for reactor operation and control.

Abstract

We investigate a nonlinear parabolic partial differential equation whose boundary conditions contain a single control input. This model describes a chemical reaction of the type `` product'', occurring in a dispersed flow tubular reactor. The existence and uniqueness of solutions to the nonlinear Cauchy problem under consideration are established by applying the theory of strongly continuous semigroups of operators. We also prove the stability of the equilibrium of the closed-loop system with a proposed feedback law. Additionally, using Lyapunov's direct method, we evaluate the exponential decay rate of the solutions.

Paper Structure

This paper contains 8 sections, 5 theorems, 44 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic notations and auxiliary results
  3. Description of the model
  4. Main results
  5. Existence and uniqueness of solution to the problem \ref{['eq1.5']}
  6. Exponential stability of the steady-state solution
  7. Numerical simulations
  8. Conclusion

Key Result

Theorem 2.1

Let $\mathcal{A}: D(A)\to X$ be a linear operator defined on a subset $D(\mathcal{A})$ of a Banach space $X$. Then $\mathcal{A}$ generates a strongly continuous semigroup of operators on $X$ if and only if:

Figures (3)

  • Figure 1: 3D plot of $w(x,t)$ for $n=1$, $\alpha=0$.
  • Figure 2: Case $n=\frac{1}{2}$ and $\alpha=\frac{1}{2}$.
  • Figure 3: Case $n=2$ and $\alpha=\frac{1}{2}$.

Theorems & Definitions (12)

  • Theorem 2.1: Lumer--Phillips theorem Pazy
  • Theorem 2.2: LiYong
  • Remark 3.1
  • Lemma 4.1
  • proof
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • ...and 2 more