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Stabilization to trajectories of nonisothermal Cahn-Hilliard equations

Behzad Azmi, Marvin Fritz, Sérgio S. Rodrigues

Abstract

In this work, it is proven the semiglobal exponential stabilization to time-dependent trajectories of the nonisothermal Cahn-Hilliard equations. In the model, the input controls are given by explicit feedback operators that involve appropriate oblique projections. The actuators are given by a finite number of indicator functions. The results also hold for the isothermal Cahn-Hilliard system. Numerical simulations are shown that illustrate the stabilizing performance of the proposed input feedback operators.

Stabilization to trajectories of nonisothermal Cahn-Hilliard equations

Abstract

In this work, it is proven the semiglobal exponential stabilization to time-dependent trajectories of the nonisothermal Cahn-Hilliard equations. In the model, the input controls are given by explicit feedback operators that involve appropriate oblique projections. The actuators are given by a finite number of indicator functions. The results also hold for the isothermal Cahn-Hilliard system. Numerical simulations are shown that illustrate the stabilizing performance of the proposed input feedback operators.

Paper Structure

This paper contains 28 sections, 5 theorems, 139 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Stabilization to trajectories
  3. Previous literature
  4. Contents
  5. Notation
  6. Functional setting and families of actuators
  7. Function spaces
  8. The actuators
  9. Feedback
  10. Stability of the difference to the target
  11. Main result
  12. Boundedness of the control input
  13. Existence and uniqueness of solutions
  14. General bounds for Galerkin approximations
  15. Further bounds for Galerkin approximations
  16. ...and 13 more sections

Key Result

Lemma 2.1

We have that the sequences $(\alpha_{M}^H)_{M\in{\mathbb N}_+}$ and $(\alpha_{M}^V)_{M\in{\mathbb N}_+}$ of constants are divergent; $\lim_{M\to+\infty}\alpha_{M}^H=+\infty$ and $\lim_{M\to+\infty}\alpha_{M}^{V}=+\infty$.

Figures (10)

  • Figure 1: Supports of actuators, for rectangular $\Omega\subset{\mathbb R}^2$. $(M_\sigma,M_\varsigma)=M^{2}(3,1)$.
  • Figure 2: Supports of actuators, for triangular $\Omega\subset{\mathbb R}^2$. $(M_\sigma,M_\varsigma)=4^{M-1}(3,1)$.
  • Figure 3: Depiction of actuators for $M=1$ (left), $M=2$ (middle) and $M=3$ (right) on the spatial grid; the red actuators influence the $z_{1{\tt c}}$ equation whereas the green actuators influence the $z_{2{\tt c} }$ equation.
  • Figure 4: Evolution of the reference $z_{\tt r}=(z_{1{\tt r}},z_{2{\tt r}})$ with initial state \ref{['ini-r']}.
  • Figure 5: Case $\lambda=0$. Free dynamics evolution of $z_{{\tt c}}$, with initial state \ref{['ini-c']}.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Remark 5.2
  • ...and 1 more