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The complex Ginzburg-Landau equation on a finite interval and chaos suppression via a finite-dimensional boundary feedback stabilizer

Dionyssios Mantzavinos, Türker Özsarı, Kemal Cem Yılmaz

TL;DR

The paper tackles the complex Ginzburg-Landau equation on a finite interval with inhomogeneous Dirichlet-Neumann boundaries, establishing local and global well-posedness via the unified transform and energy methods. It then designs a rapid boundary feedback controller that uses only a finite number of Fourier modes, implemented through a kernel-based backstepping transform and a Volterra-type integral map, to achieve exponential stabilization of the zero equilibrium. The authors provide precise conditions on the number of modes and the damping parameter to guarantee stability for both linear and nonlinear dynamics, and they validate the theory with numerical simulations that demonstrate chaotic suppression and robust stabilization. The work advances finite-dimensional boundary control for dissipative PDEs with nonlocal actuation, offering a rigorous pathway to chaos suppression in complex-Ginzburg-Landau-type systems and relevant applications in wave and reaction-diffusion contexts.

Abstract

In this paper, we study the well-posedness and boundary stabilization of the initial-boundary value problem for the complex Ginzburg-Landau (CGL) equation on a finite interval. First, we establish a local well-posedness theory for the open loop model in $L^2$-based fractional Sobolev spaces in the case of Dirichlet-Neumann type inhomogeneous mixed boundary conditions. This local well-posedness result is based on linear estimates derived by using the weak solution formula obtained via the unified transform (also known as the Fokas method). Next, we study the global well-posedness properties of the open loop model in presence of inhomogeneous boundary conditions. Then, we turn our attention to the rapid boundary feedback stabilization problem and design a nonlocal controller which uses a finite number of Fourier modes of the state of solution. This design relies on the fact that solutions of the CGL equation can be separated into a slow, finite-dimensional component and a rapidly decaying tail, with the former primarily governing long-term behavior. We determine the necessary number of modes required to stabilize the system at a specified rate. Additionally, we identify the minimum number of modes that ensure stabilization at an unspecified decay rate. These theoretical results are validated by numerical simulations. The spatiotemporal estimates established in the first part of the paper are also employed to obtain local solutions of the controlled system. The existence of global energy solutions follows from stabilization estimates, while uniqueness follows from the uniqueness of an associated initial-boundary value problem with homogeneous boundary conditions whose solutions are in correspondence with the solutions of the original system through a bounded invertible Volterra-type integral transform on Sobolev spaces.

The complex Ginzburg-Landau equation on a finite interval and chaos suppression via a finite-dimensional boundary feedback stabilizer

TL;DR

The paper tackles the complex Ginzburg-Landau equation on a finite interval with inhomogeneous Dirichlet-Neumann boundaries, establishing local and global well-posedness via the unified transform and energy methods. It then designs a rapid boundary feedback controller that uses only a finite number of Fourier modes, implemented through a kernel-based backstepping transform and a Volterra-type integral map, to achieve exponential stabilization of the zero equilibrium. The authors provide precise conditions on the number of modes and the damping parameter to guarantee stability for both linear and nonlinear dynamics, and they validate the theory with numerical simulations that demonstrate chaotic suppression and robust stabilization. The work advances finite-dimensional boundary control for dissipative PDEs with nonlocal actuation, offering a rigorous pathway to chaos suppression in complex-Ginzburg-Landau-type systems and relevant applications in wave and reaction-diffusion contexts.

Abstract

In this paper, we study the well-posedness and boundary stabilization of the initial-boundary value problem for the complex Ginzburg-Landau (CGL) equation on a finite interval. First, we establish a local well-posedness theory for the open loop model in -based fractional Sobolev spaces in the case of Dirichlet-Neumann type inhomogeneous mixed boundary conditions. This local well-posedness result is based on linear estimates derived by using the weak solution formula obtained via the unified transform (also known as the Fokas method). Next, we study the global well-posedness properties of the open loop model in presence of inhomogeneous boundary conditions. Then, we turn our attention to the rapid boundary feedback stabilization problem and design a nonlocal controller which uses a finite number of Fourier modes of the state of solution. This design relies on the fact that solutions of the CGL equation can be separated into a slow, finite-dimensional component and a rapidly decaying tail, with the former primarily governing long-term behavior. We determine the necessary number of modes required to stabilize the system at a specified rate. Additionally, we identify the minimum number of modes that ensure stabilization at an unspecified decay rate. These theoretical results are validated by numerical simulations. The spatiotemporal estimates established in the first part of the paper are also employed to obtain local solutions of the controlled system. The existence of global energy solutions follows from stabilization estimates, while uniqueness follows from the uniqueness of an associated initial-boundary value problem with homogeneous boundary conditions whose solutions are in correspondence with the solutions of the original system through a bounded invertible Volterra-type integral transform on Sobolev spaces.
Paper Structure (31 sections, 15 theorems, 390 equations, 5 figures, 3 algorithms)

This paper contains 31 sections, 15 theorems, 390 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Well-posedness of the open-loop problem
  3. Estimates for the reduced initial-boundary value problem
  4. Linear decomposition
  5. Estimates for the homogeneous linear Cauchy problem
  6. Estimates the forced linear Cauchy problem
  7. Construction of compactly supported boundary data
  8. Estimation of the forced linear CGL equation on a finite interval
  9. Local well-posedness of CGL on a finite interval
  10. Global solutions of the open-loop problem
  11. Chaos suppression
  12. Kernel and the linear target model
  13. Bounded invertibility of $T_N$
  14. Explicit representation of the boundary controller
  15. Local solutions of the non-local boundary value problem in fractional Sobolev spaces
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $s \in \left(\frac{1}{2}, \frac{3}{2}\right) \cup \left(\frac{3}{2}, \frac{5}{2}\right)$ and $p>0$ such that either $p \in 2\mathbb N$ or, otherwise, Furthermore, let $T>0$ satisfy the conditions T-into, T-into-2, T-contr and T-contr-2. Then, for initial data $u_0 \in H^s(0, L)$ and boundary data $a \in H^{\frac{2s+1}{4}}(0, T)$, $b \in H^{\frac{2s-1}{4}}(0, T)$ satisfying the compatibility c

Figures (5)

  • Figure 2.1: The straight lines \ref{['hyp-eq']} that define the region $D^+ \cup D^-$.
  • Figure 2.2: The regions of integration corresponding to the left side (shaded) and the right side (enclosed in bold) of inequality \ref{['regions-ineq']}.
  • Figure 4.1: Simulations for the stabilized nonlinear model \ref{['pde_lin_num']}.
  • Figure 4.2: Simulations for the nonlinear model \ref{['pde_nonlin_num']} with $g = 0$.
  • Figure 4.3: Simulations for the stabilized nonlinear model \ref{['pde_nonlin_num']}.

Theorems & Definitions (35)

  • Theorem 1.1: Hadamard well-posedness
  • Theorem 1.2: Global well-posedness
  • Theorem 1.3: Linear stabilization
  • Theorem 1.4: Nonlinear stabilization
  • Theorem 2.1: Space estimate
  • Remark 2.1
  • Remark 2.2: CGL versus NLS
  • proof
  • Theorem 2.2: Time estimates
  • proof
  • ...and 25 more