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Small-time global controllability of a class of bilinear fourth-order parabolic equations

Subrata Majumdar, Debanjit Mondal

TL;DR

The paper establishes new results on the small-time global controllability of a class of bilinear, fourth-order parabolic equations on the one-dimensional torus, focusing on two prototypical nonlinearities: Kuramoto–Sivashinsky and Cahn–Hilliard. It proves small-time global approximate controllability between states of the same sign using only three scalar, time-dependent control profiles, via a saturating geometric control strategy built on conjugated dynamics limits and dense saturating subspaces. It then shows small-time global exact controllability to nonzero constant states by combining a five-profile moment-based linearization with a fixed-point argument and a local exact controllability result, with KS reducible to four controls under structural adjustments. The approach highlights the robustness of geometric-saturation methods for higher-order nonlinear PDEs and suggests potential extensions to dispersive models; the results provide a pathway to exact steering to trajectories despite known obstructions for linear bilinear controls. Overall, the work advances a global controllability theory for bilinear control of fourth-order parabolic PDEs with practical implications for pattern formation and interfacial dynamics on periodic domains.

Abstract

In this work, we investigate the small-time global controllability properties of a class of fourth-order nonlinear parabolic equations driven by a bilinear control posed on the one-dimensional torus. The controls depend only on time and act through a prescribed family of spatial profiles. Our first result establishes the small-time global approximate controllability of the system using three scalar controls, between states that share the same sign. This property is obtained by adapting the geometric control approach to the fourth-order setting, using a finite family of frequency-localized controls. We then study the small-time global exact controllability to non-zero constant states for the concerned system. This second result is achieved by analyzing the null controllability of an appropriate linearized fourth-order system and by deducing the controllability of the nonlinear bilinear model through a fixed-point argument together with the small-time global approximate control property.

Small-time global controllability of a class of bilinear fourth-order parabolic equations

TL;DR

The paper establishes new results on the small-time global controllability of a class of bilinear, fourth-order parabolic equations on the one-dimensional torus, focusing on two prototypical nonlinearities: Kuramoto–Sivashinsky and Cahn–Hilliard. It proves small-time global approximate controllability between states of the same sign using only three scalar, time-dependent control profiles, via a saturating geometric control strategy built on conjugated dynamics limits and dense saturating subspaces. It then shows small-time global exact controllability to nonzero constant states by combining a five-profile moment-based linearization with a fixed-point argument and a local exact controllability result, with KS reducible to four controls under structural adjustments. The approach highlights the robustness of geometric-saturation methods for higher-order nonlinear PDEs and suggests potential extensions to dispersive models; the results provide a pathway to exact steering to trajectories despite known obstructions for linear bilinear controls. Overall, the work advances a global controllability theory for bilinear control of fourth-order parabolic PDEs with practical implications for pattern formation and interfacial dynamics on periodic domains.

Abstract

In this work, we investigate the small-time global controllability properties of a class of fourth-order nonlinear parabolic equations driven by a bilinear control posed on the one-dimensional torus. The controls depend only on time and act through a prescribed family of spatial profiles. Our first result establishes the small-time global approximate controllability of the system using three scalar controls, between states that share the same sign. This property is obtained by adapting the geometric control approach to the fourth-order setting, using a finite family of frequency-localized controls. We then study the small-time global exact controllability to non-zero constant states for the concerned system. This second result is achieved by analyzing the null controllability of an appropriate linearized fourth-order system and by deducing the controllability of the nonlinear bilinear model through a fixed-point argument together with the small-time global approximate control property.
Paper Structure (38 sections, 14 theorems, 190 equations)

This paper contains 38 sections, 14 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. System under study
  3. Small-time global approximate controllability
  4. Small-time global exact controllability to the constant states
  5. Cahn-Hilliard equation:
  6. Kuramoto-Sivashinsky equation:
  7. Structure of the paper
  8. Preliminaries
  9. Function spaces and notations
  10. Local well-posedness and semi-global stability
  11. Small-time limit of conjugated dynamics
  12. Saturating subspaces
  13. Claim:
  14. Small-time global approximate null controllability
  15. Concatenation property
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $s>\tfrac{1}{2}$ and $u_0, u_1 \in H^{s}(\mathbb T)$. Then the controlled system ctrl_prblm--fourier_mods enjoys the following small-time approximate controllability properties:

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.1: Small-time global approximate controllability
  • Theorem 1.2: Global exact controllability
  • Theorem 1.3: Global exact controllability
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • ...and 16 more