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Global controllability to harmonic maps of the heat flow from a circle to a sphere

Jean-Michel Coron, Shengquan Xiang

Abstract

In this paper, we study the global controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important properties, such as small-time local controllability, local quantitative rapid stabilization, obstruction to semi-global asymptotic stabilization, and global controllability to geodesics. Surprisingly, due to the geometric feature of the equation we also discover the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding problem for the nonlinear heat equations.

Global controllability to harmonic maps of the heat flow from a circle to a sphere

Abstract

In this paper, we study the global controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important properties, such as small-time local controllability, local quantitative rapid stabilization, obstruction to semi-global asymptotic stabilization, and global controllability to geodesics. Surprisingly, due to the geometric feature of the equation we also discover the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding problem for the nonlinear heat equations.
Paper Structure (25 sections, 24 theorems, 229 equations, 10 figures)

This paper contains 25 sections, 24 theorems, 229 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The main results
  3. Stability of the harmonic map heat flow
  4. Stabilization of the harmonic map heat flow
  5. Small-time global exact controllability between harmonic maps and an open problem
  6. Global controllability of the harmonic map heat flow
  7. The strategy for the proof of the global controllability result (Theorem \ref{['thm:globalcontrol']})
  8. The organization of the paper
  9. Some further questions
  10. Preliminary part
  11. Some basic properties of the controlled harmonic map heat flow
  12. Rotation invariance
  13. Well-posedness and continuous dependence results
  14. Controllability within each homotopy class
  15. Stabilization and closed-loop systems
  16. ...and 10 more sections

Key Result

PROPOSITION 1.3

For any initial state $u_0\in H^1(\mathbb{T}^1; \mathbb{S}^k)$, for any $\varepsilon>0$, there exists some time $T= T(\varepsilon, u_0)$ such that the unique solution of eq:freehmhfs becomes a "$\varepsilon$-approximate harmonic map" at time $T$.

Figures (10)

  • Figure 1: The controlled harmonic map heat flow from $\mathbb{T}^1$ to a torus of genus 2. The blue curve is the solution at a given time $t$, and the red part is the place where we are allowed to add extra control force.
  • Figure 2: Deformation of the curve in the same homotopy class. In this picture, the solution remains in the torus of genus 2. We observe that the state $u_1$ can be continuously deformed to the state $u_2$, but it cannot reach the state $v$. The homotopy problem states that every state is homotopic to a harmonic map. See Proposition \ref{['thm:homopotyconver']} and the paper Ottarsson for details.
  • Figure 3: The natural energy dissipation of the harmonic map heat flow from $\mathbb{T}^1$ to $\mathbb{S}^2$. The green arrow indicates the deformation of the curve (solution). See Proposition \ref{['thm:homopotyconver']} for details on this convergence result. The picture on the left shows that the solution converges to a harmonic map having $2\pi$-energy. This picture is also related to \ref{['Step1']} of Section \ref{['sec:strategy']}. The picture on the right shows that the solution converges to a constant state. This picture is also related to \ref{['Step4']} of Section \ref{['sec:strategy']}.
  • Figure 4: The local quantitative rapid stabilization and local null controllability of the equation around some given point $p\in \mathbb{S}^2$. See Theorem \ref{['thm-rapid-stab']} and Theorem \ref{['thm:nullcontrol']} for details. This picture is also related to \ref{['Step5']} of Section \ref{['sec:strategy']}.
  • Figure 5: Obstruction to uniform asymptotic stabilization. The constructed map $A_1: \mathbb{T}^1\times \mathbb{T}^1\rightarrow \mathbb{S}^2$ has nontrivial degree. This picture is related to Theorem \ref{['THM-nounifdecay']}.
  • ...and 5 more figures

Theorems & Definitions (45)

  • DEFINITION 1.1: Energy level set
  • DEFINITION 1.2
  • PROPOSITION 1.3: Ottarsson, convergence to harmonic maps of the heat flow
  • COROLLARY 1.4
  • COROLLARY 1.5: Semi-global exponential stability
  • DEFINITION 1.6
  • THEOREM 1.7: Quantitative local rapid stabilization
  • THEOREM 1.8: Quantitative local null controllability
  • REMARK 1.9
  • THEOREM 1.10: Obstruction to uniform asymptotic stabilization in $\mathbf{H}(2\pi)$
  • ...and 35 more