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Approximate control of parabolic equations with on-off shape controls by Fenchel duality

Camille Pouchol, Emmanuel Trélat, Christophe Zhang

Abstract

We consider the internal control of linear parabolic equations through on-off shape controls, i.e., controls of the form $M(t)χ_{ω(t)}$ with $M(t) \geq 0$ and $ω(t)$ with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative controls. We manage to build controls with constant amplitude $M(t) \equiv M$. In contrast, if the moving control set $ω(t)$ is confined to evolve in some region of the whole domain, we prove that approximate controllability fails to hold for small times. The method of proof is constructive. Using Fenchel-Rockafellar duality and the bathtub principle, the on-off shape control is obtained as the bang-bang solution of an optimal control problem, which we design by relaxing the constraints. Our optimal control approach is outlined in a rather general form for linear constrained control problems, paving the way for generalisations and applications to other PDEs and constraints.

Approximate control of parabolic equations with on-off shape controls by Fenchel duality

Abstract

We consider the internal control of linear parabolic equations through on-off shape controls, i.e., controls of the form with and with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative controls. We manage to build controls with constant amplitude . In contrast, if the moving control set is confined to evolve in some region of the whole domain, we prove that approximate controllability fails to hold for small times. The method of proof is constructive. Using Fenchel-Rockafellar duality and the bathtub principle, the on-off shape control is obtained as the bang-bang solution of an optimal control problem, which we design by relaxing the constraints. Our optimal control approach is outlined in a rather general form for linear constrained control problems, paving the way for generalisations and applications to other PDEs and constraints.
Paper Structure (59 sections, 35 theorems, 154 equations)

This paper contains 59 sections, 35 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Constrained internal control
  3. Control without constraints.
  4. Constrained control.
  5. Main results
  6. On-off shape control.
  7. Obstructions to nonnegative controllability.
  8. Amplitude and time optimal control.
  9. General results
  10. General hypotheses for Theorem \ref{['informal-shape']}.
  11. Elliptic operators.
  12. Proof strategy and related works
  13. Constrained controllability.
  14. Bathtub principle for appropriate costs.
  15. Bang-bang property of optimal controls.
  16. ...and 44 more sections

Key Result

Theorem A

For any $L\in (0,1)$, $T>0$, system nonnegative-controllability-heat is nonnegatively approximately controllable with controls in $\mathcal{U}_{\mathrm{shape}}^L$ in time $T$.

Theorems & Definitions (67)

  • Theorem A
  • Theorem B
  • Definition 1.1
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3: relaxed bathtub principle
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • ...and 57 more