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Optimal control of gradient flows via the Weighted Energy-Dissipation method

Takeshi Fukao, Ulisse Stefanelli, Riccardo Voso

Abstract

We consider a general optimal control problem in the setting of gradient flows. Two approximations of the problem are presented, both relying on the variational reformulation of gradient-flow dynamics via the Weighted-Energy-Dissipation variational approach. This consists in the minimization of global-in-time functionals over trajectories, combined with a limit passage. We show that the original nonpenalized problem and the two successive approximations admits solutions. Moreover, resorting to a $Γ$-convergence analysis we show that penalised optimal controls converge to nonpenalized one as the approximation is removed.

Optimal control of gradient flows via the Weighted Energy-Dissipation method

Abstract

We consider a general optimal control problem in the setting of gradient flows. Two approximations of the problem are presented, both relying on the variational reformulation of gradient-flow dynamics via the Weighted-Energy-Dissipation variational approach. This consists in the minimization of global-in-time functionals over trajectories, combined with a limit passage. We show that the original nonpenalized problem and the two successive approximations admits solutions. Moreover, resorting to a -convergence analysis we show that penalised optimal controls converge to nonpenalized one as the approximation is removed.
Paper Structure (12 sections, 8 theorems, 87 equations)

This paper contains 12 sections, 8 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. An example
  4. Preliminary lemmas
  5. Existence: proof of Theorem \ref{['theorem']}. i
  6. Well-posedness for Problem \ref{['initialproblem']}
  7. Well-posedness for Problem \ref{['epsilonproblem']}
  8. Well-posedness for Problem \ref{['epsilonlambdaproblem']}
  9. Convergence: proof of Theorem \ref{['theorem']}. ii-iv
  10. Proof of Theorem \ref{['theorem']}.ii.
  11. Proof of Theorem \ref{['theorem']}.iii.
  12. Proof of Theorem \ref{['theorem']}.iv.

Key Result

Proposition 2.1

Under assumptions (A1)-(A2), there exists $\varepsilon_0 >0$ so that for all $\varepsilon \in (0,\varepsilon_0)$ and all $u \in U$ the functional $W_{\varepsilon}(\cdot,u)$ is $\kappa_{\varepsilon}$-convex in $H^1(0,T;H)$ for some $\kappa_\varepsilon>0$. In particular, there exists a unique minimize where $(\partial \phi (y^0))^{\circ}$ is the element of minimal norm in $\partial \phi (y^0)$. As $

Theorems & Definitions (14)

  • Proposition 2.1: WED approach to gradient flows
  • Theorem 2.2: WED approach to optimal control
  • Lemma 3.1: Value of $M_\varepsilon^u$
  • proof
  • Lemma 3.2: Continuity of the map $u \mapsto y_{\varepsilon}^u$
  • proof
  • Lemma 3.3: Coercivity of $P_{\varepsilon\lambda}$
  • proof
  • Lemma 3.4
  • proof
  • ...and 4 more