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Viscous Approximation of Optimal Control Problems Governed by Rate-Independent Systems with Non-Convex Energies

Merlin Andreia, Christian Meyer

TL;DR

The work tackles optimal control for rate‑independent systems with nonconvex energies by deploying a two‑layer viscous regularization across a hierarchy of Hilbert spaces, producing a non‑smooth ODE that is easier to solve. It establishes that optima of the regularized problems converge (via subsequences) to optima of the original BV solution framework, provided the original has a time‑continuous optimal state, and it proves existence of global optima for the unregularized and regularized problems. A pivotal reverse‑approximation result constructs recovery sequences that converge to a differential BV solution, enabling strong convergence of controls and states and ensuring end‑time feasibility. The findings culminate in convergence theorems for both single and double regularizations, including a robust framework for approximating BV solutions by viscous schemes and for transferring optimality from regularized to original problems. The approach balances mathematical rigor with practical relevance for simulations of systems with rate‑independent, nonconvex energies and irreversible processes.

Abstract

We consider an optimal control problem governed by a rate-inde\-pendent system with non-convex energy. The state equation is approximated by means of viscous regularization w.r.t.\ to hierarchy of two different Hilbert spaces. The regularized problem corresponds to an optimal control problem subject to a non-smooth ODE in Hilbert space, which is substantially easier to solve than the original optimal control problem. The convergence properties of the viscous regularization are investigated. It is shown that every sequence of globally optimal solutions of the viscous problems admits a (weakly) converging subsequence whose limit is a globally optimal solution of the original problem, provided that the latter admits at least one optimal solution with an optimal state that is continuous in time.

Viscous Approximation of Optimal Control Problems Governed by Rate-Independent Systems with Non-Convex Energies

TL;DR

The work tackles optimal control for rate‑independent systems with nonconvex energies by deploying a two‑layer viscous regularization across a hierarchy of Hilbert spaces, producing a non‑smooth ODE that is easier to solve. It establishes that optima of the regularized problems converge (via subsequences) to optima of the original BV solution framework, provided the original has a time‑continuous optimal state, and it proves existence of global optima for the unregularized and regularized problems. A pivotal reverse‑approximation result constructs recovery sequences that converge to a differential BV solution, enabling strong convergence of controls and states and ensuring end‑time feasibility. The findings culminate in convergence theorems for both single and double regularizations, including a robust framework for approximating BV solutions by viscous schemes and for transferring optimality from regularized to original problems. The approach balances mathematical rigor with practical relevance for simulations of systems with rate‑independent, nonconvex energies and irreversible processes.

Abstract

We consider an optimal control problem governed by a rate-inde\-pendent system with non-convex energy. The state equation is approximated by means of viscous regularization w.r.t.\ to hierarchy of two different Hilbert spaces. The regularized problem corresponds to an optimal control problem subject to a non-smooth ODE in Hilbert space, which is substantially easier to solve than the original optimal control problem. The convergence properties of the viscous regularization are investigated. It is shown that every sequence of globally optimal solutions of the viscous problems admits a (weakly) converging subsequence whose limit is a globally optimal solution of the original problem, provided that the latter admits at least one optimal solution with an optimal state that is continuous in time.
Paper Structure (12 sections, 40 theorems, 245 equations)

This paper contains 12 sections, 40 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Standing Assumptions
  3. Parametrized Balanced Viscosity Solutions
  4. Double Viscous Regularization
  5. Existence of Optimal Controls
  6. Reverse Approximation
  7. Approximation by Regularized Optimal Control Problems
  8. A Lower Semicontinuity Result
  9. Differentiability of the $\epsilon, \delta$-Norm
  10. Continuous BV Solutions are Differential Solutions
  11. Uniqueness of Differential Solutions
  12. Weak-$*$ Convergence in $\mathrm{BV}([0,T];\mathcal{Z})$

Key Result

Lemma 2.2

Let $z_k \rightharpoonup z$ in $\mathcal{Z}$. Then $\mathcal{F}(z_k) \to \mathcal{F}(z)$ as $k \to \infty$.

Theorems & Definitions (90)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • ...and 80 more