Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Switching Point Optimization for Abstract Parabolic Equations

Christian Meyer, Alimhan Musalatov

TL;DR

This work develops a rigorous infinite-dimensional framework for switching-point optimization in abstract parabolic equations, showing that the switching-time-to-control map is not differentiable in standard Bochner spaces but is continuously Fréchet-differentiable when mapped into the dual of Hölder-in-time spaces, enabling a gradient-based proximal gradient approach. Using maximal parabolic regularity, the authors derive weak-state formulations, adjoint equations, and a complete set of first-order optimality (KKT-like) conditions, including an explicit gradient of the reduced objective. Under strong, technical conditions (Lipschitzity of $f$ and $J$, among others), they prove that the reduced gradient is Lipschitz on bounded sets and establish convergence of the proximal gradient method to stationary points, revealing the inherent non-convexity of switching-point problems in this setting. Numerical experiments with a linear heat equation confirm the theory but also show that global optimality is rarely achieved, illustrating the method’s robustness alongside its limitation in non-convex landscapes. Overall, the paper provides a solid, differentiable-operator-based framework for switching-time optimization with rigorous convergence guarantees, while clarifying the practical challenges posed by non-convexity in function-space settings.

Abstract

This work is concerned with a switching point optimization problem governed by a semilinear parabolic equation in abstract function spaces. It is shown that the switching-point-to-control mapping is continuously Fréchet-differentiable when considered with values in the dual of Hölder continuous functions in time. By treating the state equation in weak form based on the concept of maximal parabolic regularity, one can then show that the reduced objective is continuously differentiable w.r.t.\ the switching points which allows to use gradient-based method like the proximal gradient method for its minimization. In order to apply the known convergence results of this method, the gradient of the reduced objective must be Lipschitz continuous, which requires additional assumptions on the data. Numerical experiments confirm our theoretical findings, but also illustrate that such a method will in general not be able to solve the problem up to global optimality due to the non-convex nature of the switching-point-to-control map.

Switching Point Optimization for Abstract Parabolic Equations

TL;DR

This work develops a rigorous infinite-dimensional framework for switching-point optimization in abstract parabolic equations, showing that the switching-time-to-control map is not differentiable in standard Bochner spaces but is continuously Fréchet-differentiable when mapped into the dual of Hölder-in-time spaces, enabling a gradient-based proximal gradient approach. Using maximal parabolic regularity, the authors derive weak-state formulations, adjoint equations, and a complete set of first-order optimality (KKT-like) conditions, including an explicit gradient of the reduced objective. Under strong, technical conditions (Lipschitzity of and , among others), they prove that the reduced gradient is Lipschitz on bounded sets and establish convergence of the proximal gradient method to stationary points, revealing the inherent non-convexity of switching-point problems in this setting. Numerical experiments with a linear heat equation confirm the theory but also show that global optimality is rarely achieved, illustrating the method’s robustness alongside its limitation in non-convex landscapes. Overall, the paper provides a solid, differentiable-operator-based framework for switching-time optimization with rigorous convergence guarantees, while clarifying the practical challenges posed by non-convexity in function-space settings.

Abstract

This work is concerned with a switching point optimization problem governed by a semilinear parabolic equation in abstract function spaces. It is shown that the switching-point-to-control mapping is continuously Fréchet-differentiable when considered with values in the dual of Hölder continuous functions in time. By treating the state equation in weak form based on the concept of maximal parabolic regularity, one can then show that the reduced objective is continuously differentiable w.r.t.\ the switching points which allows to use gradient-based method like the proximal gradient method for its minimization. In order to apply the known convergence results of this method, the gradient of the reduced objective must be Lipschitz continuous, which requires additional assumptions on the data. Numerical experiments confirm our theoretical findings, but also illustrate that such a method will in general not be able to solve the problem up to global optimality due to the non-convex nature of the switching-point-to-control map.
Paper Structure (12 sections, 17 theorems, 171 equations, 20 tables, 2 algorithms)

This paper contains 12 sections, 17 theorems, 171 equations, 20 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation, Standing Assumptions, and Rigorous Mathematical Formulation of the Optimization Problem
  3. Sensitivity Analysis
  4. Differentiability of the Switching-Point-to-Control Operator
  5. Differentiability of the Parabolic Solution Map in Weak Form
  6. First-order Necessary Optimality Conditions
  7. Proximal Gradient Method
  8. Numerical experiments
  9. Numerical results
  10. Autonomous parabolic equations
  11. Projection onto the polyhedron $\mathcal{P}$
  12. On the lack of differentiability of objectives involving point evaluations

Key Result

Lemma 2.4

The Nemyzki operator associated to $f$, for the ease of notation denoted by the same symbol, continuously maps $L^{q}(0,T;\mathcal{D})$ to $L^1(0,T;Y_q)$. Furthermore, it is continuously Fréchet-differentiable from $L^{q}(0,T;\mathcal{D})$ to $L^1(0,T;Y_q)$ with Fréchet-derivative

Theorems & Definitions (36)

  • Remark 1.1
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.9
  • proof
  • Example 3.1
  • Lemma 3.2
  • ...and 26 more