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Asymptotic Analysis for Optimal Control of the Cattaneo Model

Sebastian Blauth, René Pinnau, Matthias Andres, Claudia Totzeck

TL;DR

This work analyzes optimal control problems constrained by the Cattaneo equation, a damped-wave model for delayed heat transfer, in the asymptotic limit $\tau\to0$ toward the parabolic heat equation. It develops energy estimates independent of $\tau$, introduces a compatibility condition to enable time-differentiation, and proves linear convergence of both forward states and the optimal-control pair to their heat-equation counterparts. The results extend to the adjoint system for the case $\nu=1$, yielding linear convergence of optimal states and controls with rates proportional to $\tau$, and are corroborated by numerical experiments. The findings bridge hyperbolic and parabolic heat models with implications for parameter identification and radiative transfer applications in medical contexts.

Abstract

We consider an optimal control problem with tracking-type cost functional constrained by the Cattaneo equation, which is a well-known model for delayed heat transfer. In particular, we are interested the asymptotic behaviour of the optimal control problems for a vanishing delay time $τ\rightarrow 0$. First, we show the convergence of solutions of the Cattaneo equation to the ones of the heat equation. Assuming the same right-hand side and compatible initial conditions for the equations, we prove a linear convergence rate. Moreover, we show linear convergence of the optimal states and optimal controls for the Cattaneo equation towards the ones for the heat equation. We present numerical results for both, the forward and the optimal control problem confirming these linear convergence rates.

Asymptotic Analysis for Optimal Control of the Cattaneo Model

TL;DR

This work analyzes optimal control problems constrained by the Cattaneo equation, a damped-wave model for delayed heat transfer, in the asymptotic limit toward the parabolic heat equation. It develops energy estimates independent of , introduces a compatibility condition to enable time-differentiation, and proves linear convergence of both forward states and the optimal-control pair to their heat-equation counterparts. The results extend to the adjoint system for the case , yielding linear convergence of optimal states and controls with rates proportional to , and are corroborated by numerical experiments. The findings bridge hyperbolic and parabolic heat models with implications for parameter identification and radiative transfer applications in medical contexts.

Abstract

We consider an optimal control problem with tracking-type cost functional constrained by the Cattaneo equation, which is a well-known model for delayed heat transfer. In particular, we are interested the asymptotic behaviour of the optimal control problems for a vanishing delay time . First, we show the convergence of solutions of the Cattaneo equation to the ones of the heat equation. Assuming the same right-hand side and compatible initial conditions for the equations, we prove a linear convergence rate. Moreover, we show linear convergence of the optimal states and optimal controls for the Cattaneo equation towards the ones for the heat equation. We present numerical results for both, the forward and the optimal control problem confirming these linear convergence rates.
Paper Structure (18 sections, 15 theorems, 108 equations, 2 figures, 1 table)

This paper contains 18 sections, 15 theorems, 108 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries & Notations
  3. Well-Posedness of the State Eqations and Optimal Control
  4. Well-Posedness of the Cattaneo Equation
  5. Well-Posedness of the Heat Equation
  6. Optimal Control Problems
  7. Optimal Control of the Cattaneo equation
  8. Optimal Control of the Heat Equation
  9. Asymptotic Analysis
  10. Compatibility Condition
  11. Linear Convergence of the states
  12. Linear Convergence of the Optimal States and Optimal Controls
  13. Numerical Results
  14. Conclusion and Outlook
  15. Proofs of Section \ref{['sec:asymptotic analysis']}
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $\tau>0$, $y_0\in H^1_0(\Omega)$, $y_1\in L^2(\Omega)$ and $u_\tau\in L^2(0,T;L^2(\Omega))$. Then, there exists a unique weak solution $y_\tau\in Y(0,T)$ of the Cattaneo equation eq:cattaneo which depends continuously on the data. In fact, there exists a constant $C=C(\tau)>0$ such that

Figures (2)

  • Figure 1: Convergence rates for the optimal controls of the Cattaneo equation.
  • Figure 2: Convergence rates for the optimal states of the Cattaneo equation.

Theorems & Definitions (27)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 17 more