Asymptotic Analysis for Optimal Control of the Cattaneo Model

TL;DR

This paper analyzes the asymptotic behavior of optimal control solutions for the Cattaneo model as the delay parameter approaches zero, demonstrating convergence to the heat equation solutions with confirmed linear rates.

Contribution

It establishes the linear convergence of solutions, states, and controls from the Cattaneo model to the heat equation as delay vanishes, supported by numerical validation.

Findings

Solutions of the Cattaneo equation converge to heat equation solutions as delay tends to zero.

Optimal controls and states exhibit linear convergence rates.

Numerical results confirm theoretical convergence rates.

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 $\tau \rightarrow 0$. 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.