  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.
