**Porretta A., Zuazua E.** Remarks on Long Time Versus Steady State Optimal Control

Mathematical Paradigms of Climate Science, Volume 15 of the series Springer INdAM Series pp 67-89, DOI: 10.1007/978-3-319-39092-5_5

**Abstract:** Control problems play a key role in many fields of Engineering, Economics and Sciences. This applies, in particular, to climate sciences where, often times, relevant problems are formulated in long time scales. The problem of the possible asymptotic simplification (as time tends to infinity) then emerges naturally. More precisely, assuming, for instance, that the free dynamics under consideration stabilizes towards a steady state solution, the following question arises: Do time averages of optimal controls and trajectories converge to the steady optimal controls and states as the time-horizon tends to infinity?This question is very closely related to the so-called turnpike property stating that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time.In this paper we focus on the semilinear heat equation. We prove some partial results and enumerate a number of interesting topics of future research, indicating also some connections with shape design and inverse problems theory.