Remarks on Long Time Versus Steady State Optimal Control
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.
KeywordsSemilinear heat equations Optimal control problems Long time behavior Steady states Controllability Observability Turnpike property
AMS subject classification:49J20 49K20 93C20 49N05
Enrique Zuazua was partially supported by the Advanced Grant
NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, the FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the MTM2011-29306 and MTM2014-52347 Grants of the MINECO, and a Humboldt Award at the University of Erlangen-Nürnberg. This work was done while the second author was visiting the Laboratoire Jacques Louis Lions with the support of the Paris City Hall “Research in Paris” program.
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