Parabolic partial differential equations optimal control infinite dimensional problems infinite time horizons ill-posedness dynamic programming
Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 39, No 3-4, (2013), 331p-354p
We review an emerging application field to parabolic partial differential equations (PDEs), that’s economic growth theory. After a short presentation of concrete applications, we highlight the peculiarities of optimal control problems of parabolic PDEs with infinite time horizons. In particular, the heuristic application of the maximum principle to the latter leads to single out a serious ill-posedness problem, which is, in our view, a barrier to the use of parabolic PDEs in economic growth studies as the latter
are interested in long-run asymptotic solutions, thus requiring the solution
to infinite time horizon optimal control problems. Adapted dynamic programming methods are used to dig deeper into the identified ill-posedness issue. 2010 Mathematics Subject Classification: 91B62, 91B72, 49K20, 49L20.