Building a bridge from optimal control problems for ODEs under state constraints to those for PDEs

Titelangaben

Pesch, Hans Josef ; Wrensch, Michael ; Bechmann, Simon ; Rund, Armin:
Building a bridge from optimal control problems for ODEs under state constraints to those for PDEs.
2015
Veranstaltung: Fakultätskolloquium der Fakultät für Mathematik, Technische Universität München , 17.06.2015 , Garching bei München.
(Veranstaltungsbeitrag: Kongress/Konferenz/Symposium/Tagung , Vortrag )

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Abstract

Already a few years after the discovery of the most useful necessary condition of optimal control theory, the maximum principle, by Pontryagin and his students Boltyanskii and Gamkrelidze in 1955, Bryson, Denham and Dreyfus suggested a method in 1963 to establish necessary conditions for state constrained optimal control problems which were especially appropriate for numerical implementations.

The talk will exhibit how their idea can be transfered to optimal control problems for partial differential equations. For the sake of simplicity, we will focus on elliptic optimal control problems with state constraints and we will see that these problems can be reformulated as topology, resp. shape optimization problems with the interface between active and inactive set as optimization variable. This type of problems leads to a new class of so-called set optimal control problems which can also be interpreted as bi-level optimization problems, more abstract as optimization problems on vector bundles. In this connection various infinite-dimensional optimization problems merge together and can be viewed from a very general abstract point of view.

Not surprisingly, this approach has its counterpart in the multipoint boundary value problem formulation for the numerical solution of constrained ODE optimal control problems by multiple shooting, which has been developed at the Technische Universität München over many years from the beginning of the 1970ies by Bulirsch and his students among whom also the speaker can be ranked.

An overview on the resulting numerical methods and some numerical results will conclude the presentation.