On the nonuniqueness and instability of solutions of tracking-type optimal control problems

Constantin Christof; Dominik Hafemeyer

doi:10.3934/mcrf.2021028

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2022, Volume 12, Issue 2: 421-431. Doi: 10.3934/mcrf.2021028

This issue Previous Article Feedback stabilization of parabolic systems with input delay Next Article Local Kalman rank condition for linear time varying systems

On the nonuniqueness and instability of solutions of tracking-type optimal control problems

Constantin Christof^1, , and
Dominik Hafemeyer^1,

1.
Department of Mathematics, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany

^*Corresponding author: Constantin Christof
^*Corresponding author: Constantin Christof

Received: July 2020

Revised: March 2021

Early access: May 2021

Published: June 2022

This research was conducted within the International Research Training Group IGDK 1754, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF) under project number 188264188/GRK1754..

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Abstract

We study tracking-type optimal control problems that involve a non-affine, weak-to-weak continuous control-to-state mapping, a desired state $ y_d $, and a desired control $ u_d $. It is proved that such problems are always nonuniquely solvable for certain choices of the tuple $ (y_d, u_d) $ and instable in the sense that the set of solutions (interpreted as a multivalued function of $ (y_d, u_d) $) does not admit a continuous selection.

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Mathematics Subject Classification: 49J27, 49K40, 49N45, 90C26.

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