Applied Mathematics & Optimization

, Volume 70, Issue 1, pp 141–164 | Cite as

Optimality Conditions and the Hamiltonian for a Distributed Optimal Control Problem on Controlled Domain

  • Bernhard Skritek
  • Tsvetomir TsachevEmail author
  • Vladimir M. Veliov


The paper investigates an optimal control problem for a distributed system arising in the economics of endogenous growth. The problem involves a specific coupled family of controlled ODEs parameterized by a parameter (representing the heterogeneity) running over a domain that may dynamically depend on the control and on the state of the system. Existence of an optimal control is obtained and continuity of any optimal control with respect to the parameter of heterogeneity is proved. The latter allows to substantially strengthen previously obtained necessary optimality conditions and to obtain a Pontryagin’s type maximum principle. The necessary optimality conditions obtained here have a Hamiltonian representation, and stationarity of the Hamiltonian along any optimal trajectory is proved in the case of time-independent data.


Optimal control Distributed control Controlled domain Pontryagin-type maximum principle Endogenous economic growth 

Mathematics Subject Classification

49K20 49K21 49J20 



This research was funded by the Austrian Science Foundation (FWF) under Grant No I 476-N13.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Bernhard Skritek
    • 1
  • Tsvetomir Tsachev
    • 2
    Email author
  • Vladimir M. Veliov
    • 1
  1. 1.ORCOS, Institute of Mathematical Methods in EconomicsVienna University of TechnologyViennaAustria
  2. 2.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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