Applied Mathematics & Optimization

, Volume 64, Issue 2, pp 287–311 | Cite as

Optimal Control of Heterogeneous Systems with Endogenous Domain of Heterogeneity

  • Anton O. Belyakov
  • Tsvetomir Tsachev
  • Vladimir M. VeliovEmail author


The paper deals with optimal control of heterogeneous systems, that is, families of controlled ODEs parameterized by a parameter running over a domain called domain of heterogeneity. The main novelty in the paper is that the domain of heterogeneity is endogenous: it may depend on the control and on the state of the system. This extension is crucial for several economic applications and turns out to rise interesting mathematical problems. A necessary optimality condition is derived, where one of the adjoint variables satisfies a differential inclusion (instead of equation) and the maximization of the Hamiltonian takes the form of “min-max”. As a consequence, a Pontryagin-type maximum principle is obtained under certain regularity conditions for the optimal control. A formula for the derivative of the objective function with respect to the control from L is presented together with a sufficient condition for its existence. A stylized economic example is investigated analytically and numerically.


Optimal control Distributed control Heterogeneous systems Endogenous domain of heterogeneity Pontryagin-type maximum principle Set-valued analysis 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Anton O. Belyakov
    • 1
  • Tsvetomir Tsachev
    • 2
  • Vladimir M. Veliov
    • 1
    Email author
  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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