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Optimal Control of Heterogeneous Systems with Endogenous Domain of Heterogeneity

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Abstract

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.

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Authors and Affiliations

  1. ORCOS, Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstrasse 8/105-4, 1040, Vienna, Austria

    Anton O. Belyakov & Vladimir M. Veliov

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., Block 8, 1113, Sofia, Bulgaria

    Tsvetomir Tsachev

Authors
  1. Anton O. Belyakov
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  2. Tsvetomir Tsachev
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  3. Vladimir M. Veliov
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Corresponding author

Correspondence to Vladimir M. Veliov.

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Communicating Editor: Frederic Bonnans.

This research was financed by Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) under grant No. MA07-002 and by the Austrian Science Foundation (FWF) under grant No. I 476-N13.

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Belyakov, A.O., Tsachev, T. & Veliov, V.M. Optimal Control of Heterogeneous Systems with Endogenous Domain of Heterogeneity. Appl Math Optim 64, 287–311 (2011). https://doi.org/10.1007/s00245-011-9140-2

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  • DOI: https://doi.org/10.1007/s00245-011-9140-2

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