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