ALBERT

Description:    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. Content Type Journal Article Pages 1-25 DOI 10.1007/s00245-011-9140-2 Authors Anton O. Belyakov, ORCOS, Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstrasse 8/105-4, 1040 Vienna, Austria Tsvetomir Tsachev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev str., Block 8, 1113 Sofia, Bulgaria Vladimir M. Veliov, ORCOS, Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstrasse 8/105-4, 1040 Vienna, Austria Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616