Optimal Control of Heterogeneous Systems with Endogenous Domain of Heterogeneity

Optimal Control of Heterogeneous Systems with Endogenous Domain of Heterogeneity 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Optimal Control of Heterogeneous Systems with Endogenous Domain of Heterogeneity

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Publisher
Springer-Verlag
Copyright
Copyright © 2011 by Springer Science+Business Media, LLC
Subject
Mathematics; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Calculus of Variations and Optimal Control; Optimization; Numerical and Computational Physics; Systems Theory, Control
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-011-9140-2
Publisher site
See Article on Publisher Site

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.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2011

References

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