A Remarkable Property of the Dynamic Optimization Extremals

Delfim F. M. Torres (delfimmat.ua.pt)

Abstract: A dynamic optimization continuous problem poses the question of what is the optimal magnitude of a choice variable, at each point of time, in a given interval. To tackle such problems, three major approaches are available: dynamic programming; the calculus of variations; and the powerful optimal control approach. At the core of optimal control theory is the Pontryagin maximum principle - the celebrated first order necessary optimality condition - whose solutions are called extremals and which are obtained through a function called Hamiltonian, akin to the Lagrangian function used in ordinary calculus optimization problems. A remarkable property of the extremals is that the total derivative with respect to time of the corresponding Hamiltonian equals the partial derivative of the Hamiltonian with respect to time. In particular, when the Hamiltonian does not depend explicitly on time, the value of the Hamiltonian evaluated along the extremals turns out to be constant (a property that corresponds to energy conservation in classical mechanics). We present a generalization of the above property. As applications of the new relation, methods for obtaining conserved quantities along the Pontryagin extremals and for characterizing problems possessing given constants of the motion are obtained.

Keywords: dynamic optimization, optimal control, Pontryagin extremals, constants of the motion.

Category 1: Nonlinear Optimization (Systems governed by Differential Equations Optimization )

Citation: Research report (Cadernos de Matemática) CM01/I-14, July/2001. Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal.

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Entry Submitted: 08/13/2001
Entry Accepted: 08/15/2001
Entry Last Modified: 08/13/2001

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