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On the optimal control of implicit systems

Published online by Cambridge University Press:  15 August 2002

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Abstract

In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differentialequation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surroundingmanifold X, we define the geometric framework of q-pi- submanifold. This is an extension of the geometric framework of pi- submanifold,defined by Rabier and Rheinboldt for determined implicit differential equations, to underdetermined implicit differential equations. With thisgeometric framework we define a class of well-posed implicit differential equations for which we locally obtain, by means of a reductionprocedure, a controlled vector field on a submanifold W of the surrounding manifold X. We then show that the implicit Lagrange problem leadsto, locally, an explicit optimal control problem on the submanifold W for which the Pontryagin maximum principle is naturally used.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 1998

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