# On the optimal control of implicit systems

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petit.dvi ESAIM� Control� Optimisation and Calculus of Variations URL� http���www�emath�fr�cocv� March ����� Vol� � ���� ON THE OPTIMAL CONTROL OF IMPLICIT SYSTEMS PHILIPPE PETIT Abstract� In this paper we consider the following problem� known as implicit Lagrange problem� �nd the trajectory x argument of min Z � � L�x� �x� dt where the constraints are de�ned by an implicit di�erential equation F �x� �x� with dim F n� q � dim x n� We de�ne the geometric framework of a q���submanifold in the tangent bundle of a surrounding manifold X� which is an extension of the ��submanifold geometric framework de�ned by Rabier and Rheinboldt for control systems� With this geometric framework� we de�ne a class of well�posed implicit di�erential equations for which we obtain locally a controlled vector �eld on a submanifold W of the surrounding manifold X by means of a reduction procedure� We then show that the implicit Lagrange problem leads locally to an explicit optimal control problem on the submanifold W � for which the Pontryagin maximum principle is naturally apply� �� Introduction We consider for the state x of R n the implicit di�erential equation F �x� �x� � �� �� �� In this equation the control u does not appear explicitly but only because there are less equations than unknowns namely F � R n � R n � R n�q where q � n �see � � � Here the control variable u belongs to R q The cost function is the Lagrangian L�x� �x� of TR n A process is a trajectory x��� belonging to C � ���� ���R n � the set of continously di�erentiable functions �resp KC � ���� ���R n � the set of continuous and piecewise di�erentiable functions AC���� ���R n � the set of absolutely continuous functions see the footnotes of the subsection � � � A trajectory x��� is admissible if x��� � a x��� � b and F �x�t�� �x�t�� � �� �t � ��� �� �resp� a�e� on ��� ���� For any admissible trajectory x��� the

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