# On the optimal control of implicit systems

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 3, page 49-81
- ISSN: 1292-8119

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topPetit, P.. "On the optimal control of implicit systems." ESAIM: Control, Optimisation and Calculus of Variations 3 (2010): 49-81. <http://eudml.org/doc/197363>.

@article{Petit2010,

abstract = {
In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differential
equation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surrounding
manifold 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 this
geometric framework we define a class of well-posed implicit differential equations for which we locally obtain, by means of a reduction
procedure, a controlled vector field on a submanifold W of the surrounding manifold X. We then show that the implicit Lagrange problem leads
to, locally, an explicit optimal control problem on the submanifold W for which the Pontryagin maximum principle is naturally used.
},

author = {Petit, P.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Implicit systems; optimal control; Pontryagin maximum principle; manifold; submanifold; subimmersion .; --submanifolds; implicit differential equations},

language = {eng},

month = {3},

pages = {49-81},

publisher = {EDP Sciences},

title = {On the optimal control of implicit systems},

url = {http://eudml.org/doc/197363},

volume = {3},

year = {2010},

}

TY - JOUR

AU - Petit, P.

TI - On the optimal control of implicit systems

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 3

SP - 49

EP - 81

AB -
In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differential
equation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surrounding
manifold 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 this
geometric framework we define a class of well-posed implicit differential equations for which we locally obtain, by means of a reduction
procedure, a controlled vector field on a submanifold W of the surrounding manifold X. We then show that the implicit Lagrange problem leads
to, locally, an explicit optimal control problem on the submanifold W for which the Pontryagin maximum principle is naturally used.

LA - eng

KW - Implicit systems; optimal control; Pontryagin maximum principle; manifold; submanifold; subimmersion .; --submanifolds; implicit differential equations

UR - http://eudml.org/doc/197363

ER -

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