On the optimal control of implicit systems

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