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The Local Stable Manifold Theorem (LSMT) is one of the most important tool in proving the existence of the gradient of the optimal cost of the Dynamic Programming Equations (DPE). The theorem has long been proven by Hartman [1964] for diffeomorphic maps. Using the techniques of Carr and Krener, we give an alternative proof of the LSMT. Now, in the course of proving the existence of the smooth solutions of the DPE, we need a LMST for nondiffeomorphic maps.

We learn that the Pontryagin Maximum Principle gives a nonlinear dynamics with mixed direction of propagation. We then calculate the corresponding dynamics that propagates forward in time; we call this dynamics the \emph{forward Hamiltonian dynamics}. However, the forward Hamiltonian matrix has to be invertible and thus $0$ cannot be a closed loop eigenvalue. Since the optimal cost calculated term by term is true for all closed loop eigenvalues with magnitude less than 1, we believe that there exists a local stable manifold for a nonlinear Hamiltonian dynamics with opposing directions of propagation.

Carmeliza Navasca tied for top honors in the Best Student Poster Prize (worth $1000) at the ACM's First Richard A. Tapia Symposium 2001, Oct 18-20, 2001 in Houston, Texas. We plan to display this poster next week on the fourth floor across 470 Kerr Hall.