Detroit, MI 48202
Speaker: William M. McEneaney, University of California, San Diego
Title: Stationary Action and Diffusion-Representation Approximations for the Schrödinger Initial Value Problem
It has recently been shown that one may apply dynamic programming to obtain stationary values of payoff functionals, yielding an association of the Hamilton-Jacobi equation to the stationary-value function. This allows one to obtain fundamental solutions to two-point boundary value problems in conservative dynamical systems, including the classical n-body problem. The approach is extended to complex-valued diffusion problems, yielding a control representation for the solution of the Schrödinger equation. The Maslov dequantization is employed, where the domain is complex-valued in the space variable. The notion of stationarity is utilized to relate the Hamilton-Jacobi form of the dequantized Schrödinger equation to its stochastic control representation. Through the use of stationarity, convexity of the payoff is not required, and consequently, there is no restriction on the problem duration. This approach is also applied to obtain new insight into classes of Schrödinger initial value problem solutions.
Prof. William M. McEneaney received his B.S. and M.S. in Mathematics from Rensselaer Polytechnic Institute in 1982 and 1983, respectively. He worked at PAR Technology and Jet Propulsion Laboratory, developing theory and algorithms for estimation and guidance applications. He then attended Brown University from 1989 through 1993, obtaining his M.S. and Ph.D. in Applied Mathematics. His thesis research, conducted under Prof. W.H. Fleming, was on nonlinear risk-sensitive stochastic control. Prof. McEneaney has since held positions at Carnegie Mellon University and North Carolina State University, prior to his current appointment at the University of California, San Diego. His recent interests have been in Stationary Action, Max-Plus Algebraic Methods for Hamilton-Jacobi-Bellman Partial Differential Equations, Risk-Sensitive and Robust Control and Estimation, and Stochastic Games.