Some problems in a stochastic setting often involve the need to evaluate the Feynman-Kac formula that follows from models described in terms of stochastic differential equations. Equivalent representations in terms of partial differential equations are also of interest, and these establish the well-known connection between probabilistic and deterministic formulations of these problems. In this Letter, this connection is studied in terms of the quadratic form associated with the Feynman-Kac semigroup. The probability measures that naturally arise in this approach, and thus define how Brownian motion is killed at a specified rate while exiting a set, are interpreted as a random time change of the original stochastic differential equation. Furthermore, since random time changes alter the diffusion coefficients in stochastic differential equations while Girsanov-type measure transformations alter their drift coefficients, their simultaneous use should lead to more tractable solutions for some classes of problems. For example, the minimization of some quadratic forms leads to solutions that satisfy certain partial differential equations and, therefore, the techniques discussed provide a variational approach for finding these solutions. © 2006 Elsevier B.V. All rights reserved.

A neighboring optimal guidance scheme is presented for a nonlinear dynamic system in the presence of modeling and measurement uncertainties as applicable to fuel optimal control of an aeroassisted orbital transfer vehicle (AOTV). Here the vehicle is expected to follow a desired reference path with minimum error. The measurements considered include tracking of altitude and altitude rate. The solution approach is based on nonlinear control optimization, deterministic perturbation, stochastic state estimation, and linearized stochastic control that leads to a closed-loop control law. The reference state trajectory and control trajectory are precomputed by solving a two-point boundary-value control problem. Numerical results show that the actual (true) state trajectory is kept near its ideal (reference) state trajectory at the expense of increasing the total cost.

A neighboring optimal guidance scheme for a nonlinear dynamic system is devised with stochastic inputs and perfect measurements as applicable to fuel optimal control of an aeroassisted orbital transfer vehicle. For the deterministic nonlinear dynamic system describing the atmospheric maneuver, a nominal trajectory is determined. Then, taking modeling uncertainties into account, a linear, stochastic, neighboring optimal guidance scheme is devised. Assuming the additive character of the stochastic effects, the optimal trajectory is approximated as the sum of the deterministic nominal trajectory and the stochastic neighboring optimal solution. Numerical results are presented for a typical aeroassisted orbital transfer vehicle.

The optimal-control problem arising in coplanar orbital transfer using aeroassist technology is addressed. The maneuver involves the transfer from high earth orbit to low earth orbit with minimum fuel consumption. Simulations are carried out to obtain a corridor of entry conditions which are suitable for flying the spacecraft through the atmosphere. A highlight of the present work is the application of an efficient multiple shooting method for handling the difficult nonlinear two-point boundary value problem resulting from the optimization procedure.