Improvement of stochastic neighbouring-optimal control using nonlinear Gaussian white noise terms in the Taylor expansions

In order to obtain an approximate solution to the optimal control of nonlinear stochastic systems, one used to suppose that the magnitude of the noise is small enough in such a manner that one can apply a linearization around the deterministic trajectory defined by the system in the absence of noise. When this assumption is not satisfied (i.e. when the magnitude of the noise is of some importance), it is necessary to improve this approximation by taking into account the nonlinear terms of the Taylorʹs expansion, so that, as a result, we are so dealing with stochastic systems subject to powers of Gaussian white noise. In an engineering mathematics framework, in order to cope with the mathematical difficulties so involved, we propose an approach via the central limit theorem. We first define the integral of powers of Gaussian white noise, whereby we can derive a generalization of the Fokker-Planck-Kolmogorov equation. Then we show how this result can be used in a variational approach to neighbouring stochastic optimal control, via the moment equations.