The spectral linear filter method for a stochastic optimal control problem of partially observable systems

In this paper, two spectral methods are presented to solve a stochastic optimal control problem of a partially observable system. These two methods work together to solve such problems. In fact, solving such problems involves two cases: obtaining the control function and simulating the partially observable system. At first, a spectral linear filter is defined as a function of time to obtain an appropriate solution for a partially observable system. This linear filter is equipped with an orthogonal basis and it is made to predict the future behavior of this system. In this method, the goal is to approximate the trend of the partially observable system. The second method is suggested to achieve the optimal control corresponding to each sample path. In this method, the spectral Fourier transform is used. These two methods are used together to solve linear and nonlinear cases. In fact, the innovative contents of this paper are both the spectral linear filter and the suggested spectral optimal control method.