On Optimal Control with Polynomial Cost Functions for Linear Systems with Time-Invariant Stochastic Parameters

This study focuses on minimizing the expectation of a polynomial cost function for linear systems with time-invariant stochastic parameters. These polynomial and time-invariant properties cause difficulties in solving this optimal control problem. Conventional approaches such as the principle of optimality are not applied to the problem due to the time-invariant stochastic parameters. Whereas various performance metrics can be expressed by the polynomial cost, it makes the problem more complicated than well-known quadratic cost functions. To overcome these difficulties, this study derives an explicit relation between the polynomial cost function and a linear feedback gain of the controller. Using this relation, a gradient method yields sub-optimal feedback gains to the problem even for the polynomial cost.