A gradient flow approach to computing a non-linear discrete time quadratic optimal feedback gain matrix

This paper proposes an approach to solving infinite planning horizon quadratic optimal regulator problems with linear static state feedback for discrete time systems. The approach is based on solving a sequence of approximate problems constructed by combining a finite horizon problem with an infinite horizon linear problem. A gradient-flow algorithm is derived to solve the approximate problems. As part of this, a new algorithm is derived for computing the gradient of the cost functional, based on a system of difference equations to be solved completely forward in time. Two numerical examples are presented.