Analytic solution to indefinite linear quadratic regulator for stochastic systems

This paper aims to deal with the indefinite linear quadratic regulator(ILQR) for stochastic systems. It provides both the analytic solution to the ILQR and the sufficient and necessary condition under which the ILQR is solvable. Different from the existed literature, we obtain the results in a novel way. In order to obtain a tighter necessary condition, we investigate the solution structure of a two-point boundary value problem for a stochastic differential equations directly but involve no complicated probabilistic derivation. Consider that there are close relationships between the linear game problem and H
_∞
control and ILQR, the idea in the paper can also be extended to solve these problems.