Second cumulant statistical control with indefinite control weight

In linear-quadratic-Gaussian control, the positive definiteness of the control weighting matrix in the cost function has been assumed, however, it has been shown that solutions do exist for indefinite control weight matrices for the linear-quadratic-Gaussian case. Here we extend the results to other statistical control methods such as second cumulant statistical control and risk-sensitive control. In this paper, we find the optimal controller where the diffusion term in the state equation depends on the control.