A discrete separation principle with a stochastic terminal constraint

Previous formulations [2] of the "Separation Theorem" for a stochastic linear system have excluded terminal conditions and instead used a quadratic performance index. It is shown herein for the discrete-control case that it is possible to have one (or possibly more) stochastic terminal condition, such that a linear combination of the expected terminal state conditioned on the current measurement set shall be a specified number, within certain broad restrictions. A quadratic performance index may also be specified. The calculus of variations is used to derive the necessary conditions, leading to the use of expected values of the costate, Lagrange multiplier and future control, all conditioned on the current measurement set. In a general example based on planar missile guidance, it is specified that the expected miss distance shall be zero, while the quadratic performance index weights the terminal Yd, i.e., rate of missile-target separation perpendicular to the original line-of-sight. A closed-form solution for the optimal control is found, with a term proportional to the expected, zero-effort, miss distance and a second term proportional to the expected, zero-effort, terminal Yd.