Optimal Control of Sampled Linear Systems with Control-Linear Noise

We address the problem of steering a stochastic linear system with control-linear noise from an initial condition to a final state while minimizing mean squared deviation from a target state. We assume that the controller can access noiseless, full-state sampled measurements at discrete times. The main contribution is to provide a complete solution to this optimization problem. The explicit properties of the solution are formulated as a Theorem and subsequently proven. We show that the optimal feedback control law is linear in the initial state, the target state, and the sampled measurements. Moreover, we show that the time-varying, matrix-valued feedback state gains can be computed offline before the system is launched. Numerical studies show that the mean squared deviation decreases dramatically as the number of measurements grows and converges to a nonzero limiting value.