LQR model with noise amplification

We examine the control of a linear system in which the noise is amplified by a quantity which is quadratic in the state and in the control. We develop the Bellman-Hamilton-Jacobi partial differential equation for our problem, assuming a quadratic cost index. We find that the optimal control can also be expressed in conventional feedback form, u^opt=k(t)x(t). We also examine a simple differential pursuit game and show how the evader can introduce noise into the system dynamics to increase the miss distance