Extended 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). The important case of k(t)=k leads to a time invariant partial differential equation for the evolution of the probability density function. The solution is discussed and the steady state case is exhibited. Our model is a continuous version of the discrete model first analyzed by Jacobson (1974), and more recently by Yat (1987). This paper is a continuation of the author´s previous papers (1990, 1995). The underlying formalism can be found in Jazwinski (1970) and Fleming et al. (1975)