In this paper an approximation scheme is developed for the solution of the linear quadratic Gaussian (LQG) control on a finite time interval for hereditary systems with multiple noncommensurate delays and distributed delay. The solution here proposed is achieved by means of two approximating subspaces: the rst one to approximate the Riccati equation for control and the second one to approximate the filtering equations. Since the approximating subspaces have finite dimension, the resulting equations can be implemented. The convergence of the approximated control law to the optimal one is proved. Simulation results are reported on a wind tunnel model, showing the high performance of the method.