A stability theory for the linear-quadratic-Gaussian problem for systems with delays in the state, control, and observations

The estimation and control of linear stochastic systems with delays in the state, control, and observations are studied. First, the infinite time deterministic optimal control problem with quadratic cost is examined. Using an appropriate notion of stabilizability and detectability, the optimal control law is obtained, and the closed loop system is shown to be L2-stable. Next, the stochastic filtering problem is studied. Under suitable assumptions of detectability and stabilizability, the filter gains are shown to converge, and the optimal stationary filter is shown to be L2-stable. Finally, by putting together the optimal control and filtering results, a stable constant stochastic control law is obtained for the linear-quadratic-Gaussian problem.