Linear quadratic regulation for linear continuous-time systems with distributed input delay

This paper is concerned with the linear quadratic regulation (LQR) problem for linear continuous-time systems with distributed input delay. In this paper, we address the LQR problem by first establishing a duality between it and a smoothing problem for an associated stochastic backward system with output delay. An analytical solution to the LQR control is then derived by solving the smoothing problem and is given in terms of the solution of Riccati type partial differential equation of the same dimension as the plant (ignoring the delays). The infinite horizon LQR problem is also considered in this paper and the convergence and stability analysis of the LQR controller is provided based on infinite dimensional system theory.