Linear quadratic optimal control for continuous-time stochastic systems with single input-delay

The paper considers the linear quadratic(LQ) control problem for the Itô-type stochastic system with input delays. Due to simultaneous appearances of diffusion terms (dependent on the system state and control input) as well as delays in the dynamic system, the problem is very involved and outstanding at present. Our main idea is to pursue the explicit optimal cost of the problem and to exploit the interplay between the original problem and its equivalent abstract description. Therefore, we not only provide a causal and adapted controller based on generalized Riccati equations with the same dimension as the system (without delays) but also present the sufficient and necessary condition under which the optimal control problem for the abstract stochastic system is solvable.