  In this paper, the reinforcement learning (RL)-based optimal control problem
is studied for multiplicative-noise systems, where input delay is involved and
partial system dynamics is unknown. To solve a variant of Riccati-ZXL
equations, which is a counterpart of standard Riccati equation and determines
the optimal controller, we first develop a necessary and sufficient stabilizing
condition in form of several Lyapunov-type equations, a parallelism of the
classical Lyapunov theory. Based on the condition, we provide an offline and
convergent algorithm for the variant of Riccati-ZXL equations. According to the
convergent algorithm, we propose a RL-based optimal control design approach for
solving linear quadratic regulation problem with partially unknown system
dynamics. Finally, a numerical example is used to evaluate the proposed
algorithm.
