  In this paper, we investigate reconfigurable intelligent surface
(RIS)-assisted communication systems which involve a fixed-antenna base station
(BS) and a mobile user (MU) that is equipped with fluid antenna system (FAS).
Specifically, the RIS is utilized to enable communication for the user whose
direct link from the base station is blocked by obstacles. We propose a
comprehensive framework that provides transmission design for both static
scenarios with the knowledge of channel state information (CSI) and harsh
environments where CSI is hard to acquire. It leads to two approaches: a
CSI-based scheme where CSI is available, and a CSI-free scheme when CSI is
inaccessible. Given the complex spatial correlations in FAS, we employ
block-diagonal matrix approximation and independent antenna equivalent models
to simplify the derivation of outage probabilities in both cases. Based on the
derived outage probabilities, we then optimize the throughput of the FAS-RIS
system. For the CSI-based scheme, we first propose a gradient ascent-based
algorithm to obtain a near-optimal solution. Then, to address the possible high
computational complexity in the gradient algorithm, we approximate the
objective function and confirm a unique optimal solution accessible through a
bisection search method. For the CSI-free scheme, we apply the partial gradient
ascent algorithm, reducing complexity further than full gradient algorithms. We
also approximate the objective function and derive a locally optimal
closed-form solution to maximize throughput. Simulation results validate the
effectiveness of the proposed framework for the transmission design in FAS-RIS
systems.
