  Integrated sensing and communications (ISAC) is an emerging critical
technique for the next generation of communication systems. However, due to
multiple performance metrics used for communication and sensing, the limited
degrees-of-freedom (DoF) in optimizing ISAC systems poses a challenge.
Reconfigurable intelligent surfaces (RIS) can introduce new DoF for beamforming
in ISAC systems, thereby enhancing the performance of communication and sensing
simultaneously. In this paper, we propose two optimization techniques for
beamforming in RIS-assisted ISAC systems. The first technique is an alternating
optimization (AO) algorithm based on the semidefinite relaxation (SDR) method
and a one-dimension iterative (ODI) algorithm, which can maximize the radar
mutual information (MI) while imposing constraints on the communication rates.
The second technique is an AO algorithm based on the Riemannian gradient (RG)
method, which can maximize the weighted ISAC performance metrics. Simulation
results verify the effectiveness of the proposed schemes. The AO-SDR-ODI method
is shown to achieve better communication and sensing performance, than the
AO-RG method, at a higher complexity. It is also shown that the
mean-squared-error (MSE) of the estimates of the sensing parameters decreases
as the radar MI increases.
