  In this paper, we consider a hybrid reconfigurable intelligent surface (RIS)
comprising of active and passive elements to aid an integrated sensing and
communication (ISAC) system serving multiple users and targets. Active elements
in a hybrid RIS include amplifiers and phase shifters, whereas passive elements
include only phase shifters. We jointly design transmit beamformers and RIS
coefficients, i.e., amplifier gains and phase shifts, to maximize the
worst-case target illumination power while ensuring a desired
signal-to-interference-plus-noise ratio for communication links and
constraining the RIS noise power due to the active elements. Since this design
problem is not convex, we propose a solver based on alternating optimization to
design the transmit beamformers and RIS coefficients. Through numerical
simulations, we demonstrate that the performance of the proposed hybrid RIS
assisted ISAC system is significantly better than that of passive RIS assisted
ISAC systems as well as ISAC systems without RIS even when only a small
fraction of the hybrid RIS contains active elements.
