This paper considers a hybrid reconfigurable intelligent surface (RIS) assisted integrated sensing and communication (ISAC) system, where each RIS element can flexibly switch between the active and passive modes. Subject to the signal-to-interference-plus-noise ratio (SINR) constraint for each communication user (CU) and the transmit power constraints for both the base station (BS) and the active RIS elements, with the objective of maximizing the minimum beampattern gain among multiple targets, we jointly optimize the BS transmit beamforming for ISAC and the mode selection of each RIS reflecting element, as well as the RIS reflection coefficient matrix. Such formulated joint hybrid-RIS assisted ISAC design problem is a mixed-integer nonlinear program, which is decomposed into two low-dimensional subproblems being solved in an alternating manner. Specifically, by using the semidefinite relaxation (SDR) technique along with the rank-one beamforming construction process, we efficiently obtain the optimal ISAC transmit beamforming design at the BS. Via the SDR and successive convex approximation (SCA) techniques, we jointly determine the active/passive mode selection and reflection coefficient for each RIS element. Numerical results demonstrate that the proposed design solution is significantly superior to the existing baseline solutions.

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