Stacked intelligent metasurfaces (SIMs) have recently emerged as a promising hardware architecture for integrated sensing and communication (ISAC), owing to their capability of performing low-power wave-domain signal processing. This paper investigates an SIM-aided multi-static ISAC system, where multiple SIM-enabled base stations (BSs) are coordinated by a central processing unit (CPU) to support downlink communication and target sensing. In the considered architecture, the BSs are configured as either transmitting or receiving nodes, enabling joint user service, target illumination, and cooperative echo processing. To characterize the interplay between communication and sensing, we formulate a joint design problem that maximizes the communication sum rate subject to target-specific sensing quality constraints, transmit power constraints, and SIM hardware constraints. The resulting non-convex problem involves the coupled optimization of transmit power allocation, transmit and receive SIM phase shifts, and centralized sensing receive beamformers. To solve this problem, we develop an efficient alternating optimization algorithm by combining weighted minimum mean-square error reformulation, generalized Rayleigh quotient maximization, and Riemannian manifold optimization. Numerical results demonstrate that the proposed design achieves a favorable communication-sensing trade-off and outperforms representative benchmark schemes.
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