An analytical and optimization framework for the joint design of communication and control (JDCC) in a center-fed pinching-antenna system (C-PASS) is proposed. Within this framework, the mathematical coupling between data throughput and closed-loop control cost is characterized for JDCC. The analysis captures the mutual interference between concurrently transmitted data and control streams during uplink state uploading and downlink command delivery. Based on this coupling, an achievable communication-control performance region is established for C-PASS. Accordingly, a throughput maximization problem is formulated to jointly optimize digital and pinching beamforming subject to control-cost constraints. To address this problem, an alternating optimization algorithm is developed based on constraint reformulation and resource decomposition. The proposed framework reveals how control requirements shape achievable data throughput, guiding the joint beamforming design. Numerical results demonstrate that 1) the proposed C-PASS design achieves 110.2\% higher frame-average throughput than conventional PASS, and 2) the proposed concurrent data-control transmission protocol increases frame-average throughput by up to 66.7\% over separated transmission.

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