The sending-end system of line-commutated converter based high-voltage direct current (LCC-HVDC) systems is vulnerable to transient voltage disturbances (TVDs), posing a significant threat to voltage stability. This paper proposes a novel strategy to maximize the dynamic voltage support (DVS) capability of LCC-HVDC systems under various TVDs. The physical mechanisms underlying DVS in LCC-HVDC systems are systematically analyzed, forming the basis for an optimization model that maximizes the DVS capability while incorporating security constraints at both the rectifier and inverter ends. To address the challenge of directly solving the model, an optimality analysis with intuitive geometric interpretations is performed. Based on these insights, a two-stage optimal DVS control strategy for LCC-HVDC systems is developed to iteratively approach the optimal solution through coordinated control of the rectifier and inverter stations. The effectiveness and superiority of the proposed strategy in supporting the sending-end system are validated through dynamic simulations, and its applicability under practical operating conditions is discussed.