Demands for rotorcraft with increased flight speed, improved operational performance and reduced environmental impact have led to a drive in research and development of alternative concepts. Compound rotorcraft overcome the flight speed limitations of conventional helicopters with additional lifting and propulsive components. Further to operational benefits, these augmentations provide additional flight control parameters, resulting in control redundancy. This work aims to investigate the impact of optimal control strategies for a generic coaxial compound rotorcraft, equipped with turboshaft engines, targeting the minimization of mission fuel burn and gaseous emissions. The direct redundant controls considered are: (a) main rotor speed, (b) propeller speed, and (c) fuselage pitch attitude. A simulation tool for coaxial compound rotorcraft analysis has been developed and coupled to a zero-dimensional engine performance model and a stirred-reactor combustor model. First, experimental and flight test data were used to provide extensive validation of the developed models. A parametric analysis was then carried out to gain insight into the effect of the redundant controls. This was followed by the derivation of a generalized set of optimal redundant control allocations using a surrogate-assisted genetic algorithm. Application of the optimal redundant control allocations during realistic operational scenarios has demonstrated reductions in fuel burn and NOx of up to 6.93% and 8.74%, respectively. The developed method constitutes a rigorous approach to guide the design of control systems for future advanced rotorcraft.