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This paper examines the performance of a dualhop, noise-limited, non-regenerative system where the relay gain is allowed to vary as an unknown deterministic function of the channel states, subject to instantaneous (i.e. peak) and average power constraints, the latter directly affecting battery lifetime in wireless systems. A convex optimization problem, in terms of a generic performance metric, is formulated and solved algorithmically for the Signal-to-Noise Ratio (SNR), Shannon rate and BPSK Bit-Error-Rate (BER) utilities. The presented model allows for arbitrary statistics of the fading channels, including correlation between them. Special emphasis is placed on the relation between the optimal solutions obtained when observing the channels of either the first or both hops. The regions in channel state space where zero and full power is allocated are determined in closed form, with numerical results being presented for the Shannon rate and BER utilities. It is observed that, for sufficiently high first hop SNR, monitoring both channels instead of just the first hop can lead to a significant performance increase.