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Advances in chip-multiprocessor processing capabilities has led to an increased power consumption and temperature hotspots. Maintaining the on-chip temperature is important from the power reduction and reliability considerations. Achieving highest performance while maintaining the temperature constraint is a challenge. We develop analytical solutions for the optimal control of frequencies for each core in a chip-multiprocessor. The objective is to reduce the makespan or the latest task completion time of all tasks. We show that the optimal frequency policy is bang-bang when the temperature constraint is not active and is exponential when the temperature constraint is active. We show that there is a significant improvement in overall throughput with our proposed solution and yet all cores operate under the thermal maximum.