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Underdamped stochastic thermodynamic engines in contact with a heat bath with arbitrary temperature profile

We study thermodynamic processes in contact with a heat bath that may have an arbitrary time-varying periodic temperature profile. Within the framework of stochastic thermodynamics, and for models of thermo-dynamic engines in the idealized case of underdamped particles in the low-friction regime, we derive explicit bounds as well as optimal control protocols that draw maximum power and achieve maximum efficiency at any specified level of power.