  We study a chain of alternating hot and cold electronic nanocavities --
connected to one another via resonant-tunneling quantum dots -- with the intent
of achieving precise thermal control across the chain. This is accomplished by
positioning the dots' energy levels such that a predetermined distribution of
heat currents is realized across the chain in the steady state. The number of
electrons in each cavity is conserved in the steady state which constrains the
cavities' chemical potentials. We determine these chemical potentials
analytically in the linear response regime where the energy differences between
the dots' resonant levels and the neighboring chemical potentials are much
smaller than the thermal energy. In this regime, the thermal control problem
can be solved exactly, while, in the general case, thermal control can only be
achieved in a relative sense, that is, when one only preassigns the ratios
between different heat currents. We apply our results to two different cases:
We first demonstrate that a "heat switch" can be easily realized with three
coupled cavities, then we show that our linear response results can provide
accurate results in situations with a large number of cavities.
