  We consider the performance of periodically driven stochastic heat engines in
the linear response regime. Reaching the theoretical bounds for efficiency and
efficiency at maximum power typically requires full control over the design and
the driving of the system. We develop a framework which allows to quantify the
role that limited control over the system has on the performance. Specifically,
we show that optimizing the driving entering the work extraction for a given
temperature protocol leads to a universal, one-parameter dependence for both
maximum efficiency and maximum power as a function of efficiency. In
particular, we show that reaching Carnot efficiency (and, hence, Curzon-Ahlborn
efficiency at maximum power) requires to have control over the amplitude of the
full Hamiltonian of the system. Since the kinetic energy cannot be controlled
by an external parameter, heat engines based on underdamped dynamics can
typically not reach Carnot efficiency. We illustrate our general theory with a
paradigmatic case study of a heat engine consisting of an underdamped charged
particle in a modulated two-dimensional harmonic trap in the presence of a
magnetic field.
