Motivated by engineering applications, we address bounded steady-state optimal control of linear dynamical systems undergoing steady-state bandlimited periodic oscillations. The optimization can be cast as a minimization problem by expressing the state and the input as finite Fourier series expansions, and using the expansions coefficients as parameters to be optimized. With this parametrization, we address linear quadratic problems involving periodic bandlimited dynamics by using quadratic minimization with parametric time-dependent constraints. We hence investigate the implications of a discretization of linear continuous time constraints and propose an algorithm that provides a feasible suboptimal solution whose cost is arbitrarily close to the optimal cost for the original constrained steady-state problem. Finally, we discuss practical case studies that can be effectively tackled with the proposed framework, including optimal control of DC/AC power converters, and optimal energy harvesting from pulsating mechanical energy sources.