Motivated by two engineering applications, we address nonlinear bounded steady-state optimal control of linear dynamical systems undergoing steady-state bandlimited periodic oscillations. The optimisation can be cast as a minimisation problem by expressing the state and the input as finite Fourier series expansions, and using the expansions coefficients as parameters to be optimised. With this parametrisation we address linear quadratic (LQ) problems involving periodic bandlimited dynamics by using quadratic minimisation with parametric time-dependent constraints. We hence investigate the implications of a discretisation of linear continuous time constraints and propose an algorithm that provides a feasible sub-optimal 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 energy harvesting from pulsating mechanical energy sources, and optimal control of DC/AC power converters.