Optimal control theory with arbitrary superpositions of waveforms

Standard optimal control methods perform optimization in the time domain. However, many experimental settings demand the expression of the control signal as a superposition of given waveforms, a case that cannot easily be accommodated using time-local constraints. Previous approaches [1,2] have circumvented this difficulty by performing optimization in a parameter space, using the chain rule to make a connection to the time domain. In this paper, we present an extension to Optimal Control Theory which allows gradient-based optimization for superpositions of arbitrary waveforms directly in a time-domain subspace. Its key is the use of the Moore-Penrose pseudoinverse as an efficient means of transforming between a time-local and waveform-based descriptions. To illustrate this optimization technique, we study the parametrically driven harmonic oscillator as model system and reduce its energy, considering both Hamiltonian dynamics and stochastic dynamics under the influence of a thermal reservoir. We demonstrate the viability and efficiency of the method for these test cases and find significant advantages in the case of waveforms which do not form an orthogonal basis.