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. Since this type of constraint is not time-local, Optimal Control Theory cannot be used without modifications. Simplex methods, used as a substitute in this case, tend to be less efficient and less reliable than Optimal Control Theory. In this paper, we present an extension to Optimal Control Theory which allows gradient-based optimization for superpositions of arbitrary waveforms. Its key is the use of the Moore-Penrose pseudoinverse as an efficient means of transforming from a time-local to a waveform-based description. To illustrate this optimization technique, we study the parametrically driven harmonic oscillator as model system and reduce its energy, considering both Hamiltonian dynamics and opensystem dynamics. We demonstrate the viability and efficiency of the method for these test cases. PACS numbers: 02.30.Yy, 02.60.Pn, 05.10.Gg Submitted to: J. Phys. A: Math. Theor. ‡ Present adress: School of Mathematical Sciences, The University of Nottingham, Nottingham, NG7 2RD, United Kingdom Optimal control theory with arbitrary superpositions of waveforms 2