A Periodic Optimal Control Problem for a Bilinear Harmonic Oscillator, with Applications in Quantum and Classical Physics

Abstract. In this paper we study and solve a periodic optimal control problem for a bilinear harmonic oscillator (equivalently, for the Hill-type equation [Formula: see text] with bounded, sign-indefinite stiffness) motivated by applications in quantum and classical physics. Although apparently simple, this optimal control problem is not easy to solve and we resort to various elaborated methods of optimal control theory. We finally show its relationships to two problems in physics: the computation of the ground state for 1D Schrödinger operators with a finite potential well, and the optimal dynamical Kapitza stabilization problem. We also briefly discuss the Floquet stability properties associated with the optimal periodic bang-bang control.