We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrödinger operator defined on a bounded interval with Dirichlet boundary conditions under an L q -norm restriction (q≥1). This is done by first extending the definition of the functional determinant to the case of L q potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all q≥1, providing a complete characterisation of extremal potentials in the case where q is one (a pulse) and two (Weierstraß’s ℘ function).