  We consider the problem of finding extremal potentials for the functional
determinant of a one-dimensional Schr\"odinger operator defined on a bounded
interval with Dirichlet boundary conditions under an $L^q$-norm restriction
($q\geq 1$). 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 $q\geq 1$, providing a complete
characterization of extremal potentials in the case where $q$ is one (a pulse)
and two (Weierstrass's $\wp$ function).
