  We consider the problem of optimal input design for estimating uncertain
parameters in a discrete-time linear state space model, subject to simultaneous
amplitude and l1/l2-norm constraints on the admissible inputs. We formulate
this problem as the maximization of a (non-concave) quadratic function over the
space of inputs, and use semidefinite relaxation techniques to efficiently find
the global solution or to provide an upper bound. This investigation is
motivated by a problem in medical imaging, specifically designing a substrate
injection profile for in vivo metabolic parameter mapping using magnetic
resonance imaging (MRI) with hyperpolarized carbon-13 pyruvate. In the
l2-norm-constrained case, we show that the relaxation is tight, allowing us to
efficiently compute a globally optimal injection profile. In the
l1-norm-constrained case the relaxation is no longer tight, but can be used to
prove that the boxcar injection currently used in practice achieves at least
98.7% of the global optimum.
