We consider the problem of optimal input design for estimating uncertainparameters in a discrete-time linear state space model, subject to simultaneousamplitude and l1/l2-norm constraints on the admissible inputs. We formulatethis problem as the maximization of a (non-concave) quadratic function over thespace of inputs, and use semidefinite relaxation techniques to efficiently findthe global solution or to provide an upper bound. This investigation ismotivated by a problem in medical imaging, specifically designing a substrateinjection profile for in vivo metabolic parameter mapping using magneticresonance imaging (MRI) with hyperpolarized carbon-13 pyruvate. In thel2-norm-constrained case, we show that the relaxation is tight, allowing us toefficiently compute a globally optimal injection profile. In thel1-norm-constrained case the relaxation is no longer tight, but can be used toprove that the boxcar injection currently used in practice achieves at least98.7% of the global optimum.
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