  Flavor-changing neutral currents are not present in the Standard Model at
tree level and are suppressed in loop processes by the unitarity of the
Cabibbo-Kobayashi-Maskawa matrix; the corresponding rates for top quark decay
processes are experimentally unobservable. Extensions of the Standard Model can
generate new flavor-changing neutral current processes, leading to signals
which, if observed, would be unambiguous evidence of new interactions. A
dataset corresponding to an integrated luminosity of 36.1 fb$^{-1}$ of $pp$
collisions at a center-of-mass energy of $\sqrt{s}=$ 13 TeV recorded with the
ATLAS detector at the Large Hadron Collider is used to search for top quarks
decaying to up or charm quarks with the emission of a Higgs boson, with
subsequent Higgs boson decay to final states with at least one electron or
muon. No signal is observed and limits on the branching fractions
$\mathcal{B}(t \to Hc) <$ 0.16% and $\mathcal{B}(t \to H u) <$ 0.19% at 95%
confidence level are obtained (with expected limits of 0.15% in both cases).
