  It was recently demonstrated that, when coupled to N=1 supergravity, the
Dirac-Born-Infeld (DBI) action constructed from a single chiral superfield has
the property that when the higher-derivative terms become important, the
potential becomes negative. Thus, DBI inflation cannot occur in its most
interesting, relativistic regime. In this paper, it is shown how to overcome
this problem by coupling the model to one or more additional chiral
supermultiplets. In this way, one obtains effective single real scalar field
DBI models with arbitrary positive potentials, as well as multiple real scalar
field DBI inflation models with hybrid potentials.
