  In this paper we consider an impulsive extension of an optimal control
problem with unbounded controls, subject to endpoint and state constraints. We
show that the existence of an extended-sense minimizer that is a normal
extremal for a constrained Maximum Principle ensures that there is no gap
between the infima of the original problem and of its extension. Furthermore,
we translate such relation into verifiable sufficient conditions for normality
in the form of constraint and endpoint qualifications. Links between existence
of an infimum gap and normality in impulsive control have previously been
explored for problems without state constraints. This paper establishes such
links in the presence of state constraints and of an additional ordinary
control, for locally Lipschitz continuous data.
