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

No infimum gap and normality in optimal impulsive control under state constraints