Higher-Order Normality and No-Gap Conditions in Impulsive Control with $L^1$-Control Topology

In optimal control, extending the class of admissible controls is a common strategy to guarantee the existence of optimal solutions. However, such extensions may introduce a gap between the infimum of the original problem and the minimum of the extended one, especially in the presence of endpoint constraints. Since Warga's seminal work, normality of first-order necessary conditions for extended minimizers has been recognized as a sufficient condition to avoid this phenomenon, though it is far from being necessary. In this paper, we consider impulsive extensions of control-affine systems with unbounded controls. We establish that a notion of \textit{higher-order normality}, based on iterated Lie brackets of the systems vector fields, suffices to prevent an infimum gap. The key novelty of this manuscript consists in showing that this holds under a local topology defined by the $L^1$-distance between controls, rather than the more common $L^\infty$-distance between trajectories. Among the reasons that motivate the interest in this issue, let us mention that a counterexample by R. B. Vinter shows that for a different extension -- based on convexification of the velocity set -- a local extended minimizer that is normal with respect to the $L^1$-norm of the controls may still exhibit a gap. Our method relies on set-separation techniques. Such an approach makes it possible to derive higher-order conditions and to exploit the corresponding notion of higher-order normality.