Higher-Order Normality and No-Gap Conditions in Impulsive Control with $L^1$-Control Topology
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arXiv
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| Format: | Preprint |
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2026
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| author | Motta, Monica Palladino, Michele Rampazzo, Franco |
| author_facet | Motta, Monica Palladino, Michele Rampazzo, Franco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06427 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher-Order Normality and No-Gap Conditions in Impulsive Control with $L^1$-Control Topology Motta, Monica Palladino, Michele Rampazzo, Franco Optimization and Control 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. |
| title | Higher-Order Normality and No-Gap Conditions in Impulsive Control with $L^1$-Control Topology |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2603.06427 |