In optimal control theory one sometimes extends the minimization domain of a given problem, with the aim of achieving the existence of an optimal control. However, this issue is naturally confronted with the possibility of a gap between the original infimum value and the extended one. Avoiding this phenomenon is not a trivial issue, especially when the trajectories are subject to endpoint constraints. However, since the seminal works by Warga, some authors have recognized ``normality"" of an extended minimizer as a condition guaranteeing the absence of an infimum gap. Yet, normality is far from being necessary for this goal, a fact that makes the search for weaker assumptions a reasonable aim. In relation to a control-affine system with unbounded controls, in this paper we prove a sufficient no-gap condition based on a notion of higher order normality, which is less demanding than the standard normality and involves iterated Lie brackets of the vector fields defining the dynamics.

Unbounded Control, Infimum Gaps, and Higher Order Normality

Motta, Monica;Rampazzo, Franco
2022

Abstract

In optimal control theory one sometimes extends the minimization domain of a given problem, with the aim of achieving the existence of an optimal control. However, this issue is naturally confronted with the possibility of a gap between the original infimum value and the extended one. Avoiding this phenomenon is not a trivial issue, especially when the trajectories are subject to endpoint constraints. However, since the seminal works by Warga, some authors have recognized ``normality"" of an extended minimizer as a condition guaranteeing the absence of an infimum gap. Yet, normality is far from being necessary for this goal, a fact that makes the search for weaker assumptions a reasonable aim. In relation to a control-affine system with unbounded controls, in this paper we prove a sufficient no-gap condition based on a notion of higher order normality, which is less demanding than the standard normality and involves iterated Lie brackets of the vector fields defining the dynamics.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3449924
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