Infimum Gaps for Limit Solutions

Abstract

After recalling the notion of \(\mathcal {L}^{1}\) limit solution for a dynamics which is affine in the (unbounded) derivative of the control, we focus on the possible occurrence of the Lavrentiev phenomenon for a related optimal control problem. By this we mean the possibility that the cost functional evaluated along \(\mathcal {L}^{1}\) inputs (and the corresponding limit solutions) assumes values strictly smaller than the infimum over AC inputs. In fact, it turns out that no Lavrentiev phenomenon may take place in the unconstrained case, while the presence of an end-point constraint may give rise to an actual gap. We prove that a suitable transversality condition, here called Quick 1-Controllability, is sufficient for this gap to be avoided. Meanwhile, we also investigate the issue of trajectories’ approximation through implementation of inputs with bounded variation.

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Correspondence to Franco Rampazzo.

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This work is supported by the EU under the 7th Framework Programme FP7-PEOPLE-2010-ITN - Grant agreement number 264735-SADCO and by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), Italy.

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Aronna, M., Motta, M. & Rampazzo, F. Infimum Gaps for Limit Solutions. Set-Valued Var. Anal 23, 3–22 (2015). https://doi.org/10.1007/s11228-014-0296-1

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Keywords

  • Optimal control
  • Impulsive control
  • Lavrentiev phenomenon
  • Controllability

Mathematics Subject Classification (2010)

  • 49N25
  • 49N60