Set-Valued and Variational Analysis

, Volume 23, Issue 1, pp 3–22 | Cite as

Infimum Gaps for Limit Solutions

  • M.-Soledad Aronna
  • Monica Motta
  • Franco Rampazzo


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.


Optimal control Impulsive control Lavrentiev phenomenon Controllability 

Mathematics Subject Classification (2010)

49N25 49N60 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • M.-Soledad Aronna
    • 1
  • Monica Motta
    • 2
  • Franco Rampazzo
    • 2
  1. 1.IMPARio de JaneiroBrazil
  2. 2.Dipartimento di MatematicaUniversità di PadovaPadovaItaly

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