Volume 29, 2023
|Number of page(s)||20|
|Published online||19 January 2023|
Univ Brest, UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique,
6 Avenue Victor Le Gorgeu,
2 Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121 Padova, Italy
*** Corresponding author: email@example.com
Accepted: 25 November 2022
We consider a Bolza type optimal control problem of the form
where Λ(s, y, u) is locally Lipschitz in s, just Borel in (y, u), b has at most a linear growth and both the Lagrangian Λ and the end-point cost function g may take the value +∞. If b ≡ 1, g ≡ 0, (Pt, x) is the classical problem of the Calculus of Variations. We suppose the validity of a slow growth condition in u, introduced by Clarke in 1993, including Lagrangians of the type and and the superlinear case. We show that, if Λ is real valued, any family of optimal pairs (y*, u*) for (Pt,x) whose energy Jt(y*, u*) is equi-boundcd as (t, x) vary in a compact set, has L∞ – equibounded controls. Moreover, if Λ is extended valued, the same conclusion holds under an additional lower semicontinuity assumption on (s, u) ↦ Λ(s, y, u) and requiring a condition on the structure of the effective domain. No convexity, nor local Lipschitzianity is assumed on the variables (y, u). As an application we obtain the local Lipschitz continuity of the value function under slow growth assumptions.
Mathematics Subject Classification: 49N60 / 49K05 / 90C25
Key words: Regularity / Lipschitz / uniform / growth
© The authors. Published by EDP Sciences, SMAI 2023
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