Uniform boundedness for the optimal controls of a discontinuous, non-convex Bolza problem

• Published in: ESAIM. Control, optimisation and calculus of variations Vol. 29; p. 12
• Main Authors: Bettiol, Piernicola, Mariconda, Carlo
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 2023
• Subjects: Calculus; Calculus of variations; Continuity (mathematics); Convexity; Cost function; Mathematics; Optimal control; Validity; Regularity of minimizers; Optimal Control
• ISSN: 1292-8119; 1262-3377; 1262-3377
• DOI: 10.1051/cocv/2022079

We consider a Bolza type optimal control problem of the form [see formula in PDF] Subject to: [see formula in PDF] 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, ( P t, 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 [see formula in PDF] and [see formula in PDF] and the superlinear case. We show that, if Λ is real valued, any family of optimal pairs ( y *, u *) for (P t,x ) whose energy J t ( 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.