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# The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives

• We prove validity of the classical DuBois-Reymond differential inclusion for the minimizers $y(\cdot)$ of the integral

$\int_{a}^{b}L( x( t) ,x^'( t)) d\,t,\text{ \ }x\( \cdot) \in W^{1,1}((a,b) ,\mathbb{R}^{n}) ,\text{ \ }x(a)=A\,x(b) =B\ \$(*)

whose velocities are not a.e. constrained by the domain boundary.
Thus we do not ask ( as preceding results do) the free-velocity times

$T_{f ree}:=\{ t\in[ a,b] :y^'( t) \in$int$\text{ }dom\ L( y\( t) ,\cdot) \}$

to have "full measure"; on the contrary, "positive measure" of $T_{f ree}$ suffices here to guarantee the above necessary condition.
One main feature of our result is that $L( S,\xi) =\infty$ freely allowed, hence the domains $dom$$L( S,\cdot)$ may be e.g. compact and (*) can be seen as the variational reformulation of general state-and-velocity constrained optimal control problems.
Another main feature is the clean generality of our assumptions on $L( \cdot) :$ any Borel-measurable function $L:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow[ 0,\infty]$ having $L( \cdot,0)$ $lsc$ and $L( S,\cdot)$ convex $lsc$ $\forall\,S.$
The nonconvex case is also considered, for $L( S,\cdot)$ almost convex lsc $\forall\,S.$

Mathematics Subject Classification: Primary: 49J05, 49J15, 49J24, 49J45, 49K05, 49K15, 49K24.

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