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

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  • 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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