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

We consider a Bolza type optimal control problem of the form \begin{equation}\min J_{t}(y,u):=\int_t^TΛ(s,y(s), u(s))\,ds+g(y(T))\tag{P$_{t,x}$}\end{equation} Subject to: \begin{equation}\label{tag:admissible}\tag{D}\begin{cases} y\in AC([t,T];\mathbb R^n)\\y'=b(y)u\text{ a.e. } s\in [t,T], \,y(t)=x\\u(s)\in \mathcal U\text{ a.e. } s\in [t,T],\, y(s)\in \mathcal S\,\,\forall s\in [t,T], \end{cases} \end{equation} 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 running cost function $g$ may take the value $+\infty$. If $b\equiv 1$ and $g\equiv 0$ problem (P$_{t,x}$) is the classical one of the calculus of variations. We suppose the validity a slow growth condition in $u$, introduced by Clarke in 1993, including Lagrangians of the type $Λ(s,y,u)=\sqrt{1+|u|^2}$ and $Λ(s,y,u)=|u|-\sqrt{|u|}$ and the superlinear case. If $Λ$ is real valued, any family of optimal pairs $(y_*, u_*)$ for (P$_{t,x}$) whose energy $J_t(y_*, u_*)$ is equi-bounded as $(t,x)$ vary in a compact set, has $L^\infty$ -- equibounded optimal controls. If $Λ$ is extended valued, the same conclusion holds under an additional lower semicontinuity assumption on $(s,u)\mapstoΛ(s,y,u)$ and on the structure of the effective domain. No convexity, nor local Lipschitz continuity is assumed on the variables $(y,u)$. As an application we obtain the local Lipschitz continuity of the value function under slow growth assumptions.