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

We consider a Bolza type optimal control problem of the form min Jt{y,u) :=jf A(s, y(s), u(s)) ds + g{y(T)) Subject to: y G W1,1([t,T];Rn) y = b(y)u a.e. 5 G [t,T], y(t) = x u(s) G U a.e. 5 G [t,T], 1/(5) G 5 Vs G [t,T],where A(s.y.u) is locally Lipschitz in s, just Borel in (y,u), b has at most a linear growth and both the Lagrangian A and the end-point cost function g may take the value +00. 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 A(s,y, u) = \J 1 + M2 and A(s.y.u) = |u| - y/\u\ and the superlinear case. We show that, if A is real valued, any family of optimal pairs (ym,u.) for (Pt,x) whose energy Jt(y~, is equi-bounded as (t,x) vary in a compact set, has - equibounded controls. Moreover, if A is extended valued, the same conclusion holds under an additional lower semicontinuity assumption on (s, u) 1-> A(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.