UNIFORM BOUNDEDNESS FOR THE OPTIMAL CONTROLS OF A DISCONTINUOUS, NON-CONVEX BOLZA PROBLEM

We consider a Bolza type optimal control problem of the form ZT min J(t)(y, u) := integral(T)(t )lambda(s, y(s), u(s)) ds + g(y(T)) Subject to: integral y is an element of W-1,W-1([t, T]; R-n) y ' = b(y)u a.e. s is an element of [t, T], y(t) = x u(s) is an element of u a.e. s is an element of [t, T], y(s) is an element of S (sic)s is an element of [t, T], where lambda(s, y, u) is locally Lipschitz in s, just Borel in (y, u), b has at most a linear growth and both the Lagrangian lambda and the end-point cost function g may take the value +infinity. If b = 1, g = 0, (P-t,P-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 lambda(s, y, u) = root 1 + |u|(2) and lambda(s, y, u) = |u| -root|u| and the superlinear case. We show that, if lambda is real valued, any family of optimal pairs (y(*), u(*)) for (P-t,P-x) whose energy J(t)(y(*), u(*)) is equi-bounded as (t, x) vary in a compact set, has L-infinity - equibounded controls. Moreover, if lambda is extended valued, the same conclusion holds under an additional lower semicontinuity assumption on (s, u) (sic) lambda(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.