Necessary conditions on the extremum for convex differential inclusions with phase constraints (Q1076330)

Let us consider the following optimization problem: Minimize \(\int^{1}_{0}f(t,x(t))dt\) s.t. \(\dot x(\)t)\(\in a(t,x(t))\) a.e., \(x(0)=x_ 0\) and x(t)\(\in G(t)\), where \(f:[0,1]\times R^ n\to R\) is a convex integrand, a is a multivalued mapping which is measurable in t and convex in x and G is continuous with G(t) closed, convex and int G(t)\(\neq \emptyset\) for \(t\in [0,1]\). Using a family of perturbed problems, the author gives necessary conditions of Fritz John type for optimality, and shows that these conditions are also sufficient when the multiplier of the objective function is nonzero.