First- and Second-order Necessary Conditions for Control Problems with Constraints

Second-order necessary conditions are developed for an abstract nonsmooth control problem with mixed state-control equality and inequality constraints as well as a constraint of the form G(x, u) e T, where T is a closed convex set of a Banach space with nonempty interior. The inequality constraints g{s, x, u) < 0 depend on a parameter 5 belonging to a compact metric space S. The equality constraints are split into two sets of equations K(x, u) = 0 and H(x, u) = 0 , where the first equation is an abstract control equation, and H is assumed to have a full rank property in u . The objective function is maxteT f(t, x, u) where T is a compact metric space, / is upper semicontinuous in t and Lipschitz in (x, u). The results are in terms of a function a that disappears when the parameter spaces T and S are discrete. We apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints. Thus we obtain a generalization and extension of the existing results on this problem.