Necessary conditions for optimal control problems with different constraints

Second order necessary conditions are presented 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)∈ Γ, where Γ is a closed convex set of a Banach space with nonempty interior. The inequality constraints g(s, x, u)⩽0 depend on a parameter s 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 max _t∈Tf(t, x, u) where T is a compact metric space, f is upper semicontinuous in t and Lipschitz in (x, u). The results are in terms of a function σ 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