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 ) ∈ Γ G(x,u) ∈ Γ , where Γ Γ is a closed convex set of a Banach space with nonempty interior. The inequality constraints g ( s , x , u ) ⩽ 0 g(s,x,u) 0 depend on a parameter s s belonging to a compact metric space S S . The equality constraints are split into two sets of equations K ( x , u ) = 0 K(x,u) = 0 and H ( x , u ) = 0 H(x,u) = 0 , where the first equation is an abstract control equation, and H H is assumed to have a full rank property in u u . The objective function is max t ∈ T f ( t , x , u ) {max t ∈ T}f(t,x,u) where T T is a compact metric space, f f </inline-for