Second-order necessary conditions in optimal control of evolution systems (Q2108525)

The paper deals with a semilinear equation in an abstract Banach separable space referred to the infinitesimal generator of a strongly continuous semigroup. For this equation, the authors consider a control problem where the control variable is in the nonlinear term, the cost functional depends on the end-value of the state equation, and where some constraint conditions for the state function, in the form of inequalities, are given. The aim of the paper is to obtain some second-order necessary conditions for this type of problems without to assume any structural condition for the optimal controls. The paper also provides some applications of these abstract results to parabolic and hyperbolic PDE.