Necessary conditions for an infinite time optimal control problem (Q1820422)

Let there be given two functions \(f: J\times R^ n\times B\times U\to R^ n\) and \(H: B\to R\) where \(J=[0,\infty)\), B is a normed space and U is a given set. The following problem is considered: Minimize H(p) assuming that the functions x(\(\cdot)\) and u(\(\cdot)\) and the parameter p verify the following conditions \(x'(t)=f(t,x(t),p,u(t))\) a.e. on J, \(x(0)=0\), \(\lim_{t\to \infty}x(t)=0\) and for every \((x,p)\in R^ n\times B\) the function \(f_ u: J\to R^ n\), \(f_ u(t)=f(t,x,p,u(t))\) is locally integrable on J. Necessary optimality conditions for this problem are derived. As an application an optimal control problem with infinite horizon is adapted to this frame.