Degenerate optimal control problems with state constraints (Q2706141)

It is considered the optimal control problemNEWLINENEWLINENEWLINE(P) minimize \(g(x(0),x(1)) \) over \(x\in W^{1,1}([0,1];R^{n})\) satisfying \(\dot{x}(t) \in F(t,x(t))\) almost everywhere, \((x(0),x(1))\in C_{0}\times C_{1}, x(t)\in A\) for all \(t\in [0,1]\) for which the data comprises a function \(g: R^{n}\times R^{n}\to R,\) closed sets \(A,C_{0},C_{1}\) in \(R^n\) and a multifunction \(F:[0,1]\times R^{n}\rightsquigarrow R^{n}.\) NEWLINENEWLINENEWLINEHere \(W^{1,1}([0,1];R^{n})\) denotes the Banach space of absolutely continuous \(R^{n}\) valued functions on the interval \([0,1]\), with norm \(\|x\|_{W^{1,1}}=|x(0)|+ \int_{0}^{1}|\dot{x}(t)|dt.\) NEWLINENEWLINENEWLINEIn this paper the necessary conditions for problem (P) are obtained.