Second-order necessary conditions for differential inclusion problems (Q1330919)

The author considers the following optimal control problem \[ \text{minimize}\quad g(x(T)) \quad\text{subject to } x'(t)\in F(t,x(t)),\;x(0)\in C_ 0,\;x(T)\in C_ 1, \] where \(g\) is Lipschitz continuous, \(C_ 0\) and \(C_ 1\) are closed and convex sets in \(\mathbb{R}^ n\), \(F\) is compact- and convex-valued, integrably Lipschitz in \(x\) and measurable in \(t\) in a tube around a local optimal solution. Second-order necessary conditions are obtained by a suitable approximation of the Clarke normal cone to the reachable set. The approach is based on a paper by \textit{R. T. Rockafellar} [Math. Oper. Res. 14, No. 3, 462-484 (1989; Zbl 0698.90070)].