Lipschitzianity of optimal trajectories for the Bolza optimal control problem (Q850813)

The following non-smooth optimal control problem is considered \[ \dot{x}(t)=f(t,x(t),u(t)),\quad u(t)\in U(t),\;t\in I=[0,T], \;(x(0),x(T))\in Q,\;x(t)\in K,\;t\in I, \] \[ \int_{0}^{T}L(t,x(t),u(t))\,dt+g(x(0),x(T))\rightarrow \min, \] where \(U:I\rightarrow 2^{Z}\) is a measurable set valued map with closed non-empty images; \(K\subset R^{m},Q\subset R^{m}\times R^{m}\) are closed sets; \(Z\) is a complete separable metric space; further, the functions \(f(t,.,u),L(t,.,u)\) and \(g\) are locally Lipschitz; the functions \(L(.,x,.)\) and \(f(.,x,.)\) are Lebesgue-Borel measurable on \(\{ (t,u)\in I\times Z:u(t)\in U(t)\}.\) New criteria are established for normality of the maximum principle, for Lipschitz continuity of optimal trajectories, for essentially boundedness of optimal controls. Moreover, for a special case the existence of a minimizing sequence is proved.