Optimization problem with phase constraints at different times (Q795321)

Consider the following problem of optimal control: \(\dot x(t)=f(t,x,u)\), 0\(\leq t\leq T\), \(x(0)=x_ 0\), \(x(T)=x_ T\), \(\int^{T}_{0}\phi(t,x,u)dt\to \min_{u}\). Here x(t) is a state vector and \(u\in U\) is a control. Some constraints on the state vector are imposed at two times \(t_ 1,t_ 2\in [0,T]: \gamma_ 1(x(t_ 1),x(t_ 2))\leq 0,\quad...,\quad \gamma_ l(x(t_ 1),x(t_ 2))\leq 0.\) A necessary condition for optimality is proved in the form of a maximum principle. A two-dimensional example is investigated in details.