Necessary optimality conditions for optimal control problems with intermediate constraints (Q1972768)

Given the controlled object \[ \dot{x} = f(x,u,t),\quad t \in [t_{1}, t_{m}], \quad t_{1} < t_{m},\tag{*} \] with constraints \[ K_{1}(p) \leq 0, \quad K_{2}(p)=0, \quad R_{1}(x,u,t) \leq 0, \quad R_{2}(x,u,t)=0, \] and the criterion of optimality \(K_{0}(p)\rightarrow \min\), where \(x \in E^{n}\), \(u \in E^{r}, \dim p= m(n+1)\), the vector \(p=(t_{1},\dots{},t_{m},x_{1},\dots{},x_{m})\), \(x_{i}=x(t_{i}),\;i=1,\dots{},m,\;t_{1} \leq \dots{} \leq t_{m}\). The constraints \(K_{1}, K_{2}\) are intermediate, but \(R_{1}, R_{2}\) are mixed. The paper deals with the optimal control problem of the system (*), however, the main aim is to obtain the first and second order necessary conditions for an optimal controlled process by intermediate constraints.