The optimal control problem for differential inclusion with state constraint. Smooth approximations and necessary conditions of optimality (Q2771538)

This article deals with the optimal control problem: NEWLINE\[NEWLINE\dot x\in F(x), \quad t\in[t_1,t_2];NEWLINE\]NEWLINE NEWLINE\[NEWLINEp\in P,\quad p=(x_1,x_2),\quad x_1=x(t_1), \quad x_2=x(t_2);NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(t)\in G\;\forall t\in[t_1,t_2]; \quad J(p)\to\min,NEWLINE\]NEWLINE where \(x\in R^{n}\), \(F\) is a many-valued mapping, \(P, G\) are closed subsets from \(R^{2n}\) and \(R^{n}\), respectively; \(J\) is a locally Lipschitz function. The author gives the necessary optimality conditions for the considered problem. The proof of these optimality conditions is based on approximation of the given problem by a sequence of classical optimal control problems without state constraints. The sufficient conditions of regularity of measure from the maximum principle are presented. An approach to modeling and approximation of the optimal control problem with state constraints is discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00042].