Extremal problems for differential inclusions with state constraints (Q2784569)

The present paper is devoted to the study of the following extremal problem: NEWLINE\[NEWLINE \dot{x}\in F(x),\quad t\in I=[t_1,t_2]; \tag{1} NEWLINE\]NEWLINE NEWLINE\[NEWLINE p\in P,\quad p=(p_1,p_2),\quad x_1=x(t_1),\quad x_2=x(t_2);\tag{2} NEWLINE\]NEWLINE NEWLINE\[NEWLINE x(t)\in G\quad \forall t\in I;\tag{3} NEWLINE\]NEWLINE NEWLINE\[NEWLINE J(p)\to \min. \tag{4} NEWLINE\]NEWLINE Here \(x\in \mathbb{R}^{n}, F\) is a multivalued mapping, \(P,G\) are closed subsets of \(\mathbb{R}^{2n}\) and \(R^{n}\), respectively, \(J\) is locally Lipschitzian function. The minimum in the problem (1)-(4) is sought in the class of Lipschitz vector-valued functions \(x\); each of these functions is defined on its own time interval \(I.\) NEWLINENEWLINENEWLINEThe main objective of this paper is to obtain the most complete necessary optimality conditions for the problem (1)-(4).NEWLINENEWLINEFor the entire collection see [Zbl 0983.00023].