An optimal control problem for a differential inclusion with state constraints. Smooth approximations and necessary optimality conditions (Q5948395)

The main object studied in this paper is the following extremal problem: \[ \dot{x}\in F(x),\quad t\in I=[t_1,t_2]; \tag{1} \] \[ p\in P,\quad p=(p_1,p_2),\quad x_1=x(t_1),\quad x_2=x(t_2);\tag{2} \] \[ x(t)\in G\quad \forall t\in I;\tag{3} \] \[ J(p)\to \min. \tag{4} \] Here \(x\in \mathbb{R}^{n}, F\) is a multivalued mapping, \(P,G\) are closed subsets of \(\mathbb{R}^{2n}\) and \(\mathbb{R}^{n}\), respectively, \(J\) is a 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 \(I.\) The necessary optimality conditions for the problem (1)--(4) are stated and discussed. The sufficient conditions for regularity of the measure that enters relations of the maximum principle are presented, too.