On the existence of a solution for some systems of partial differential equations --- Necessary conditions for optimality (Q1813765)

The author considers the following system of partial differential equations \[ \partial w^ i(x)/\partial x^ i=\sum^ n_{j=1}a^ i_ j(x)w^ j(x)+\bar f^ i(x)\leqno (1) \] with the boundary conditions \[ w^ i(x^ 1,\ldots,x^{i-1},0,x^{i+1},\ldots,x^ n)=\varphi^ i(x^ 1,\ldots,x^{i-1},x^{ i+1},\ldots,x^ n) \] for \(i=1,\ldots,n\), \(x=(x^ 1,\ldots,x^ n)\in K^ n=\times^ n_{i=1}[0,1]\). Assuming that the functions \(a^ i_ j\), \(\bar f^ i\), \(\varphi^ i\) are integrable, the author establishes sufficient conditions for the existence and uniqueness of the solution to the boundary problem (1), (2) in the Caratheodory sense. The solution of the considered system is presented as a functional series. Furthermore the author finds a necessary condition for the existence of an optimal control.