On the structural representation of \(S\)-homogenized optimal control problems (Q2735867)

It is investigated the question of \(S\)-homogenization of a family of optimal control problems of the type NEWLINE\[NEWLINE\inf I_\varepsilon(x,y),NEWLINE\]NEWLINE NEWLINE\[NEWLINEA_\varepsilon(x,y)= f_\varepsilon, \qquad F_\varepsilon(x,y)\geq 0,NEWLINE\]NEWLINE where \(A_\varepsilon\), \(F_\varepsilon\) are nonlinear operators, which may arbitrarily depend on \(\varepsilon\), \(I_\varepsilon\) is a cost function and \(\varepsilon\) denotes a ``small'' multiparameter of a set \(E\), partially decreasing by ordered \((0\leq \varepsilon\) for every \(\varepsilon\in E\) and 0 is the minimal element in \(E)\). NEWLINENEWLINENEWLINEThe existence of a strongly \(S\)-homogenized optimal control problem is investigated and several important variational and topological properties of it are obtained. A formula for the representation of the homogenized problem is given.