Аннотация
In this paper, we consider the optimal-control problem for a system whose behavior is governed by a nonlinear elliptic-type equation in the absence of constraints that guarantee unique solvability of the boundary-value problem. The nonlinear term in the equation is assumed to be nonlinear. Insolvability of the optimization problem is admitted. Using the penalty method with smooth approximation of the system operator, and also the Tikhonov method, we pass to some solvable variational problem, for which necessary extremum conditions are established. We show that the solution of the obtained problem presents, in a sense, an approximate solution to the initial problem.