Maximum principle for optimal control of non-well posed elliptic differential equations
The authors study optimal control problems governed by non-well-posed similinear elliptic differential equations. So the state equations are governed by nonmonotone operators and admit more than one solution. They ask to minimize the cost functional over all pairs \((y,u)\) where \(u\) is a control in the set of controls and \(y\) is a minimal positive solution of the state equation corresponding to \(u\). They introduce two kinds of new penalty functionals via which one can transform the original optimal control problem into optimization problems by thinking of the state equations as sets of constraints mixed by the state and the control. They use Ekeland's variational principle and some perturbation to obtain the necessary conditions for the solution of the optimization problem, then by passing to the limit they obtain the maximum principle for the original optimal control problem.
similinear elliptic differential equations
scientific article; zbMATH DE number 1880086
scientific article