Maximum principle via singular perturbations

The paper is concerned with necessary optimality conditions for parabolic boundary control problems. Its main purpose is to provide a regularization technique via singular perturbations to obtain optimality conditions for the time optimal control problem. The considered system is nonlinear and consists of a controlled coupled ODE/PDE. Systems of this type frequently arise in modelling population dynamics in a contaminated environment. For the regularization Ekeland´s variational principle along with a singular perturbation technique is used. For this purpose a new trajectory on an additional exterior domain, whose size may be considered as a singular perturbation parameter, is introduced. The technique allows us to obtain necessary optimality conditions without involving boundary data