Optimality conditions for distributed parameter systems using Dubovitskii–Milyutin's theorem with incomplete information about the initial conditions

In this paper, we consider an optimal control problem for a system described by a linear partial differential equation of the parabolic type with Neumann's boundary condition. We impose some constraints on the control. The performance functional has the integral form. The control time $T$ is fixed. The initial condition is not given by a known function but belongs to a certain set (incomplete information about the initial state). To obtain optimality conditions for the Neumann problem, the generalization of the Dubovitskii–Milyutin theorem was applied. The problem formulating in this paper describes the process of optimal heating, of which we do not have exact information about the initial temperature on the heating object. We present an example in which the admissible controls and one of initial conditions are given by means of the norm constraints too.