Control problem for a parabolic system with disturbances and possible changes in dynamics

The problem of control of a parabolic system, which describes the heating of a given number of non-uniform rods, is considered. The control is point heat sources, which are located at the ends of the rods. Some boundary conditions are affected by uncontrolled disturbances. The density functions of the internal heat sources of the rods are not exactly known, but the segments of their change are given. We admit that at some moments of time, changes may occur in the equations describing the dynamics of the controlled system. These moments of time are not known in advance. The goal of choosing the control is to ensure that at a fixed moment of time, the weighted sum of the average temperatures of the rods belongs to a given segment for any admissible realizations of disturbances, unknown functions and moments of change in the dynamics. After the change of variables, this problem is reduced to a one-dimensional control problem with disturbance. Necessary and sufficient termination conditions are found.