Optimization in a nonlinear parabolic system with a control in the coefficients

Two extremal problems are considered for a system described by a nonlinear equation of parabolic type with a control in the coefficients. Since the nonlinearity of the system is sufficiently strong, it is difficult to establish differentiability of the state function with respect to the control, and this is an obstacle to getting optimality conditions by the standard methods. The low regularity of the optimality criteria does not permit use of the penalty method. The apparatus of extended differentiability is used to overcome the difficulties arising. It is proved that extended differentiability holds for the state function with respect to the control, and this leads to the necessary optimality conditions.